0.0 Introduction

Years of technical only explanations in trying to teach computer aided design (CADesign) have been found by this author to be not very effective resulting in digital future shock. Most digital based design applications in the last twenty years have not reached beyond the dimensional boundaries of paper based processes and their attending cultures and social structures. Thus, this chapter takes a unique approach by presenting a digital way of thinking and techniques for the digital based three dimensional computer aided design process. So, we will begin with an introduction to the digital way that includes digital ethics, responsibilities, 3D dimensional orientation and description of the differences between paper and digital design processes.

0.1 The Digital Way

The most difficult, expensive, time consuming to develop, and important part of all digital systems is the human part of the system. The human part of most digital systems is usually defective or broken. Because of the lack of understanding of digital technologies, most digital systems are applied in the same way as the past information technologies they replace. The result is the growth of digital systems being blocked by technically obsolete constructs and old social structures of the past paper, radio and video based analogue information systems. The human part of current digital systems has yet to make the transition to digital based, interactive networked information systems. An example of the inappropriate application of current paper social structures to digital are the present copyright and patent laws. Because of the nature of digital data,  these are difficult or impossible to enforce with regard to digital. Digital data is physically easy to copy and modify. Ownership of digital data can not be rigorously proved. Digital data costs almost nothing to copy. If you share digital data with a friend, you still have it, unlike an apple or loaf of bread. Your friend can easily add to the data and modify the shared data. Because of the dependencies in the complex world of digital data, the natural tendency is to share data or be excluded from evolving systems. The more you share digital data, the more there is. The more there is, the more it is used and the more digital technology grows. An example of this phenomenon is the great digital growth in the United States (US) in the 1980s. In 1981, IBM introduced its personal computer (PC) for use in the home, office and schools. Due to the open architecture of the IBM PC, the 1980's saw an expansion in computer use as clones of the IBM PC made the personal computer even more affordable. The number of personal computers in use more than doubled from 2 million in 1981 to 5.5 million in 1982. Ten years later, 65 million PCs were being used. The growth happened because of the freedom to share and modify digital data, processes and technical information. All of the technical information for the hardware and the source code for software was free and open. It is a very simple idea -- if digital technology is open, can be freely used, modified, and shared by everyone, then digital technology will grow very quickly. This digital reality is in conflict with traditional social structures that currently close digital technology by high costs through copyrights, patent laws, and other forms of traditional ownership.

0.2 Common Good Public License

Digital technologies and its effect on humans and their social structures should be of the greatest value and interest to the scientist and academic person, but receives little or no attention in comparison to technical issues. Because digital technologies are based on global sharing of digital information, those creating and using new digital technologies have the obligation to support: digital freedom, human rights and a sustainable future. Because digital computations of the personal computer are predicted to equal capabilities of the human brain in the next twenty years (The Age of Sprirtual Machines, Ray Kurzweil, Viking/Penguin Group, New York (1999) ISBN 0-670-88217-8) a deeper commitment to ethics and global openness by the creators and users of digital technologies is required to secure a safe, stable future for all. The development of digital technologies under the Common Good Public License (formerly known as the Greater Good Public License - in short, CGPL - http://www.ggpl.org) agreement, for example, and its three provisions where all developers and users agree to use the technology to promote digital freedom, human rights and a sustainable future, is not only possible but it is the digital future.

0.3 How to Use

You are free to use this text and any ideas contained here, if you agree to share this knowledge freely and use it to promote human rights and a sustainable future. Please use this text in a special way. The person reading these words should doubt, question and reflect upon how their senses, perception and reason may deceive them. The person learning the digital way needs to learn to think in a very special and systematic way about computers and other digital devices. The following are the three steps in learning to think in this special way. You must walk these steps many time before you will understand the digital way.
  • First - define digital.
    • A person should carefully describe the nature of digital materials.
  • Second - understand digital processes.
    • A person should try to understand the logic for the existence of digital
      processes stemming from the nature of digital materials.
  • Third - derive first principles of digital.
    • When a person understands the nature of digital materials (size, weight,
      speed, and cost) and the nature and relationship of materials to processes,
      it is then possible by logical reasoning to reveal the first principles in the
      development and application of digital devices and understand the
      emerging digital culture and social structures that arise from the use of
      digital devices. This special systematic way of thinking does not rely on
      our physical senses, but on complex logic sometimes called scientific
      thought, which requires "rigorous proof" of an idea.

0.4 How We are Deceived

An idea such as the sun moves around the earth must be proven by rigorous logic, observation and testing not just by simple observation. An example of how we are deceived is shown by the fact that we can not feel the movement of the earth. However we are told that the earth is moving in an orbit around the sun, that the earth and its orbit around the sun, the sun and the rest of our solar system are moving at great speeds through space, but we can not feel the speed at which we are moving or easily see any movement. We see clearly the sun moving across the sky, and with no other information we would reason that the sun revolves around the earth, but we are deceived. Currently we see that a computer screen displays information that was in the past displayed on a piece of paper, and we would reason that it is the same as a piece of paper and we can use the computer in the same way as we use paper, but we are deceived about computers and what they have to offer. Fully understanding computers and the digital way and what the future holds requires a person to doubt the current use of computers and ask:
  • What are digital materials?
  • What is digital culture?
  • What are digital social structures / laws ?
  • How long does digital data last?
The answers are not what you would expect. How much will change in future years will surprise most everyone and may be difficult to deal with.
If you have any questions, please mailto:carl@applied3d.org

0.5 Chapter Description

The Computer Aided Design (CAD) study in this chapter gives an overview of underlying general principles concerning multi-dimensional modeling including orientation, operations and applications. This chapter's text includes the following:
  • Discussion about digital materials and processes.
  • Presentation of the Cartesian coordinate system.
  • Handedness and orientation in three dimensional space.
  • Concepts of the multidimensional digital design process.
  • The differences between: Visual Representation, Boundary Representation (B-rep), Constructive Solid Geometry (CSG) Representation, Function Representation (F-rep)

0.6 Chapter Objectives

The first objective is to provide a digital world view in the discussion and demonstration of multidimensional modeling which shows the differences between paper and digital material and processes. The second objective is to demonstrate basic methods used in multidimensional modeling. The third objective is the realization that the user interface must be as simple as possible so as not to interfere with the creative thought process of the designer. The fourth objective is to outline a set of basic functions for a 3D CAD modeling system which can be used as a specification for multidimensional modeling systems of the future.

0.7 Paper Design Overview

The paper based design process for a new product model takes a long time. It involves many people typically working in a large hierarchical social structure divided into three functional design phases: conceptual design by experts in a research group, working design by an engineering department, and production design by the manufacturing department. Paper design involves people with various levels of different skills and academic degrees such as in physics and mathematics. It also involves highly skilled drafters needing years of technical training. Drafting on paper is the principal process. Drafting is done for the purpose of control and verification that the design will work . The actual verification of the design is best done by producing a product prototype. This requires yet a different set of highly skilled people. When the product is scheduled for production it often needs modifications so it can be produced effectively.

0.8 Digital Design Overview

Digital design based synthetic simulation allows levels of interactive design detailing and verification not available in the physical prototyping processes. A hierarchical structure involving many highly skilled people doing many disparate tasks is not necessary. The long time from conceptual to production designs is drastically reduced from years to months and from months to days. The process is only hindered by the ease of navigation in Cartesian space that imposes demands for a highly interactive and yet simple, intuitive human interface. Digital designing offers so much more than paper and physical processes, yet the change over to digital design is very slow because of lack of understanding and overcoming entrenched paper based hierarchical social structures of the past millennia. There are still technical issues to be addressed. However the development and implementation of multidimensional modeling is more of a social problem than a technical problem.

Note: At the end of the chapter there is a list of software resources and exercises. Three dimensional visual spatial navigation and modeling is understood through experience and reflection on that experience. It is recommended that you use the HTML version on the CD with animated GIFs and the free programs provided.

1.0 Social Discontinuity

At each new level of energy input to a given system, that system may cross boundaries of discontinuities creating instabilities. The invention of digital devices and the rate of technological advances stemming from these devices are creating social discontinuities that often do not allow for recognition of the capabilities or allow for the full use of such digital technology in certain sectors.  This is especially true in CAD.

1.1 Paper and Pencil

Sometimes inventions do not flow out of previous inventions. This adds to the confusion in transitioning to the new technology. The pencil did not evolve into a computer. While both are tools that serve a common need for human communication, one did not evolve from the other. A comparison between pencil and a computer seems to hardly have any relevance because there is such a great difference. The paper and pencil are very simple objects and are easy to use for simple communications. Because of these limitations, drafting, which relies heavily on the use of paper and pencil is difficult, complex, slow and inaccurate compared to computer based engineering and design practices. Although computer based design and engineering has its own complexities that need to be overcome, it is relatively easy, fast, precise and practically unlimited in nature. Again a comparison between an imprecise analog line drawn on paper and a digital model of a line on the windshield of your computer design system is so great that a comparison should seem to be irrelevant or even silly. However most people do not make the realization that a line in the computer is dynamic: able to be copied perfectly, rotated, shortened, lenghtened, divided, offset, mirrored, arrayed, has no width yet carries sixteen decimal places of accuracy.  In fact, it is functionally alive; there is no parallel on paper.

