Visualizing 3D

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Visualizing 3D

• Between Measurement and Illusion
• Dan Collins
• VizProto

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• Visualization is.
• Visualization is a method of computing. It
  transforms the symbolic into the geometric,
  enabling researchers to observe their simulations
  and computations. Visualization offers a method
  for seeing the unseen. It enriches the process of
  scientific discovery and fosters profound and
  unexpected insights. In many fields it is already
  revolutionizing the way scientists do science.
• SIGGRAPH proceedings, 1987. B. McCormick, T.
  DeFanti, and M. Brown MCC87

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Euclid300 B.C.

• Euclid's "Elements," written about 300 B.C., a
  comprehensive treatise on geometry, proportions,
  and the theory of numbers, is the most long-lived
  of all mathematical works. This manuscript
  preserves an early version of the text. Shown
  here is Book I Proposition 47, the Pythagorean
  Theorem the square on the hypotenuse of a right
  triangle is equal to the sum of the squares on
  the sides. This is a famous and important theorem
  that receives many notes in the manuscript.

• Pythagorean Theorem

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Euclids description of the Pythagorean theorem
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Euclid300 B.C.

• Euclid also wrote Optica, the first text on
  geometrical optics, in which he defines the terms
  visual ray and visual cone.
• He noted that light travels in straight lines and
  described the law of reflection. He believed that
  vision involves rays going from the eyes to the
  object seen and he studied the relationship
  between the apparent sizes of objects and the
  angles in which they meet at the eye.

Raphael, The School of Athens, 1509, Fresco,
Vatican, Rome
Detail showing Euclid with his students
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Pythagorus580-520 B.C.

• Pythagorus was a mathematician who made
  important contributions to geometry. "He was a
  Greek philosopher and religious leader who was
  responsible for important developments in the
  areas of mathematics, astronomy, and music
  theory. He was also a healer, a wrestler, and was
  politically active. He founded a philosophical
  and religious school which has come to be known
  as the Pythagorean Society.
• The Pythagoreans saw that many things in the
  universe were related in ways that could be
  stated in numbers. They reasoned that numbers
  must be the 'stuff' philosophers were looking
  for. The universe including man is a closed
  system. Both can be understood by the relation of
  the parts. These relations can be expressed in
  terms of numbers. These ideas led them to believe
  that if one could penetrate the secrets of
  numbers, he would penetrate the secrets of the
  universe and the destiny of man. This led to the
  careful study of geometry, the highest form of
  mathematics.

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PantheonRome, Italy, 118 to 126 ADArchitect
unknown
Exterior view of the Pantheon in modern day Rome
Interior view of the Pantheon Giovanni Paolo
Panini, c. 1750
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The Pantheon and the Neo-Pythagoreans

• The Roman pantheon can be considered an
  architectural image of the Greek Pythagorean
  cosmos, a "living organism" with a
  mathematically-proportioning "soul" and
  unchanging, "eternal" consonant-symphonic ratios.
  To generate harmony, the laws of arithmetic,
  geometry, astronomy and musical-proportions are
  fused. It "resembles the heavens", but is a
  resemblance based on mathematical knowledge, a
  summary of the ancient quadrivium.
• --Girt Sperling
• The quadrivium was the higher division of the
  seven liberal arts in the Middle Ages, composed
  of geometry, astronomy, arithmetic, and music.

Section showing pythagorean ratios at work in the
Pantheon.
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Before Perspective

• Perspectival errors appear in paintings usually
  done before 1400.
• The perspective lines usually converge, but not
  to a single point and not on the horizon.
• Initial word panel of Psalm from the Kaufmann
  Haggadah. Spain, late 14th C.

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Brunelleschi1377-1446

• Brunelleschi designed the stupendous dome which
  crowns the cathedral in Florence, a work which
  occupied him intermittently from 1417 to 1434.
  The technical difficulties involved in erecting
  the new dome underscore an important aspect of
  his talents he was a daring innovator, with a
  solid knowledge of math and mechanics.

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Brunelleschi

• He developed many important construction methods
  as well as contributing to the evolution of
  perspective. His mathematical work led to the
  invention of linear perspective.

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Brunelleschi

• Filippo Brunelleschi was the first to carry out a
  series of optical experiments that led to a
  mathematical theory of perspective.. Brunelleschi
  used his training as a gold smith to apply a
  silver background on a painted panel, allowing
  the color of the sky and passing clouds to become
  part of the painting as seen by the viewer. This
  was an attempt at a perspective painting and
  interactive art. The panel was constructed with a
  hole at the vanishing point. The reflection of
  the image was viewed in a mirror through the
  hole, giving an illusion of depth.
• http//library.thinkquest.org/3257/illusion.htmlp
  eep

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Brunelleschi devised a method of perspective for
architectural purposes he is said by Manetti to
have made a ground plan for the Church of Santo
Spirito in Florence on the basis of which he
produced a perspective drawing to show his
clients how it would look after it was built.
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Masaccio1401-1428

• Masaccio's Trinity, 1427-28 Santa Maria Novella,
  Florence (6.67 x 3.17 m) is often used to
  illustrate the early culmination of mathematical
  perspective experiments.

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Alberti1404 - 1472

• ALBERTI'S WINDOW
• The traditional form of pictorial representation
  using perspective methods developed by
  Renaissance artists is sometimes referred to as
  Alberti's Window.
• This is because, in his treatise Della pittura,
  On Painting, 1435-6, the Classical theorist and
  painter Leon Battista Alberti noted that, when he
  set out to paint a scene on a panel, he assumed
  the picture would represent the visible world as
  if he were looking through a window. Some artists
  did, in fact, create grids across the opening of
  a window and transfer the scene to a gridded
  canvas as compelling evidence that western
  perspective was a natural form of representation.

Albertis fenestre (window) or velo
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Alberti

• Alberti's Construction System
• 1. B-one braccio module (one third of the height
  of a man). The base of the picture is divided
  into braccia. The height of the man at the front
  plane of the picture gives the level of the
  horizon, H.
• 2. The braccio divisions are joined to the
  perspective focus, V, to give the orthogonals.
• 3. In side elevation, lines are drawn from
  braccio divisions behind the picture plane P to
  the eye at E. The points of intersection on P are
  noted.
• 4.The levels of the points of intersection are
  marked at the side of the picture plane, and
  locate the horizontal divisions of the tiles. Z
  is the 'distance' point, though Alberti only
  mentions using one diagonal to check the
  construction.

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Paolo Uccello

• Among the best examples of early uses of linear
  perspective is Paolo Uccello's fresco of the
  "Deluge" in Florence, completed about 1448. Here
  linear perspective is used to present an
  elaborate architectural setting. The real object
  of fascination, however, is Uccello's rendering
  of the mazzocchi, the curious checkered hats, of
  which there are two in "The Deluge. Ucello had
  actually drawn such wonderful polyhedral forms in
  studies of perspective drawings, and these
  clearly demonstrate the mastery he had of the new
  mathematical techniques.

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Piero della Francescac.1420 - 1492

• The culmination of the mathematical theory of
  perspective with a philosophical program of the
  most intense and religious order comes with the
  work of Piero della Francesca. His St. Anthony's
  Polyptich, in Perrugia, shows how masterfully he
  was able to use the new theory of perspective.

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Leonardo1452 - 1519

• Perspective is nothing else than the seeing of an
  object through a sheet of glass, on the surface
  of which may be marked all the things that are
  behind the glass --Leonardo da Vinci
• Leonardo studied optics from both the scienitific
  and the artistic points of view. He believed that
  painting should be considered a Liberal Art
  because it was based on mathematically derived
  perspective theory and satisfied the primary
  sense of sight. Da Vinci realized that unless a
  person viewed a painting through a peephole, the
  visual image would be different than the image
  the artist painted.