1.2 Faking a Reality

The paper drafting process and procedures based on the simplicity of paper and pencil have severe dimensional and dynamic limits in comparison to the computer modeling process. Paper drafting is two dimensional media. It fakes three dimensions through front/side/top projections or 3/4 views drawn with illusive isometric perspective, or fake exploded views. The skill and training requirements needed for designing using drafting procedures are very high. Six years of math to be an engineer plus two years of drafting classes equals a total of eight years of training. The basic tools used in paper drafting procedure are very simple in comparison to the computer design tools. However in application, the engineering and design based drafting is far more complex, difficult and slow to use in comparison to computer based engineering and design. The paper drafting process is very inaccurate because of the imprecision of the materials and the need for working to scale, the computation of that scale and dimensions by hand-held calculator and the use of complex geometrical projection drafting procedures required in faking a reality of three dimensions. The computer drafting process where the computer is used as a simple substitute for paper and pencil uses the exact same procedures as the paper drafting process and is as inaccurate and slow if not slower than paper drafting. The users of a computer paper drafting system will say they are much faster using paper and pencil in comparison to using a computer. The computer they say slows them down and is too complex to set up and maintain. They very often laugh and say that the computer is just a very expensive electric eraser. They will admit that it is a little faster in the creation of labels and drawings are cleaner and look better. The fact is that if they use the computer as a substitute for paper, they are quite correct in their observations and should limit the use of computers in their operations and should continue to use their traditional drafting processes. The typical computer drafting workstation includes a electronic drawing pad, plotter, a hand-held calculator, pad of scratch paper, pencils, technical pens, drafting board and all of the other supporting equipment and tools to do cut and paste of paper drawings. On the other hand a computer aided design workstation requires only software, PC, mouse, keyboard and a plotter when the output is paper drawings.

1.3 Paper Computers and CAD

A computer is a very powerful general-purpose tool, that can be used as a simple substitute for pencil and paper. The Macintosh computers were designed to be easily used as a substitute for pencil and paper. So of course the Macintosh by many of its users is seen as a typewriter and used only as a typewriter for the sole purpose of making paper documents. Macintosh users typically do not know how to share files or how to use any of the advanced capabilities of a computer. In the same way, many current computer aided design systems are actually computer aided drafting systems. Their user interface has been designed to be similar to the use of a piece of paper and like the Macintosh applied to drafting as a simple substitute for paper and pencil using the drafting processes and procedures that have been developed over the last 400 years. The people who use computers as a substitute for paper and pencil have no idea how to really use the powerful capabilities offered by computers.

1.4 The Acronym CADesign

That the use of computers as a substitute for paper causes confusion between drafting and multi-dimensional computer design is reflected in the understanding of the acronym CAD itself. Many who use the paper processes for design believe the acronym CAD stands for Computer Aided Drafting (CAD), but the acronym stands for Computer Aided Design (CAD) as first claimed by the United States Air Force materiel's lab. The users of CADrafting do their calculations with a hand-held calculator and call what they are doing drawings, and these drawings are as full of mistakes as the paper drawings drawings that preceded them.. In comparison, the users of CADesign do their calculations using the computer to create geometric constructions through the view screen into multidimensional space. The users of CADesign are not drafting and faking dimensions. They are modelers modeling precise multidimensional, mathematical models.

1.5 The Extinction of Drafting

Computer Aided Design (CADesign) did not evolve from drafting even if both serve as design tools and the processes are confused by most. Designing with a computer by creating models in multidimensional space requires very different procedures, processes and rules than traditional drafting. The CADesign processes are as different from the drafting processes as a pencil is from a computer. The CADesign system uses digital media to create a precise mathematical model in multidimensional space and then create a flat projection from that model in order to produce an accurate set of paper drawings with a plotter. Yet many people continue to use CADrafting which is only a substitute for paper and pencil and nothing more and reject the computer's capabilities, dismissing the advantages of multidimensional space as too complicated. However, as the power of the PC is increasing, CADesign technology is prevailing, particularly with a new generation of computer users, pointing to the eventual extinction of CADrafting.

1.6 Synthetic Realities

CADesign in contrast to CADrafting is a multi-dimensional digital media, offering direct manipulation of 3-dimensional objects. Models can be made interactive by giving the viewer the ability to fly through buildings exploring its design, or to zoom inside a machine examining the relationship of its parts. Models can be animated moving them through a fourth dimension - time. A designer or engineer using CADesign only needs to learn a small set of simple rules in comparison to drafting. Because all mathematical operations can be done visually through geometric constructions, the designers and engineers do not need the extensive formal training previously required. A person can become a master designer or engineer in six months rather than eight years.

1.7 Useless Textbooks

Most of the training given in CADesign is almost always based on learning the user interface. In reviewing thirty five CAD textbooks on Autodesk's AutoCAD software application, each text book began by stating that the user should consider the computer screen to be a piece of paper. None of the textbooks gave anymore than a history of the development of AutoCAD commands. None of the textbooks demonstrated how to solve necessary engineering calculations by doing geometrical constructions. Most of the textbooks consisted of exercises for each command, but they only dealt with two dimensional applications. Yet none of the exercises took into account the rules of drafting and the drafting specifications for the creation of paper drawings. None of the textbooks offered a systematic approach to creating models in Cartesian space and then being able to use the models to create paper drawings.

1.8 CADesign, a Cyberspace Ship

The computer screen of a CADesign's workstation is not a piece of paper! It is the windshield of a cyberspace ship - the face shield of a small personal virtual jet ski surfing, probing, and exploring the vast virtual universe. Understanding the true nature of the digital materials and processes as practiced in CADesign systems is essential to making the most effective use of these systems. CADesign is not merely a substitute for drawing on paper. CADesign is a new digital media device. It is a multidimensional modeling space that allows you to make simple virtual bit stream - blocks, spheres, nuts, bolts, pistons, molecules, cells, so on - realities that can evolve into complex virtual and synthetic life streams - ever more complex assemblages of data. This digital flux will expand our consciousness and create a network of synthetic awareness of the world unlike anything we have ever known.

1.9 The Digerati

Only a few people who are digitally literate have come to understand and develop an intuition concerning the social impact of digital materials and processes, but even they most often consider synthetic worlds to be parallel worlds to the natural world with little connection between the two. However the practice of CADesign demonstrates that synthetic worlds are very much a part of the natural world and the natural world will be greatly affected by synthetic objects created with these systems. CADesign systems are the critical link, bridge or transportation device that connects digital realities to the natural world.

1.10 Future Stock

The new digital structures are creating future shock. The confusion between CADrafting and CADesign is an example of this future shock. Legal ownership of digital materials, 0's and 1's, and their associative digital data, bit streams and processes cannot be rigorously proven or verified causing misunderstanding as to value. For example, Netscape, though highly successful, never created a positive cash flow for its investors. The failed dot coms caused financial instability of the stock market. Digital bit streams are more illusive than water or air, where only the ownership of the land through which they flow can be proven by traditional law. Digital material and its attending virtues are bringing to question the validity of ownership, and this suggests that ownership should be replaced with stewardship and service. The value of digital materials depends upon their use, and unlike analog materials their uses are unlimited. Unlike the analog process of copying, where the copy degrades from the original and the process is thereby limited, the digital process of copying does not degrade and costs very little in comparison to the analog process. The extreme difference between the digital and analog processes is expressed in the basic conflict between the idea of openly sharing digital information and the concept of proprietary ownership. Currently, companies use the CADesign process to create proprietary designs and patents on which they depend to create value on the stock market. They have no interest in sharing that information with anyone. Therefore most of the software companies that provide CADesign systems have no interest in their systems having any compatibility with any other systems. 