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Dürer1471-1528

• One of several machines invented by Dürer for
  making perspectival drawings consisted of a
  needle driven into the wall and a piece of string
  and a hinged frame. The piece of string has a
  pin on one end and a weight on the other between
  the eye of the needle and the object is placed a
  wooden frame within which every point can be
  determined by two movable threads crossing each
  other at right angles. When the pin is put on a
  certain point of the object the place where the
  string passes through the frame determines the
  location of that point within the future picture.
  This point is fixed by adjusting the two movable
  threads and is at once entered upon a piece of
  paper hinged to the frame and by a repetition of
  this process the whole object may be transferred
  gradually to the drawing sheet.

A woodcut from Albrecht Dürer's treatise on
measurement Underweysung der Messung, 1527
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Dürer1471-1528

• The Perspectograph is an instrument that allows
  the user to obtain, point by point, a correct
  perspective drawing of a three dimensional
  object. Perspectographs were used by painters and
  sceno-graphers in 16th and 17th cent. (and as
  early as the 15th cent. by Alberti). Some types
  of Perpespectograph are very simple (as these
  reproduced in Dürer's xylographies), some types
  are rather complex. In this model of Dürer's
  perspectograph, an observer looking through the
  ocular sees the pattern drawn on the vertical
  table exactly superimposed on the pattern drawn
  on the horizontal table.

Dürer's Perspectograph, early 16th c. (replica)
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Palladio1508-1580

• The art historian Rudolph Wittkower writes, "The
  conviction that architecture is a science, and
  that each part of a building, inside as well as
  outside, has to be integrated into one and the
  same system of mathematical ratios, may be called
  the basic axiom of Renaissance architects." Many
  modern authors have analyzed Wittkower's thesis
  that harmonic proportions derived from musical
  scales played a central role in the minds and
  designs of Renaissance theorists and architects.
  Central to this debate is Palladio's oeuvre--his
  architecture and his Quattro libri (four books).
• --Stephen R. Wassell

Elevation and plan of a typical Palladian villa.

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vitruvius c. 90-20 B.C.E.

• Images by Cesare Cesariano (1521)
• This is a profusely illustrated edition of the
  most famous of antique texts on architecture, The
  Ten Books on Architecture. It was known
  throughout the Middle Ages, in multiple copies
  and probably versions.

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Hans Holbein, The Ambassadors, 1536
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(No Transcript)
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Francesco Borromini 1599 - 1667
This architectural tromp l'oeil of an actual
"perspective" collonade in the Palazzo Spada, was
fashioned by Galileo's contemporary, Borromini in
1653. This is actually an illusion, played with
the help of mathematical perspective. The trick
is revealed in the image at the right where two
figures of equal height show the perspective at
work. The image in the center is a modern CAD
rendering.
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Diderot 1713 - 1784

• Denis Diderot was the creator of the first
  Encyclopedia in 1751. More then 160 authors
  contributed to the encyclopedia. By 1789 there
  were nearly 16,000 copies sold. The pope placed
  the encyclopedia on the Index of Prohibited
  books.
• In his discussions on art, he provides the
  broader social context for the arts. In his
  entries on Art, he describes the origin of the
  sciences and arts, their distribution into
  liberal and mechanical arts, the goal of the
  arts, and his own project for a general treatise
  on the mechanical arts.
• We began by making observations on the nature,
  service, usage, qualities of beings of their
  symbols then we gave the name of science or of
  art or of discipline in general, to the center or
  unifying point to which we related the
  observations that we had made, to form a system
  of either rules or instruments, of rules
  tending towards the same goal because that is
  what a discipline is in general. (ART, in Diderot
  d'Alembert, 1751-1772, Vol. 1, p. 713)