1.11 Virtual Networked Organisations

In the digital world no one has ownership and the idea of product is being replaced with the idea of service. This is really not much different than the actuality of business today. Most people who conduct business do not own patent rights or copyrights but are simply providing a service. The overwhelming efficiency and convenience of CADesign will promote the development of direct manufacturing. Consider the following future scenario. Choose an object that you want from swarms of cyberbody information. The object will modify itself to suit your needs and directly manufacture itself. Materials will be recycled. This will be a reproduction service having little or nothing to do with ownership of the original design, tooling and or production facilities. The design and production processes will become self modifying. How then can one judge who owns a patent or copyright to a design created by automated processes. Businesses of the future will be digital structures called Virtual Networked Organisations (VNOs), organic growth models featuring no central control or job descriptions.
[ For further study on the subject of VNOs, please refer to the paper: "Management and Virtual Decentralised Networks: The Linux Project" by George N. Dafermos, First Monday, volume 6, number 11 (November 2001), URL: http://firstmonday.org/issues/issue6_11/dafermos/index.html ]

1.12 Why Study CADesign?

Because of its application value to design and drafting, CADesign software applications continue to be some of the most expensive and profitable personal computer software in the world. The profit in CADesign software is second only to operating system software. At the start of the PC revolution in the United States, AutoCAD software by Autodesk became the de facto standard for many years in personal computer computer aided design software and was second to Microsoft in profit measured in billions of dollars. Profit however is not the reason to study CADesign. Multidimensional modeling and synthetic simulation are some of the most challenging and important technologies in the field of computer software.

1.13 Control the Replicators

The importance of computer modeling / CADesign is only understood by a few people who have been in the field of manufacturing and production involving the application of numerical control (NC) machines or other digitally controlled processes. Studies made about the development of flexible automation show that multidimensional and accurate digital data and speed of processing is the limiting factor in the design, development, manufacturing and implementation of a cycle of new devices. The length of this cycle was critical to the US Air Force who needed a new fighter plane to fight a jungle war in Vietnam. The lead time was 5 to 7 years for the development of a fighter. Acquiring the data needed for the NC equipment and the associated tooling to assemble the fighter were the bottleneck in development. It is this author's opinion and experience that the big four companies of the US military industrial complex have blocked the development of small, just in time manufacturing units and flexible manufacturing, because this type of development threatens their combined monopoly on the manufacturing of military equipment. As the power of personal computers continues to increase, so does the value of virtual models. Virtual models will quickly be pushed to the higher level of synthetic models. Automatic digital design agents capable of design verification and design modification for conservation of materials will become available. Three dimensional sub micro printing will create nano machines and thin film flexible digital devices.
Co2 laser printing The current rapid prototype machines using only one type of material will evolve to use many types of materials, and rapid prototyping will turn into direct manufacturing. Laser Sintering will be able to create solid 3D objects, layer by layer, from plastic, metal, or ceramic powders that are "sintered" or fused using CO2 laser energy. The inherent versatility of this technology allows a broad range of advanced rapid prototyping and direct or rapid manufacturing applications to be addressed. Objects can be designed and made on demand when needed with this type of replication. Copyright and patent laws from the centralized point of view block the research and development of this type of technology. Clearly new social paradigms are needed to allow and support development of these emerging digital technologies.
Modeling systems of the future will not be used to create paper drawings but the actual objects of design. The modeling systems will need to be much more accurate than current systems and to be free for peer review and modification.

2.0 Theoretical Orientation

Theoretical orientation includes the history of the Cartesian coordinate system, the basic theoretical concepts behind the Cartesian modeling system, and definitions of the Cartesian coordinate system.  Also covered in this section are a digital learning approach, para-solids modeling kernels, Constructive Solid Geometry (CSG), and Function Representation (F-rep).

2.1 The Cartesian Coordinate System

The Cartesian coordinate system is procedural method used to describe the dimensions of a synthetic space. The Cartesian coordinate system is attributed to/documented by Rene Descartes (1596-1650).
Rene Descartes (1596-1650)
Descartes is most famous for having written a relatively short work, Meditationes de Prima Philosophia (Meditations On First Philosophy), published in 1641, in which he provides a philosophical groundwork for the development of the sciences. However, Descartes was a maverick, a freelancer with no academic or political ties to universities. Descartes radically asserted all existing knowledge rests on the unstable foundation of Aristotelian physics based on our senses, perception and reason, which deceive us. Cartesian physics is a system of synthetic reasoning; knowledge starts with a first principle and proceeds mathematically through a series of deductions, reducing physics to mathematics. The properties of bodies in Cartesian physics are measurable specifically on ratio scales, and hence are subject to mathematical rendering. The Cartesian philosophy is the logical referencing of quantitative nodes of knowledge, establishing these quantitive nodes in a procedural relationship to create a cellular system of thinking. One should start by systematically doubting everything and find the first principle of knowledge trusting only the procedures of logical thought.
Descartes' Cartesian philosophy of reference is expressed in the latin phase, Cogito, ergo sum (l think, therefore I exist). This phrase is the point of origin from which he derived the rest of the philosophy. Descartes expressed Cartesian science in the establishment of the first principle single point of origin from which he developed procedures by which to study a three dimensional synthetic space and three dimensional virtual objects inside that space, thus linking geometry to algebra and physics to mathematics forever. Descartes did so by defining the Point of Origin and procedurally referencing it to three infinite lengths of cords called the coordinate axis labeled X ,Y, Z axis, that are mutually perpendicular to each other and bisect each other at the point of origin. The infinite axis having equal divisions of negative and positive values which originate from a single point of bisection at the point of origin.
X Y Z axis
Therefore the Point of Origin has a set of coordinate axis values called coordinates of 0,0,0. Once the Point of Origin 0,0,0 has been established along the three axis, this creates the synthetic existence of an infinite grid of cube shaped cellular space called the Cartesian Coordinate Space.

This synthetic space allows for the development of mathematical modeling and the study of three dimensional virtual objects. Rene Descartes' quantitative philosophy of synthetic cellular reasoning succeeded in overthrowing a qualitative system of natural reasoning philosophy of Aristotelian physics that was centuries old. With Cartesian space (X,Y,Z), Function Representation (F-rep) - f (X,Y,Z,...N), and the ever increasing computational power of computers, we are ready to remove the rectilinear limits of virtual objects in Cartesian space and model dimensions beyond our imagination.

2.2 Digital Learning

The natural process of learning involves associating new things to something that is familiar or looks similar. Visually a line looks the same on a piece of paper as it does on a computer screen. The comparison of a line on a piece of paper and on a computer screen is the same as the comparison of a person who is dead and a person who is alive. Please do not think that you will save time by scanning the dead lines from paper drawings into digital alive lines on a computer screen. This is much like trying to raise the dead. Yes, all of the paper drawings you have are a thing of the past and need to be made into models. The results of scanner / raising the dead will be nightmarish zombies creating manufacturing horror stories equal to the B grade movie "Night of the Living Dead". Remember the letter D in the acronym CAD is for Design, not drafting. Thinking about using the computer screen as a piece of paper is approaching the computer from a 2D drafting paradigm instead of 3D modeling, design and simulation paradigm. So please resist the natural tendency to think of the the computer screen as a piece of paper. Please think of the computer aided design system as a cybership named CADesign and the screen, as a windscreen, the face shield of CADesign through which you can travel the vast reaches of multidimensional space. Refer to the CADesign's process as modeling and the results as a model. A drawing is limited 2D lines and paper attempting to fake 3D. If you are drawing with a computer then you are using a paper design process that does not offer the advantages of the digital design processes. A person, who is using a calculator instead of CADesign's systems, drawing fake dimensions on a paper screen, is not clear on the basic concepts of using digital technology and therefore cannot see nor use the incredible mathematical intelligence of CADesign's systems that awaits their fingertips. Design calculations can be done easily by using constructive modeling (the intelligence of CADesign's systems) in multidimensional space. By watching the calculations unfold before your eyes, there it is on CADesign's windscreen if you dare to explore constructive modeling solutions for a needed design calculation. The use of constructive modeling on CADesign's windscreen is a system of using math visually. The proper use of CADesign's systems allows you to access CADesign's mathematical intelligence which mentors you in proper design replacing years of tedious mathematical training. CADesign's command and control systems has in line mathematical functions for all other calculations that can not be solved constructively. CADesign's sixteen place internal results does not need to be rounded off for the human operator. The hand typed input from the eight place display of a hand held calculator is some kind of a cruel joke. A designer/engineer at a CADrafting workstation (a CPU with multiple parallel processing floating point math units) using a pencil, pad of paper and a hand held calculator for their design calculations is sadly unaware of the joke.

2.3 Para Solid

The word "Para" means like something, but not really that something. "Para Solid" modeling means its like solid modeling, but not solid modeling. The base or core mathematical representation is not to be confused with visual representation. Visual representations, wire frame and surface polygonal meshes, are used to visualize the mathematical representation. It is important to learn the difference between solid constructive geometry, implicit surfaces, volume modeling and the boundary representation polygon based procedures used by most CAD systems. Early modeling systems were written complete with a user interface, core processing and output routines by each company. However many well known 3D CAD systems which claim to be "Solid Modelers" are now constructed based on kernels obtained from third parties, notably the ACIS (Spatial Technology, Inc.) and Parasolid (Unigraphics Solutions,  Inc.) kernels which only use some of the solid modeling procedures. They typically do not retain any of the mathematical primitives or history of procedures used, and the output from these kernels are "polygonal meshes with holes" models. In fact, they are boundary representations, not "solids". It is a fictitious stretch of marketing imagination for most well known 3D CAD systems to use the term solid modeler and to refer to the models created as solid models. The so called "solid modeler" systems currently sold today are most likely not!. Please note: ACIS and Parasolid kernels do allow provisions for the retention of some type of procedural history. What that procedural history includes is not clear and how the various companies implement the kernels is not clear. A review of a recent release of one CAD system shows that the " para solid models" can be edited. This implementation of Parasolid shows a history tree of operations. Mathematical representation by the "para solid" model kernels are improving, but the models are still polygonal surface models closely tied to the visual representation of the surface. However because these improvements are not open for review, no one can understand how good they are. Furthermore if they are closed systems, they can not be compatible with other CAD systems.

2.4 Open Procedural History

An open procedural history or list of commands to create an entity is usually much more compact than the completed model and is extremely important to the migration of digital data to other systems. Furthermore, if constructed properly, it contains all the information that is needed for the final geometry, including the representations of solidity and volumetrics. It has long been known that such a historical list (actually a tree) of commands is a valid model in its own right. The highly accurate input that forms an entity's history that can be used to adjust the level of detail, and answer the same sort of questions as the final model provides (e.g., is this point inside or outside the solid?) without actually constructing a boundary model for visual representation at all. The substantially significant advantages in the use of  procedural history include providing a robust data structure providing stability and verifiable process procedure and accuracy. Making a boundary based representation model is complicated, and inevitably inaccuracies creep in;  in particular, the edges of a boundary model often deviate slightly from the surfaces that they are bounding. It is extremely difficult to stop these errors from affecting subsequent calculations. On the other hand, working directly from the history with what we will refer to as a Constructive Solid Geometry (CSG) model, we are using the 'raw ' input and geometry. The ACIS and Parasolid modeling kernels of course use mathematical functions; however it is anyone's guess what functions are used because the source code is not open for peer review. Furthermore the vendors implement the kernels differently. Most do not retain any history at all, but fool their clients into thinking they are using "solids" and CSG when they are not. Therefore without a history or a CGS procedural tree, most of the models being created on most current modeling systems are dependent on the system that created them and are subject to being lost when any of the complex parts of the original system change. It is a sad fact that most all of the 3D models without a history of procedures or originating input could be useless within as little as one to five years and, without a doubt,  will not have any value in fifty. A procedural history of the construction of 3D models is of key importance to verify your 3D models, share your models between systems and be able to migrate the 3D models to future systems, thus protecting your investment of time and labor in creating the models.

Solid primitivesThe six CSG primitives, upon which Boolean operations can be performed in CADesign.

2.5 Constructive Solid Geometry (CSG)

Solid modeling programs that have a proven set of mathematical operations and retain a history of the mathematical operations and other procedures stored in a CSG tree (that can be traversed and modified by the users to verify the results) are recommended for many reasons. The following is an example of the need to verify data using a CSG tree. The three-quarter view and plan view diagrams below of a CSG tree of a solid model show the model's primitives on a round pad, and the Boolean operations performed on the primitives are shown on a binary fork in the branches of the CSG tree. There are a total of nine primitives and eight Boolean operations.
The model's primitives are made of eight cylinders and one cube. Starting from the top left moving to the right and down in the bottom left diagram, the following describes the union and subtraction of the first four primitives. In the first two sets of cylinders with one horizontal and one vertical cylinder in each set, the horizontal and vertical cylinders in each set are joined together with a union operation to form two solid cylindrical plus symbols. The small diameter shape is subtracted from the larger diameter shape to form a hollow shape that resembles two sections of pipe cut and welded together to form a plus symbol.

Theoretically with a proven set of given mathematical operations, changing the order of operations will always give the same mathematical results. However if you look closely at the hidden view diagrams above of just the top part of the CSG tree that has five primitives and four Boolean operations, you will notice that the one on the left has different operations from the one on the right. The one on the left has two Unions (U) and a Subtraction (S), whereas the one on the right has two subtractions and one union, in the first three operations. You will also notice the last/fourth operation (at the bottom of the tree), a subtraction of a cylinder, leaves an opening allowing you to see inside both of them. You will see the hidden view of the one on the right confirms that the model is not correct. Using the wire frame views of the models, you will be able to see in the diagram below that the error actually starts with the union operation previous to this one, but you cannot see the error in the hidden view above. The modeling operations of the one on the right is a more natural order of operations, using the subtraction operations first to create horizontal and vertical sections of pipe and then a union to join them together.  However, this order of operations creates an error as seen above.

2.6 CSG Procedural Error
Four wire frame visualizations of solid models are shown above. The first wire frame visualization of two tubes welded together starting at the top left of the diagram is the result of subtraction of a two small cylinders from two large cylinders creating two tubes followed by the union of the two tubes into the geometric shape of a plus, and the wire frame visualization clearly shows an error. The second wire frame at the top right shows the subtraction of a fifth large cylinder through the center of the plus shape and clearly shows that subsequent operations carry the error forward,  and the error grows. The third wire frame visualization at the bottom left is the result of the union operation of two large cylinders and two small cylinders into a large and small plus shape followed by the subtraction of the small plus shape from the large shape, and it is clear there is no error. The fourth wire frame view, bottom right,  is the result of the subtraction through the center of the plus shape by a large cylinder and clearly shows that there is no error. Again this type of error due to change in order of procedures is clearly very wrong. The software is quite flawed. In the future world of direct manufacturing, this type of error will be unacceptable and must be discovered by design checking agents.

Note: Do not assume your modeling to be accurate. The example above shows that without understanding complex mathematics, you can visually see the error in the model. The power to visually check complex mathematical models is one of the greatest benefits of using CADesign. The solid models in the examples above were created using AutoCAD Rel. 12 software by AutoDesk with Advance Modeling Extensions (AME). AME is the solid modeling extension of AutoCAD Rel. 12 and was one of the very few commercially available solid modeling programs. AME and the CSG tree were dropped by AutoDesk in Rel. 13. However, AutoDesk still claims that AutoCAD Rel. 13 and beyond are "Solid Modelers" and that simply is not true in this author's opinion.

2.7 Function Representation ( F-rep)

HyperFun Project (http://www.hyperfun.org) is a free software development project for functionally based shape modeling, visualization and animation. The project is based on a so-called function representation (F-rep) (http://wwwcis.k.hosei.ac.jp/~F-rep/ )of geometric objects and supporting software tools built around the HyperFun language. In F-rep, complex geometric objects are constructed using simple ones (primitives) and operations on them. Any object in three-dimensional space is defined by a function of point coordinates F(x,y,z). This continuous real-valued function is positive inside the object, negative outside, and takes zero value on its surface. Similarly, a multidimensional object is defined by a function of several variables F(x1, x2, x3, ..., xn). For example, an object changing in time can be defined by F(x,y,z,t) with t representing time. In HyperFun, the functional expressions are built with using conventional arithmetic and relational operators, standard functions, built-in special geometric transformations and F-rep library functions. 
HyperFun is the next step in CADesign development, as it also allows the mathematical definition of any number of attributes of an object such as materials, color, texture, hardness, softness and so on.  In creation of a CAD system with HyperFun, the mathematical modeling will be retained apart from the visual representation.   Mathematical representation being separated from visual representation and the processes open allow the mathematical representation to be done on any platform now or in the future. HyperFun has been in academic research and development for many years, but the application side of development has just started. At present, HyperFun modeling tools are still limited. There is not at this time a robust user interface available. However development plans to create synthetic CAD are underway and actual development should start soon.

3.0 Technical Orientation

In this section, we will discuss technical issues such as the definition of the basic elements of a CADesign system, CAD models, normal / natural orientation, mirror writing, CADesign conventions in Cartesian space, wire model viewing, User Coordinate System (UCS) icon.

3.1 CADesign System

A robust computer based modeling system creates a mathematical model not a drawing. Therefore, you need the latest and greatest computer you can lay your hands on. A plotter is necessary to output paper drawings. You should always have a small printer for data dumps to check files and operations. Do not purchase a tablet. You only need a keyboard and a mouse. High resolution screens are not recommended, because the lines on the hi-res screen become very thin and cause eye strain. I recommend the Linux operating system and Varkon CAD software (http://www.microform.se) as your best purchase. For rendering software, POV-Ray (http://www.povray.org) is great. However it uses the left hand rule Y up, but uses the right hand rule for rotation. You need to set up a transformation of the data to use POV-Ray with your CADesign models which will be right hand rule Z up. A review of Linux CAD showed it to be extremely poor. Also, get a really good chair and make sure you have good support for your arms.