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Contour plot . Map of Paris by L. L. Vauthier
(1874), showing population density by contour
lines, the first statistical use of a contour
map. This approach to representing multivariate
data arose from the use of contour maps in
physical geography showing surface elevation
(first published in 1752 by Buache), which became
common in the early 19th century. It was not
until 1843, however, that this idea was applied
to data, when Léon Lalanne constructed the first
contour plot, showing the mean temperature, by
hour of the day and by month at Halle (lower
left). Lalanne's data formed a regularly-spaced
grid, and it was fairly easy to determine the
isolines of constant temperature. Vauthier
generalized the idea to three-way data with
arbitrary (x,y) values in his map of the
population density of Paris. http//www.math.york
u.ca/SCS/Gallery/ noframes.html
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This figure (showing the population of Sweden
from 1750-1875 by age groups) by Luigi Perozzo,
from the Annali di Statistica, 1880, is a very
early example of a 3D stereogram. Perozzo's
figure is also notable for being printed in color
in a statistics journal, and in a way which
enhances the perception of depth.
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Etienne-Jules Marey1830- 1906
Etienne-Jules Marey, 1830-1906, was among the
pioneers of dynamic graphics and the graphical
representation of movement and dynamic phenomena.
This image, from Marey's La méthode graphique
dans les sciences experimentales (1876, p. 150)
compares the time course of respiration of a
person at rest and under exertion, using a
pen-recording device to plot the traces over
time.
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Mapping the London Underground

• Harry Beck's 1933 diagram of the 7 lines of the
  London Underground, although geographically
  inaccurate, provides a coherent overview of a
  complex system. (See map at upper left).
• With excellent color printing, classic British
  railroad typography (by Edward Johnson), and, in
  the modern style, only horizontal, vertical, and
  45 degree lines, the map became a beautiful
  organizing image of London.
• For apparently quite a number of people, the map
  organized London (rather than London organizing
  the map). Despite 70 years of revision due to
  extensions of the Underground and bureaucratic
  tinkering (the marketing department wrecked the
  map for several years), the map nicely survives
  to this day.
• Compare map from late 1920s at lower left.

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The History of CAD

• 25 years ago, nearly every drawing produced in
  the world was done with pencil or ink on paper.
  Minor changes meant erasing and redrawing while
  major changes often meant recreating the drawing
  from the scratch. If a change to one drawing
  affected other documents you were dependent upon
  having someone manually recognize the need to
  make the changes to the other drawings and to do
  so.
• CAD has fundamentally changed design and the
  way we visualize 3D.

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Dynamic Visualization

• Visualization of Storm patterns combines 3D
  graphics and actual metrics

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Visualizing the Web Gunilla Elam,Warriors of
the Net, 1999

• Elams background is in fine arts and she also
  did research into the social aspects of computing
  and networking technologies at the Ericsson
  Medialab and now works as a designer at a startup
  venture called AirClic. Of the many challenges in
  making Warriors of the Net, Elam says that, The
  hardest part was without question to simplify the
  structure into an understandable, easy to grasp
  concept. I had not been going into the tech part
  of the Internet much before starting with this,
  so the way we did it was Tomas filling me up with
  as much information I could handle, then let me
  think about it for a while and melt it down to a
  level where anyone would be able to understand
  it.

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Decision Theater at ASU

• East Valley Water Forum (EVWF)
• The EVWF is a regional cooperative of water
  providers who are working with Arizona Department
  of Water Resources (ADWR) with support from the
  Bureau of Reclamation to develop data driven
  scenarios about ground water policy issues under
  a variety of drought scenarios. Their work with
  the Decision Theater will assist them in
  developing informed planning decisions as the
  east portion of the Salt River Valley continues
  its explosive growth. Key collaborators K.
  Sorenson (City of Mesa) and D. Mason (ADWR).

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Decision Theater at ASU

• Urban Heat Island (UHI)
• The UHI explores and models heat retention in the
  Phoenix metropolitan area. The effect of UHI
  during Arizona summers has been a 12 degree rise
  in night time low temperatures in the last 20
  years. Scientists have developed predictive
  models based on dynamic changes in land use that
  can help planners and decision makers better
  understand the UHI phenomenon. The goals are to
  understand probable impacts of UHI on planning
  urban systems (such as electrical capacity to
  accommodate increased power use for air
  conditioning) and to explore the effectiveness
  and impact of potential solutions for mitigation.

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Decision Theater at ASU

• Environmental Fluid Dynamics Program
• Typically, computational fluid dynamics models of
  atmospheric events are presented as numeric data
  or 2 dimensional graphics. Data from a Defense
  Threat Reduction Agency (DTRA) funded project to
  simulate anthrax release in Oklahoma City has
  been modeled and visualized as a 3D animation.
  This work provides a foundation for developing
  interactive scenarios to study the effects of
  wind direction, wind speed, and building design
  on dissemination of bacterial agents. The
  research permits informed training of emergency
  response teams to real natural or man made
  emergencies.

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The History of CAD (pre-1970)

• The first graphic system was in mid 1950 the US
  Air Force's SAGE (Semi Automatic Ground
  Environment) air defense system. The system was
  developed at MITs Lincoln Laboratory. The system
  involved the use of CTR displays to show
  computer-processed radar data and other
  information.
• In 1960, Ivan Sutherland used TX-2 computer
  produced at MIT's Lincoln Laboratory to produce a
  project called SKETCHPAD, which is considered the
  first step to CAD industry.
• In 1960 McDonnell Douglas Automation Company
  (McAuto) founded. It will play a major role on
  CAD developments.
• The first Computer-Aided Design programs used
  simple algorithms to display patterns of lines at
  first in two dimensions, and then in 3-D. Early
  work in this direction had been produced by Prof.
  Charles Eastman at Carnegie-Mellon University,
  the Building Description System is a library of
  several hundred thousands architectural elements,
  which can be assembled and drawn on screen into a
  complete design concept.
• In mid 1960 large computers characterized the
  period, vector display terminals and software
  development done in assembly language. The only
  significant attempt to create a commercially CAD
  system was Control Data Corporation's
  Digigraphics division, a successor to the
  previously mentioned ITEK. The system costs half
  million dollars and were sold in few units.
• In 1968 Donald Welbourn had the vision to see the
  possibility of using computers to assist pattern
  makers to solve the problems of modelling
  difficult 3D shapes. Today we take for granted 3D
  modelling, in 1968 only crude 2D drawing systems
  were available using terminals linked to large
  main frame computers.
• David Evans and Ivan Sutherland founded in 1968
  Evans and Sutherland.
• In 1969 were founding Computervision and Applicon
  companies. Computervision was created to produce
  systems for production drafting and in the same
  year it sold the first commercial CAD system to
  Xerox.

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The History of CAD (1970-1980)

• At the end of 70s a typical CAD system was a
  16-bit minicomputer with maximum of 512 Kb memory
  and 20 to 300 Mb disk storage at a price of
  125,000 USD.

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The History of CAD (1980-1990)

• 1981 Computer graphics from Cornell University
  founded 3D/Eye Inc., a pioneered 3D and graphics
  technology. Unigraphics introduced the first
  solid modeling system, UniSolid. It was based on
  PADL-2, and was sold as a stand-alone product to
  Unigraphics.
• 1982 CATIA Version 1 is announced as an add-on
  product for 3D design, surface modeling and NC
  programming. Mini computers with much more power
  at less cost started to appear. This was a major
  step forward and by 1984 the technology began to
  be competitive with traditional methods. For many
  years aircraft had of course been designed using
  computers, but now it was becoming possible to
  economically design saucepans and other domestic
  products with complex 3D shapes using a computer.
  Autodesk was founded by sixteen people in April
  1982 in California by initiative of John Walker
  in idea to create a CAD program for a price of
  1000 to can run on PC. John Walker has been
  running Marinchip Systems for two years before.
  In November at COMDEX trade show in Las Vegas was
  demonstrated the first CAD program in the world
  that runs on PC. This was the initial release of
  AutoCAD and deliveries begun in December.
• 1983 Unigraphics II introduced to market
• 1984, a Hungarian physicist, Gabor Bajor,
  smuggled two Macs into his country. At the time,
  ownership of personal computers was illegal under
  Communist rule. Using Pascal, he and a teenager,
  Tamas Hajas worked to write a 3D CAD program for
  the Mac which will be the beginning of Graphsoft
  Company. Drafting capabilities are added to CATIA
  in 1984, enabling it to function independently of
  CADAM. The first Autodesk Training Centre. In
  October AutoCAD version 2 (Release 5) with text
  improvements, DXFIN and DXFOUT commands, new
  Inquire commands, Object Snap, named views,
  Isometric capabilities and new Attribute
  features.
• 1988 Surfware Inc., ships the first version of
  SurfCAM, a CAD/CAM program.
• 1989 Parametric Technology ships the first
  version of Pro/ENGINEER.