3.2 CAD Models

A model, such as Sazaedo shown in the image to the left, is made up of objects which are made from blocks which contain entities. Entities are the basic primitives of the system used to construct blocks and objects. Depending on what are the basic primitives provided by a given CAD system, entities might be points, lines, arcs , circles. There is no standard convention for the naming of different elements used in a given modeling system other than the naming conventions of the Cartesian coordinate system. A point is a set of X,Y,Z coordinates. A line has two points, a start point and an end point; so it has direction and rotation; therefore a line is a vector. An arc can be defined by a start point, midpoint, end point and has a direction of rotation around the center point not to be confused with the midpoint. A circle has a center point and a distance. A polyline has an unknown number of points that define both lines and arcs. Polylines can be open or closed entities. If a polyline is a closed entity, it can be extruded into a solid object. Other basic entities are 3D faces, polygons and polymeshes. Only three serious CAD systems have higher level CSG entities. There are several experimental CAD systems that use higher level function representation entities. From blocks or groups of entities objects are created. In the Sazaedo model, the compound complex shape of the lower roof could not be created with CSG primitives. Also the spiraling over hanging roof and internal ramp could not be created with CSG primitives, because they also have compound complex shapes. Compound complex shapes have surfaces that move in all three directions (X, Y and Z)at once.  That is to say that the X, Y and Z values for any given set of points on the surface will be different.  All other parts of the model including the top roof are CSG based entities. 
IGES (Initial Graphics Exchange Standard) and  STEP (International Standard for the Exchange of Product Model Data) are standards for the sharing of modeling data. However they are closed standards and, in the IGES case, limited and, in the STEP case, not implemented.  So there is no standard data file format used in the CAD industry for accurate three dimensional mathematical (CSG) models which can be used to transmit the logical structure of the  model, but only disconnected surface data.  The Sazaedo model is a historical  digital preservation work. Therefore, mathematical definitions, accuracy of the model and a history of the logical structure of the building are important information from the historical preservation viewpoint.  Only the CSG based entities will be able to survive over time in the current CADesign environment. The AutoCAD DXF, a DIF type file format, that has become a de facto standard for the export and import of three dimensional polygon surfaces. The DIF file format is type of binary file format. It is binary in that it has two lines of text for each data record. The first line of text is a code that tells what type of data is in the second line of text. Examples: Code 0 is the start of a new entity. Code 10 is the start point of a line. Code 11 is the end point of a line. Most CAD files have data tables of various settings for the operation of the system. The settings necessary to setup AutoCAD for various applications can be extensive. AutoCAD takes 250 keystrokes for the average setup of a given application.

3.3 Normal / Natural Orientation

In theory we can define and use any orientation for modeling because mathematical transformations are so simple and easy to do. Mathematical theorists and some computer scientists insist orientation makes no difference if the orientation is first defined. This is true in theory but not in practice. Usage reveals that humans naturally establish one normal orientation and can not think and work in different orientations without experiencing confusion and making mistakes. The use of orientations other than the normal orientation in theoretical work obscures understanding and has even been used as a form of encryption.
Our hands are mirrored structures of each other, and yet there are only a few exceptional people who can easily handle or adjust to a stated orientation were the input or output of a system is mirrored and rotated without making mistakes. It is interesting to note that at an early age some children are naturally ambidextrous and will easily handle writing with both hands and can mirror write with either hand. To read what they have written, one must hold the writing up to a mirror. However the children see no difference at all, until it is pointed out to them and they are taught the difference. Leonardo Da Vinci, an Italian Renaissance artist persecuted for his knowledge and creative ideas, protected himself by keeping his notes and journals from being easily read by mirror reading and writing.

3.4 Mirror Writing Exercise

To understand how important the use of proper natural / normal orientation is please take out a piece of paper and pencil try doing the following tasks. First, try writing with your left hand if your a right handed person. If you are left handed person, you do not need to do this because you live in a world of devices created for only right handed people. Now please try some mirror writing with both the left hand and right hand for a few minutes. Once you have tried the ambidextrous exercises, you will understand how easy it is to get confused and how hard it is to work in a different orientation. When possible, it would be better if we all work with an established and agreed upon orientation, because the confusion in trying to use different orientations is very great.

3.5 CADesign Orientation Defined

The orientation in this chapter on modeling systems uses the Cartesian Coordinate System having X, Y and Z axis with normal / natural orientation. Normal / natural orientation is right handed where: the X axis is positive movement to your right, the Y axis is positive movement to the left , XY axis create a plane referred to as "the plane" or "the ground plane" that is normal to the pull of gravity, and the Z axes is positive movement in the upward direction, having vertical orientation to the pull of the gravity of the earth and where the positive Z direction is against gravity. This is the normal and natural orientation used in drafting, and CADesign, because it is the orientation used in the fields of aviation, engineering, architecture and manufacturing for the last four centuries.

3.6 Wired Model Viewing

Unfortunately some people do not have the ability to visualize wire frames images in 3D at all. Working on wire frame images from the bottom view is not recommended, because it causes visual confusion as we shall see below.  In working with wire frame, one should use 3/4 views, which are 3D views, for designing and editing. Those who attempt to design in  "plan views" (views from the top) or "elevation views" (views from the side) are not working in 3D, but rather in 2D.  They will have difficulty selecting a vertex, because they will not be able to tell if the vertex is the one near to them or far from them. However, plan and elevation views are useful for checking the model. The following example shows the visual problems associated with viewing a 3/4 view wire frame.

Which wire frame view is from the top and which is from the bottom?
The two images to the right and the left are different views of a single 3D model visualized in wire frame. The two views are of a rectangular box that is 4 x 5 x 6 units in size that is a 3/4 view from the -1x -1y +1z octant reveling the front, top and left hand side and a 3/4 view of the same model from the bottom -1x, -1y, -1z. Please note: If the model were a cube instead of a rectangle shown in  3/4 views from the top (+1z) and from the bottom (-1z), we would see exactly the image. Therefore we would not be able to tell the top of a cube from the bottom of a cube. This problem is solved by using an icon to give visual indication of orientation.

3.7 XY Coordinate Icon

Proper normal orientation is both the visual and theoretical frame work on which you create a model. Modeling with wire frame visualizations of complex mathematical models is confusing at best and not even possible, if a standard normal orientation and some type of visual cue for orientation is not used. The "User Icon" is a dynamic symbol that is a visual cue for the user as to the orientation of the User Coordinate System relative to the World Coordinate System, the model and the user's view. The User Icon indicates the location of 0,0,0 or the point of origin for the User Coordinate System, the general orientation of the User Coordinate System and the positive and negative orientation of the Z axis in relation to the screen as shown below. Typically the User Icon is not used in computer drafting which is 2D . This is unfortunate because the ability to have access to a temporary point of origin / a user origin point is extremely useful in designing in 2D. The default state for the User Icon in AutoCAD is "on".  It has been by the author's experience that 95% of CAD installations do not use the User Icon; it is turned off, as the users do not understand its function.
A 3D Designer Must Use a UCS Icon
The addition of a coordinate icon helps. Now, can you tell which wire frame is being viewed from the top and the bottom? If you can not, that is OK because you can train yourself to be able to see wire frame 3/4 views in 3D. A designer can not design in 3D if he does not turn on the coordinate icon feature. Please note several things about the coordinate icon. 1 - This icon displays a W and that means it is in the world coordinate view. 2 - There are small tick marks at the intersection of the two arrows; this means that the icon is on the origin point. 3 - In the left hand view you will notice that two lines are missing from the icon that form a square in the view on the right; this means that Z is pointing away from you.
Has the visual cues of the User Icon helped you to tell which wire frame you are viewing from the bottom? Working on wire frame images from the bottom view is not recommended, because it impairs the user's ability to visualize the 3D wire frame model without confusion.

Below are images that show only six of the possible eight states of the User Icon's visual cues. Two images are missing. Which ones are missing? Can you list all eight states of the UCS icon? Read the three numbered statements above again very carefully and you should be able to figure it out.

3.9 Normal Orientation Exercise

Three points define a plane, a point and a line define a plane, and two lines define a plane - are basic Euclidean axioms used in plane geometry. Using the axiom two lines define a plane, we will create two 3D models of the X, Y and Z axis of the Cartesian system - one with a sheet of paper and the second with our right hand.
1st model
Take out a sheet of paper and thin wooden pencil. Now place a point in the middle of the paper and label it O for origin. Now draw a line from the origin point in the direction to your right and label it X. Now draw a second line from the origin point perpendicular to the first line in the forward direction and label it Y. With these two perpendicular lines, the first plane of the Cartesian system called "the plane" is made. The term "the plane" always refers to the X, Y plane or ground plane that is normal to the pull of gravity. Now finish the representation by piercing your paper from the back side at the origin with the point of your pencil pointing upwards from the front surface of the paper normal to "the plane".  Please be careful - but look at the paper with the point of the pencil pointing toward you so it looks like the image to the left.  In drafting, this is called the default view or unless otherwise noted the top down or "plan view" of "the plane". You might think of this view as a map or a layout of the ground plane with you in the center.  This is the creator orientation or "God Orientation".   Place this model in your left hand and create the second model.
2nd model
Now, hang your right hand downward with hand open and your fingers pointing toward the ground plane. Next, rotate your wrist so that the palm of your hand is facing in the forward direction. Bend your right arm at your elbow 90 degrees so that the palm of your hand is now facing upwards. Your right hand and your thumb should be pointing in the positive direction of the X axis or movement to the right and your index finger is pointing in the positive Y axis or pointing forward. Now, bend the third finger upward. It is now pointing in the Z axis.  Rotate the right arm at the elbow so your right hand is directly in front of you just level with your elbow and keeping your right hand as shown in the image.  In 3D design, this is "the view", "the artists view" or in drafting "the three quarter view".  "The view" is the normal viewing of real objects that is the most common view experienced by people. It is a view of an object from a natural body position for normal people.  Now move and rotate the paper model to a position that is the same orientation as your right hand.  This is the "Right Hand Rule". 

The above 3D models, one with paper and pencil and the other with the right hand, are anthropomorphic based orientation for 3D modeling and viewing - that is most efficient for humans.  In fact, if you do not use this view, you probably will not be able to model in 3D.  This orientation is called "the right hand rule" with the normal natural orientation of Z up (Zup). However Zup is the not the
natural orientation for some people like programmers.  They work visually in a virtual world of a computer screen where the forward movement of the mouse in the real world is transferred to an upward movement of the mouse cursor.  So the natural orientation for them is Y up (Yup). The difference in orientation between the Yup people and the Zup people causes conflict. Orientation confusion in the real world where gravity is a serious matter can lead to serious mistakes.

3.10 Virtual World Orientation

Virtual world orientations are in conflict with the real world. Unfortunately the orientations used in virtual worlds are not just the relatively simple Zup Yup conflict of a 90 degree rotation, but the virtual orientation is often mirrored from right to left as well. Programmers who spend most of their time viewing and navigating the abstract virtual world of cyberspace and very little time building physical things in the real world of gravity are not aware of right handed orientation or the need for a person modeling to read wire frame models with normal views that match aviation and engineering standards. Such programmers have written all of the basic Open GL libraries (http://www.opengl.org) with the virtual world orientation of left hand rule and Y up. Novice programmers who are not aware of the problem will of course write simple programs using the the left hand rule Y up orientation of the Open GL libraries. A very famous program, POV-Ray, uses the left hand rule Y up orientation and the right hand rule for vector rotation. VRML (Virtual Reality Modeling Langauage) uses the orientation of right hand rule and Y up, and the HyperFun program began life with the left hand rule Y up orientation. HyperFun, looking toward synthetic simulation, did not want to be in conflict with the real world and modified their orientation. The Y up (Yup) or Z up (Zup) problem is the most difficult to change in computer science because all the books about Open GL talk about the Z buffer as a depth buffer, which has become standard nomenclature for the virtual world. Open GL libraries were given the orientation left hand rule with X as width, Z as depth and Y as vertical height. These alternate views are placing serious barriers to the visual bridge between the real and virtual worlds and causing a great deal of disorientation and confusion. Mirrored and rotated systems, including VRML standard language and POV-Ray, come from not trying to find a first principle with regard to normal orientation in the real world. People creating such systems usually start out by not being aware that their virtual world orientation is in conflict with real world orientation conventions of the last 400 years in engineering, manufacturing, and aviation. Programmers continue to use the left hand rule which is a mirror of the right hand rule and/or a Yup orientation which is a 90 degree around the X axis from the real world normal orientation of Zup.

The following is a quote from the POV-Ray web site:
"there is so much controversy about the right hand rule we decided to use the left hand rule"

3.11 Octant Space

The X, Y axis divides a given space into four sections, called quadrants. The addition of the Z axis divides the quadrants to create octants.
If we pick a point O as origin and draw two perpendicular lines through the origin, we create "the plane." The two lines are labeled X and Y axis. The term "the plane" always refers to the X, Y plane or ground plane that is said to be horizontal to or normal to the pull of gravity. In the practice of drafting, the default view shown to the left is the view from the top down or "plan view" of "the plane." You might think of the plan view as a layout or a map of the ground plane. The plane's X, Y axis create four divisions of coordinate space called the "quadrants." The quadrants are named for their signs. The 1st quadrant is called the plus plus quadrant (+X +Y axis). Proceeding in a counter clockwise rotation, the 2nd quadrant is the minus plus quadrant (-X +Y axis). The 3rd quadrant is the minus minus quadrant (-X -Y axis) and the 4th quadrant is the plus minus quadrant (+X -Y axis). A move to the right is in the positive direction of X. A move to the left is in the negative direction of X. A move forward is in the positive direction of Y. A move backward is in the negative direction of Y.
The third axis, the Z axis, is added mutually perpendicular through the point of origin of the X, Y axis. The a third dimension Z is know as height or elevation in aviation. The Z axis creates eight divisions of three dimensional space called "octants." The positive direction of the Z axis is up against the pull of gravity. Adding the Z axis defines two more datum planes, the X, Z and the Y, Z. When viewing these datum planes from a perpendicular angle to the surface of the planes, the views are refereed to as plan elevation views in the fields of engineering, architecture and aviation. The figure to the right showing all of the octants is known as a 3/4 view. Now we have created a synthetic space, and every point in that space has a unique triplet of cartesian coordinates labeled as X, Y, Z . It is understood that if given a list of three numbers, they are the Cartesian coordinates X, Y, Z  - - and they are used to determine a point in Cartesian space applying a number as movement in space along X, Y, Z axis. The 1st octant is +X +Y +Z. The 2nd octant is -X +Y +Z. The 3rd octant is -X -Y +Z. The 4th octant is +X -Y +Z. The 5th octant is  +X +Y -Z. The 6th octant is -X +Y -Z. The 7th octant is -X -Y -Z. The 8th octant is +X -Y -Z.

4.0 Cartesian's Navigation Systems

In this section we discuss the subject of Cartesian navigation or viewing in-depth. Examples below will show you how to view and change views of multidimensional models in the virtual space created by the Cartesian coordinate space system on the windscreen of your computer modeling system. ("Cartesian's windscreen" where Cartesian is the name of a new imaginary multidimensional spacecraft to be constructed.) We also present an imaginary user interface to describe the subject of viewing in-depth in the text below, as if it were an existing command where the use of two letter acronyms is the keyboard entry which brings on screen "Cartesian's" visual interactive interface for the command mode. The ease and skill with which one can view the development of a 3D model is an essential part of being able to model in Cartesian space. A person's modeling ability is only limited by their ability to navigate Cartesian space. Exploration and development of effective navigational systems for Cartesian space has just begun. The complete user control and ease of navigation in Cartesian space is the most important aspect of multidimensional modeling.

4.1 The Five Systems

We will begin with an outline of all of the parts necessary for a robust Cartesian based navigation system and then discuss the addition of a multidimensional interface to navigation systems. Modeling systems must define and keep track of at least two Cartesian coordinate systems, the world and a user view. Most robust systems track and use at least four to five different types of coordinate systems:
  1. World coordinates
  2. User view coordinates
  3. User or creation coordinates
  4. Entities coordinates
  5. Groups of entities coordinates

4.2 World Coordinate System (1)

The World Coordinate System is a single point of origin / base / handle / for the entire model and for the other coordinate systems. However it is possible to create a simple system using only World and User View Coordinate System(s). An example of this type of system is the HyperFun polygonizer; such systems most usually depend on mathematical definitions and are important teaching tools that provide an intuitive feel for the relationship between math, geometry, programming and computer visualizations. These types of systems are not useful for modeling large complex objects or designs, because they are text editor based. Viewing from only one View Point (VP) and the point of focus or the View Target (VT) will allow us to easily see and verify the results editing a single simple object using a text editor, as shown in the HyperFun example below.  To do visual based editing rather than text based editing, one needs the other coordinate systems as well.

4.2.1 HyperFun System

HyperFun (http://www.hyperfun.org) is a simple system of navigation for simple constructions. However, in concept, the system can mathematically model any level of detail.
Text File
HyperFun uses one and a half (1 1/2) coordinate systems. The world coordinate system is almost the same as the the view target (VT) or point of interest. The user can not move the VT or point of interest and therefore is unaware of it. Only the user view point can be moved by the user.
image of modle VT - point of interest can not be moved. This system can not be used for large models with many objects. The text file is shown to the left. Next to that at the command prompt, the program is called by "hfp txt.hf -g 35" and the image of the model appears in the foremost screen shot. The user view can be rotated and zoomed with the mouse.