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The History of CAD (1990-1995)

• 1990 McDonnell Douglas (now Boeing) chooses
  Unigraphics as the corporate standard for
  mechanical CAD/CAM/CAE. Autodesk ships Animator
  Pro, a 2D painting and animation program for DOS.
  By 1993 over 15,000 copies have been sold
  worldwide.
• 1991 Microsoft developed Open GL for use with
  Windows NT. Open GL is an API procedural software
  interface for producing 3D graphics and includes
  approximate 120 commands to draw various
  primitives such as points, lines, and polygons.
  Also includes support for shading, texture
  mapping, anti-aliasing, lighting and animation,
  atmospheric effects such as fogging and
  simulation of depth-of-field. Open GL, developed
  by Silicon Graphics, is a standard for the 3D
  color graphics programming and rendering.
• 1992 Autodesk ships 3D Studio version 2 for DOS.
  Autodesk ships AutoCAD Release 12 for DOS in
  June. Includes AutoCAD SQL Extension
  (ASE)/Autodesk SQL Interface (ASI) that lets you
  establish links between AutoCAD and an SQL
  database. Advanced Modeling Extension (AME)
  release 2.1 is supported by Release 12, with
  region modeling and new solid primitives. AutoCAD
  Render is included with AutoCAD.
• 1993 The first AutoCAD (Release 12) for Windows
  platforms. It required 8 MB RAM and 34 MB Hard
  Drive space for complete installation. The
  Windows version of AutoCAD includes 36 icons
  toolbox, allows multiple AutoCAD sessions,
  separate Render window, support for Windows GUI,
  DDE and OLE, as well as Drag-and-Drop and Bird's
  Eye view capabilities. The AutoCAD main menu has
  been eliminated After initial configuration,
  AutoCAD displays the graphics screen. AutoCAD 12
  for Windows was one of the most successful CAD
  programs ever
• 1994 MiniCAD version 5. Hewlett Packard ships
  version 3.5 of PE/Solid Designer, its high end
  Solid Modeling.
• 50,000 seats installed to date.
• 1995 CATIA-CADAM AEC Plant Solutions are
  announced. This next generation object-oriented
  plant modeling system enables powerful
  knowledge-based engineering capabilities that can
  dramatically streamline the process of plant
  design, construction and operation. It brings the
  power of "smart" applications to the desktop with
  next generation object-oriented modeling. IDEAS
  Master Series version 2.1 from SDRC. Mazda Motors
  Corp. will install 2,400 seats of this product.
  Parametric Technology ships Pro/E version 15, the
  first parametric modeling CAD/CAM program and the
  first high-end 3D solid modeling package
  available on NT platforms.

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The History of CAD (1996-99)