The VT (point of view) is just off center from the world coordinate system so that when the user view is rotated around a sphere, some movement is able to be detected due to this slight offset.

4.3 User View System (2)

The User View Coordinate System is usually defined by one point, a direction or point of interest and rotation that creates a cone and plane of view. This is needed so that the user can change views and work from various viewing angles. The editing of a single object can be done interactively on your windscreen using the rendered visualizations of the object. However when you use the visualization to edit even a single simple object, it is done much more efficiently with the use of wire frame visualizations (X-ray vision) rather than solid rendered visualizations. While seated at Cartesian's controls, you can issue a command for a re-scanning of the mathematical model. You can either use normal vision and view the solid object, or you can give yourself Superman's X-ray vision and render the Cartesian object transparent so that you can see all boundary points and edges.

solid visualization

X-ray visualization

4.4 Superman's X-ray Vision

A 3/4 view from 225 degree & 35.8 degree
Using X-ray vision provided by Cartesian's windscreen conveniently and efficiently allows you to make interactive mathematical modifications to the object. Cartesian's digital agents do the calculations for you. However, when you use X-ray vision, you are using a very unnatural view, and, as shown to the left, certain X-ray viewing angles (even simple ones) can be very confusing and almost impossible to utilize. The image on the left is a wire frame model of 3D cube of 5x,5y,5z units. It looks like a flat 2D six sided figure. Can you visualize the image in 3D and see it as a cube.? A solution as mentioned before is the use of a user icon. However it is very hard to read upside down, and with many complex models the icon becomes difficult to use. The icon helps but by itself alone is not a good solution. Therefore it is necessary (when using X-ray vision to edit objects) to be able to easily understand your orientation to the objects and the surface and distance relationships between the objects being viewed.

4.5 Autonomic Autopilot Controls

Cartesian's autonomic autopilot controls will have autonomic movements, surface probing , and visual acuity systems with standard tele-transport positioning portals. We will describe only two of the proposed navigational controls all ready proven to aid your visualization processing of wire frame models. One is the use of autonomic movements or (very slight rhythmic real time changes in the user view). All living creatures have continuous autonomic movements that aid the creature's perception. The second is the use of standard View Points (VP) so that you may instantly tele-transport to a position in Cartesian space. Below are the proposed four standard 3/4 views for modeling in three dimensions. The first view is "the view" V1 shown on the left. This is the approximate view point of a person viewing an object in their right hand with: elbow bent at a 90 degree angle, the palm up, hand open, the front of the object aligned with the X axis of the thumb, the arm rotated so the object is directly in front of the person's waist.
  • View 1 (V1) is a VP from the 3rd or ( - - +) octant.
  • View 2 (V2) is a VP from the 4th or ( + - +) octant.
  • View 3 (V3) is a VP from the 1st or ( + + +) octant.
  • View 4 (V4) is a VP from the 2nd or ( - + +) octant.
Note: The 1st (+++) & 5th (++-) octants are used for modeling space. The 5th (++-) octant is for objects that are below ground level.

4.6 Normalcy in Viewing
guage VP V1 is a common and familiar natural viewing of an object, because it is a viewing point as if the object were being held in your hand. We will call this "The Normal View" to the plane. V1 is said to be the most normal (as in normalcy) because it is the most common familiar natural viewing of an object as experienced by most people. "The Normal View" turns out to be extremely important, because it is the best angle for X-ray viewing of a 3D model without experiencing visual confusion. The parallel to the ground plane viewing angle of normalcy for VP V1 from the ground plane is 35 degrees plus or minus 10 degrees. VPs V2 V3 and V4 are less normal than V1, but are very good angles for X-ray viewing also. VPs V1 V2 V3 and V4 establish a normal cruising range for viewing a object while modeling. A person modeling tries to keep the view as normal as possible to avoid visual trouble. The dial gauge type diagrams at the left give a read out in degrees of angle for user view point V1 and also show the angles of V2, V3, V4.

4.7 Rotation of User View
View One (V1)
View Two V2
In the images to the left, the user view is rotated - the object is not. The object is not moving in relationship to the World Coordinate System. If the User Icon is not turned on, there is no way to know if the view is being moved or the object is being moved . The novice user who usually turns off the User Icon, not understanding its importance, becomes confused and lost between the movement of the View Point and the movement of the object.
View Three (V3)
View Four (V4)
The object is rendered to avoid visual confusion. However one needs to use the hidden lines (lines that have been hidden from view) during the modeling process. Therefore, rendering is for visual verification only  -  not to be used during the actual modeling of the model.
Please note: The View Target 0,0,0 is laterally moved in the images shown to the left using the pan or the zoom window command.

4.8 View Target Movement

There is some normalcy in viewing angles. However you might have noticed by now that we have only moved the View Point VP and not the View Target VT. VT remains unchanged; this is unnatural and strange to most people and hard to get use to at first. You can experience and understand how strange this is by walking around a room keeping your head and eyes on one spot (the VT) in the middle of the room. The only way to move the VT point is to pan or zoom to an object. However when you rotate the VP, the VT point moves back to the origin point 0,0,0 zooming out to the extents of the model, requiring the person modeling in 3D to zoom in selecting with a window an area to zoom into. In a large model, this is a very slow process for the computer to regenerate an image of the entire model, then the window selected. A nightmare exclaims! the person translating this text who has had no experience with CAD. Why? someone might ask. The answer is: The indirect movement of the VT in current CAD system interfaces is a 2D drafting legacy. The interface is a 2D paper interface. This viewing is not so strange or unnatural to the drafting person who works on flat two dimensional drawings. His/her viewing angle is "normal" or 90 degrees from the surface of the paper on which he works. The use of a paper interface by current CAD systems for 3D modeling is irritating at best, though it does allow the construction of large models with many objects. In the real world, the user VT point is always moving (the turning of your head), whereas the user View Point is more difficult to move - you must move your body.

4.9 Navigation Conclusion

The development of a CAD interface from drafting standards is a force and fake application of paper drafting design applied to digital design. A true digital design interface has little or nothing to do with drafting. When a 3D model is complete, it is sometimes projected into a 2D surface model where some drafting types of operations are applied for additions and adjustment in order for the 2D projection to be printed on paper. The current CAD interfaces with the paper like qualities are unlike that of game simulation movements that provide a natural interface; it is very easy for a user to quickly understand how to navigate in a simulation system. A simulation game type system of navigation is proposed as the core of Cartesian's control system.

The pure visualization power of "Cartesian's windscreen" stemming from the qualities of digital material has no equivalency to the visualization capabilities of paper materials and processes. However a subjective personal comparison can be made about interactive digital and paper based processes. Sitting at Cartesian's helm with digital agents at your command is much more exciting, rewarding and enjoyable than sitting at a computer used as a substitute for pencil and paper. Trying to fit paper processes into the digital world is not much fun. Now sitting at a drafting board using simple tools and materials to hand create a drawing has a calming, meditative aspect that is very enjoyable, but it is not as efficient nor as precise as digital based design processes.

5.0 Modeling

The common starting assumption for novice users and designers of CADesign systems is that of a "new clean sheet of paper," i.e. a blank screen; unfortunately this is a paper legacy concept that is misleading in the digital environment . Digital systems are based on sharing and modifying pervious work. Unfortunately the CAD (drafting and design) world is very slow in becoming digital because of patents and copyrights. So, the sharing of models has not become prevalent or technically high level. However, "reuse" was one of the basic reasons behind CADesign systems, since they allow creation of blocks of parts, save them and insert them into different models. It was thought that digital based designs would be much more persistent than paper designs, because digital data does not degrade when copied or wear out like paper. However, digital data persistence has turned out to be a big problem for complex reasons not covered here. The starting assumption should always be - - to start a new model based on a previous model or parts, that creation is the evolution of existing models. In the author's seventeen years of consulting and installing CADesign systems, the creation of base prototype models to create new models was the key to any client's success. After several years of using a CADesign system, you may come to realize that the moving, rotation and editing of models are the most used and important modeling and design functions in the CADesign system. However, in general, the most current CADesign systems are designed for a "new fresh blank screen," because programmers who design CADesign systems do not use them for long periods of time in production oriented environments. What is: digital material? digital culture? and digital arts? "What does it mean to be digital" is key to being able to successfully apply digital arts to the design process.

5.1 User Coordinate System (3)

The User Coordinate System is where creation/editing is done. It can be viewed as a local coordinate system created for the ease of editing objects in space. It is the space you are viewing through "Cartesian's windscreen." In single object models, the user coordinate will be the same as the World Coordinate System, but in large models, almost never the same. The proper normal position for the User Coordinate System during the creation of an entity using a "god orientation" is the following: the X axis origin starting point is placed at the lower left front, the positive direction of the X axis is to the lower right front and the Y axis positive direction is down the right side of the entity and Z is up. This is extremely important, because the User Coordinate System position during the creation of an entity becomes the Entity Coordinate System.