• 1996 Solid Edge version 3 from Intergraph hits
  the market at the price of around USD 6000.
  SolidWorks Co. ships Solid Works, an ambitious 3D
  package based on Parasolids modeling Kernel. It
  comes with a good complex surface modeling and a
  good graphical user interface. 3D/EYE Inc., ships
  Tri Spectives Technical version 2, a modeling,
  illustration and animation program for Windows
  platforms, at a very low price. Lightscape
  version 3, a high-end rendering and animation
  package, comes with IES photo-metric data
  capabilities. IES (Illuminating Engineers
  Society) is the industry standard for describing
  the shape and intensity of light energy
  distribution froma light source, ray tracing,
  natural light according to location and
  orientation of the building. Lightwave 3D version
  5 and 5.5 from New Tek, a high-end rendering,
  modeling and animation program. AutoCAD LT 95.
  Diehl Graphsoft released MiniCAD 6 for Windows,
  the first cross-platform version of MiniCAD.
  Pro/E version 17 with a new module which allows
  files to be exported into VRML file format for
  display on the Internet.
• 1997 Autodesk ships 3D Studio MAX release 2 and
  a cut-down version called 3D Studio Viz. EDS
  introduces a number of new industry-leading
  capabilities with its new version of Unigraphics,
  including WAVE - which will enable the
  definition, control and evaluation of product
  templates - considered the most important new
  technology affecting the CAD/CAM/CAE industry in
  the next five years. First version of IDEAS
  Artisan Series from SDRC, fully compatible with
  Master Series, priced at USD 5,000. Form Z, a
  solid and surface modeler, first available only
  for Mac platforms, debuts on Windows market.
• 1998 Autodesk Architectural Desktop - integrated
  architectural solution based on AutoCAD 14. First
  version of IronCAD for VDS market. Autodesk ships
  3D Studio MAX version 2.5 Lightwave 3D version
  5.6 from New Tek, comes with Procedural shades
  for snow, water and rust, Stereoscopic rendering,
  SkiTracer image warping for real time
  visualization of generated sky, and more. Solid
  Edge version 3 from Intergraph with more than 150
  new features. Solid Works 98 adds 150 new
  capabilities.
• 1999 CATIA Version 5 for native Windows NT and
  UNIX. Lightwave 3D version 6 from New Tek. Think3
  entry in the CAD market with thinkdesign, the
  first mechanical design software product to offer
  the power of parametric solids, advanced
  surfacing, wireframe and 2-D drafting, all in one
  environment. VectorWorks replaces MiniCAD. 3D
  Studio MAX cumulus 29 of the entire 3D-animation
  market and 38 of the 3D PC markets.

43
From Euclid to Desargues

• Euclid's Optica, c. 300 B.C., was the first text
  on geometrical optics, in which are defined the
  terms visual ray and visual cone.
• Vitruvius' Ten Books on Architecture which
  appeared about 25 B.C., was the only book on
  architecture to survive from antiquity. It
  profoundly influenced Renaissance architecture
  and thinking, including that of Alberti, who
  quoted Vitruvius in his Della pittura. Vitruvius
  wrote Perspective is the method of sketching a
  front with the sides withdrawing into the
  background, the lines all meeting in the center
  of a circle. Unfortunately he didn't elaborate on
  that. Elsehere, Vitruvius' reference to Greek and
  Roman stage design, implied an understanding of
  the vanishing point.
• Ptolemy's Optica, c. 140 A.D., was another early
  text on geometrical optics, and included theories
  on refraction. The centric ray is defined by
  Ptolemy as the ray that does not get refracted.
  The centric ray, we'll see, is important in the
  theory of perspective. In his Geographia, c. 140
  A.D., Ptolemy applies the principles of geometric
  optics to the projection of the spherical surface
  of the earth onto a flat surface, to produce a
  map. He is said to have made the first known
  linear perspective construction for drawing a map
  of the world. Ptolemy apparently knew about
  perspective, but applied it only to maps and to
  stage designs.
• Galen's De usu partium, c. 175 A.D., contains an
  early but erroneous description of how the eye
  creates images. The book was still important,
  however, as a stepping stone in the development
  of the theory of perspective.
• From Islam, Alhazen's Perspectiva, c. 1000 A.D.,
  was an important compendium on optics. It
  integrated the works of Euclid, Ptolemy, and
  Galen.
• Roger Bacon's Opus Majus, c. 1260 A.D., included
  a section on optics, whose geometric laws, he
  maintained, reflected God's manner of spreading
  His grace throughout the universe.
• John Pecham's Perspectiva communis, c. 1270 A.D.,
  was another treatise on optics that was widely
  available during the Renaissance.
• Blasius of Parma's Quaestiones perspectivae, c.
  1390 A.D., was a popular adaptation of the works
  of Bacon and Pecham.

44
From Euclid to Desargues

• We are all familiar with Euclidean geometry and
  with the fact that it describes our
  three-dimensional world so well. In Euclidean
  geometry, the sides of objects have lengths,
  intersecting lines determine angles between them,
  and two lines are said to be parallel if they lie
  in the same plane and never meet. Moreover, these
  properties do not change when the Euclidean
  transformations (translation and rotation) are
  applied. Since Euclidean geometry describes our
  world so well, it is at first tempting to think
  that it is the only type of geometry. (Indeed,
  the word geometry means measurement of the
  earth.) However, when we consider the imaging
  process of a camera, it becomes clear that
  Euclidean geometry is insufficient Lengths and
  angles are no longer preserved, and parallel
  lines may intersect.
• Perspective is an example of the geometric
  operation of projection and section where
  projection lines from the outline of an object to
  the eye are sectioned or cut by a picture plane.
  This has roots in the conic sections, where
  projection lines from a circle to a point form a
  cone, which is then sectioned by a plane to give
  a circle, ellipse, parabola, or hyperbola,
  depending on the angle of the cutting plane.
  These ideas were expanded by Gerard Desargues
  (1593-1662), architect/engineer, into the branch
  of mathematics called projective geometry.
• Projective geometry is a branch of mathematics
  that deals with the relationships between
  geometric figures and the images, or mappings, of
  them that result from projection. Common examples
  of projections are the shadows cast by opaque
  objects, motion pictures, and maps of the Earth's
  surface.

Projection of one line onto another
Central projection of one plane on another
45
Frank Lloyd Wright

• Wright used nature as the basis of his
  geometrical abstraction. His objective was to
  conventionalize the geometry which he found in
  Nature, and his method was to adopt the abstract
  simplification which he found so well expressed
  in the Japanese print. Therefore, it is not too
  shocking perhaps that in this quest his work
  should foreshadow the new mathematics of nature
  first put forth by Benoit Mandelbrot fractal
  geometry.
• --Leonard K. Eaton

Floor plan from a late Wright residence.
46
Platonic Solids

• The so-called Platonic Solids are regular
  polyhedra. Polyhedra is a Greek word meaning
  many faces. There are five of these, and they
  are characterized by the fact that each face is a
  regular polygon, that is, a straight-sided figure
  with equal sides and equal angles.
• The Greeks, who were inclined to see
  mathematics as something of a religious truth,
  found this business of there being exactly five
  Platonic solids very compelling. The philosopher
  Plato concluded that they must be the fundamental
  building blocks the atoms of nature, and
  assigned to them what he believed to be the
  essential elements of the universe. He followed
  the earlier philosopher Empedocles in assigning
  fire to the tetrahedron, earth to the cube, air
  to the octahedron, and water to the icosahedron.
  To the dodecahedron Plato assigned the element
  cosmos, reasoning that, since it was so different
  from the others in virtue of its pentagonal
  faces, it must be what the stars and planets are
  made of.

47
references

• General Science and Art http//library.thinkques
  t.org/3257/
• Digital Design Media http//www.gsd.harvard.edu/
  malcolm/DDM/GALLERY/15.01_1956.gif
• Durer http//www.newcastle.edu.au/department/fad
  /fi/woodrow/durer-c.htm
• General Information on Perspective
  http//www.newcastle.edu.au/department/fad/fi/wood
  row/an-persp.htm
• Leonardo http//www.mos.org/sln/Leonardo/Leonardo
  sPerspective.html
• Alberti http//www.leonet.it/culture/nexus/98/Pa
  squale.html
• Pantheon http//www.leonet.it/culture/nexus/98/S
  perling.html
• Palladio (Stephen Wassell) http//www.leonet.it/
  culture/nexus/98/Wassell.html
• Brunelleschi http//www.cuny.edu/multimedia/arsn
  ew/arch1.html