5.2 Moving Objects

The following is the movement of objects in relationship to the World Coordinate System. The movement along the X, Y, and Z axis is repeated by two key commands entered at a command line prompt. Motion studies show that using the command line prompt with one or two keys representing the command followed by the space bar as an enter key is the quickest form of entry possible at the present time. However, a virtual reality interface with the use of first person tools may represent even a faster from of command and control for modeling systems of the future, a far more intuitive interface.

Move Right (MR) is +X

Move Left (ML) is -X

Move Down (MD) is -Z

Move Forward (MF) is +Y

Mover Back (MB) is -Y

Move Up (MU) is +Z

Commands to move an object are the following: Move the block to the Right (MR) moves in the positive direction of X; Move Forward (MF) moves in the positive direction of Y; Move Left (ML) moves in the negative X; Move Back (MB) moves in the negative Y; Move Upward (MU) moves in the positive Z; Move Down (MD) moves in the negative Z. Please note that the move commands above are from the current position and rotation of the UCS. The move command is used very often; it is more convenient to think in right, left, forward, back, up and down. Note the movement is based on the user's orientation or "god orientation" not the entity's orientation or "actors orientation." The Move Right command in the command line mode looks like this at the user prompt:
MR in the <P>lus / Neg <X> /Y/Z axis by the EditNumber (EN) / <9.999999999> or select:
Objects <P>revious/1/2/3...or select:

5.3 Positive Rotation
In CAD positive rotations are counter-clockwise around a vector if the positive direction of the vector is pointing toward you. In CAD a line has a start point and an end point, i.e. direction or vector and the positive rotation around the vector follows the conventions of the right hand rule. To help you find the positive rotation of a line using your right hand, follow the directions below:

* Open your right hand
* Stick out your right thumb
* Aim your right thumb in direction of the end point of the line
* Curl the fingers of your right hand around the line

The direction of the curl of your right hand fingers around the line is in the positive rotation. The uncurling of your right hand fingers is in the negative rotation.

5.4 Rotation of Objects

+ Rotation X axis (RX)

+ Rotation Y axis (RY)

+ Rotation Z axis (RZ)
The rotate commands by default rotate in the positive direction or Counter Clockwise (CCW) direction. The negative direction is a Clock Wise (CW) direction. The Rotate command rotates objects; it does not rotate the view or the Cartesian Coordinate System. The rotate commands are as follows: rotate the block around the X axis is (RX); rotate around the Y axis is (RY); rotate around the Z axis is (RZ). Please note that the rotate commands above are from the current UCS position. The images above are from the WCS, but could just as easily have been from some User Coordinate System or an entity coordinate position. The single key command (R)otate provides the user with rotate mode options that become defaults for the RX, RY, RZ commands. Rotate is not used as much as the move command. The Rotate command in the command line mode looks like this at the user prompt:
RZ = X/Y/<Z> axis Neg/<P>lus by Entity/World/Current <C> by <90>:
Objects <P>revious /1/2/3... or select:

5.5 Entity Coordinate System (4)

The Entity Coordinate System is really an entity creation coordinate system with a "god orientation," which is the position of the User Coordinate System when the entity was created. The "god orientation" being the orientation of the user / creator's right, left, forward, back, up and down. Entity Coordinate System is not really properly named, because it is really just the position of the User Coordinate System given to the entity for the convenience of editing in CADesign systems and is really just a way to move back into a convenient modeling / creation position. In animation systems, the entity / actor has its own "actor orientation" system which is moved from the lower left hand corner of the entity to the center of the actor, and the Entity Coordinate System is rotated 180 degrees from the "god orientation" to the "actor orientation." In animation systems, the user / director directs the movement of an entity / "actor orientation" using the entity / "actor orientation" rather than the "god orientation."  This feature is very convenient for animation.  Animation systems also have creation systems, in which unfortunately they continue to use the "actor orientation" which is not easy to use for the creation of an entity. The use of both orientations, the god for creation and the actor for object movements in one system, turns out to work very well. You might notice for this chapter that the move command uses the "god orientation" to avoid confusion of mixing systems in theory. However, in real applications the mixed use of both the god and actor orientations is quite natural and actually avoids confusion. Depending on the CAD system chosen, you may or may not have access to the Entity Coordinate System. Most people are not aware of this system or its different orientations. However it is a very powerful system and will be one of the core systems necessary for Cartesian's navigation and visualization systems.
Alice (http://www.alice/org) is a 3D Interactive Graphics Programming Environment for Windows 95/98/NT built by the Stage 3 Research Group at Carnegie Mellon University. The Alice project is a public service to the wider computing and artistic communities. The current version of Alice authoring tool is free to everyone and runs on computers that are commonly available for reasonable prices.  Worlds created in Alice can be viewed and interacted with inside of a standard web browser once the Alice plug-in has been installed.  Alice is primarily a scripting and prototyping environment for 3D object behavior, not a 3D modeler;  this makes Alice much more like LOGO (http://el.media.mit.edu/logo-foundation/logo/turtle.html) than AutoCAD.  By writing simple scripts, Alice users can control object appearance and behavior, and while the scripts are executing, objects respond to user input via mouse and keyboard. 

You can create 3D models using a web browser with Teddy2.  Alice runs Teddy2, a sketched based 3D modeling tool developed by Takeo Igarashi at the University of Tokyo.  During the creation process, the "god orientation" is used. However, at the end of the process when you save the model, Teddy2 has you orient the front of the object to face you.  Saving the model in that position gives the object the "actor orientation" shown in the image at the left.  So objects in Alice have their own sense of direction. This makes the control of the object very easy. Try loading Alice and using their tutorial to create a 3D object by sketching.

5.6 Group Coordinate System (5)

Every group or block of objects has an insert point or base point assigned to it by which to insert, move or rotate it as a group. This coordinate system is an essential system. Without this system, one can not insert a model or part of a model into the model being worked on. It is the system that turns all of the rest of the models into a library of parts or blocks that can be used repeatedly. It is a digital extension of yourself that you can share digitally with others and giving you value in the digital world. Digital modeling is part of the growing landscape of digital containing digital swarms of information ... to be used from any location on the Net.

6.0 Exercises

Knowledge provided in this CAD chapter is a summery of twenty years of experience in computer aided design. Much of the information contained here is not found in any other educational texts of which I am aware. The goal of this chapter is to transfer the knowledge needed to set up a digital framework for doing the exercises below and teaching yourself a CAD system of your choice. It would be a waste of my time and yours to present knowledge, information, and exercises that are common place and can be found on the WWW easily. Instead I urge you to access that easily found information yourself using the WWW. Learning the line, circle command in one particular CAD system is best done by yourself using that particular system's online help and tutorial. What is missing from the online help systems is why and how to apply digital systems to 3D design. I have found only one such book, AutoCAD 3D Design and Presentation, M. Bousquet and J. Hester, New Riders Publishing, Carmel, Indiana, USA (1991) ISBN 0-934035-81-4. Though it does not have an educational focus or an overall approach, it does give step by step procedures for AutoCAD as a 3D design system. Follow the exercises below and you will be well on your way to CAD mastery.

Exercise 1

  1. Take out a piece of paper and pencil.
  2. On one half of a sheet of paper, draw the Cartesian Coordinate System's X, Y, Z axis from the - - + view (V1).
  3. Divide the axis in the positive direction into 10 equal segment starting from the origin point.
  4. Label the drawing with the first principle of Cartesian existence.
Exercise 2
  1. Using the axis drawing from Exercise 1 - construct a 4,5,6 unit size box on the axis starting at 0,0  * Tip: Graph out the following points on the axis drawing and then connect the lines:  0,0,0  4,0,0   4,5,0   0,5,0   0,0,6  4,0,6   4,5,6   0,5,6 
  2. Label the eight corners of the 4,5,6 box
  3. Relative from point 4,5,6 - place a point at 4,4,4
Exercise 3
  1. Install a CADesign program of your choice on a system of your choice. 
  2. Create a 3D model of the paper drawing done in Exercises 1 and 2.
  3. Reflect on and evaluate the CADesign software program.
Exercise 4
  1. Load Alice and use Teddy 2 to sketch a soft object / toy.  http://www.alice.org/
  2. Load FreeCAD - use the tutorial and evaluate the addition of reverse kinematics to CAD.  http://www.askoh.com/
  3. Load HyperFun - use the HyperFun tutorial and evaluate the concept of F-rep.  http://www.hyperfun.org/
Exercise 5
  1. Get a package of Lego blocks and calipers with LED read out and make precise 3D models of them with a CADesign of your choice.


Computer Aided Design's eXtended Dimensions

by Carl Vilbrandt