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I N T R O D U C T I O N T O C O M P U T E R G R A P H I CS

Andries van Dam September 16, 2008 3D Viewing I #/39

3D Viewing I

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From 3D to 2D: Orthographic

and Perspective ProjectionPart 1

History

Geometrical Constructions

Types of Projection

Projection in Computer Graphics

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Drawing as Projection

Painting based on mythical tale as told byPliny the Elder: Corinthian man tracesshadow of departing lover

Detail from The Invention of Drawing, 1830:

Karl Friedrich Schinkle (Mitchell p.1)

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Early Examples of Projection

Plan view (orthographic projection) fromMesopotamia, 2150 BC: earliest knowntechnical drawing in existence

Greek vases from late 6th century BC showperspective(!)

Roman architect Vitruvius wrotespecifications of plan with architecturalillustrations, De Architectura (rediscoveredin 1414). The original illustrations for thesewritings have been lost.

Carlbom Fig.1-1

Theseus Killing theMinotaur by the

Kleophrades Painter

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Most Striking Features of Linear

Perspective || lines converge (in 1, 2, or 3 axes) tovanishing point

Objects farther away are moreforeshortened(i.e., smaller) than closerones

Example: perspective cube

edges same size,with farther ones smaller

paralleledgesconverging

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Early Perspective

Ways of invoking three dimensional space:shading suggests rounded, volumetricforms; converging lines suggest spatialdepth of room

Not systematiclines do not converge tosingle vanishing point

Giotto, Franciscan Rule Approved, Assisi, Upper Basilica, c.1295-1300

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Setting for Invention of

Perspective Projection The Renaissance: new emphasis onimportance of individual viewpoint andworld interpretation, power of observationparticularly of nature (astronomy, anatomy,botany, etc.)

Massaccio Donatello

Leonardo

Newton

Universe as clockwork: intellectualrebuilding of universe along mechanical

lines

Ender, Tycho Brahe and Rudolph II in Prague (detail of clockwork), c. 1855url: http://www.mhs.ox.ac.uk/tycho/catfm.htm?image10a

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Brunelleschi and Vermeer

Brunelleschi invented systematic method ofdetermining perspective projections (early1400s). Evidence that he createddemonstration panels, with specific viewingconstraints for complete accuracy ofreproduction. Note the perspective is

accurate only from one POV (see LastSupper)

Vermeer createdperspective boxes wherepicture, when viewed through viewing hole,had correct perspective

Vermeer on the web:

http://www.grand-illusions.com/articles/mystery_in_the_mirror/

http://essentialvermeer.20m.com/

http://brightbytes.com/cosite/what.html

Vermeer, The Music Lesson, c.1662-1665 (left) and reconstruction (right)

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Hockney and Stork

An artist named David Hockney proposed thatmany Renaissance artists, including Vermeer,might have been aided by camera obscura whilepainting their masterpieces, raising a bigcontroversy

David Stork, a Stanford optics expert, refutedHockneys claim in the heated 2001 debate aboutthe subject among artists, museum curators and

scientists. He also wrote the article Optics andRealism in Renaissance Art, using scientifictechniques to disprove Hockneys theory

Hockney, D. (2001) Secret Knowledge:Rediscovering the Lost Techniques ofthe Old Masters. New York: Viking Studio.Stork, D. (2004) Optics and Realism in Renaissance Art. Scientific American12, 52-59.

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Alberti

Published first treatise on perspective, DellaPittura, in 1435

A painting [the projection plane] is theintersection of a visual pyramid [viewvolume] at a given distance, with a fixedcenter [center of projection] and a

defined position of light, represented by artwith lines and colors on a given surface[the rendering]. (Leono Battista Alberti(1404-1472), On Painting, pp. 32-33)

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The Visual Pyramid and Similar

Triangles Projected image is easy to calculate basedon height of object (AB)

distance from eye to object (CB)

distance from eye to picture (projection)plane (CD)

and using relationship CB: CD as AB: ED

CB : CD as AB : ED

object

picture plane

projected object

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The Visual Pyramid and Similar

Triangles Cont. The general case: the object were consideringis not parallel to the picture plane AB is component of AB in a plane parallel to the picture

plane

Find the projection (B) of A on the line CB. Normalize CB

dot(CA, normalize(CB)) gives magnitude, m, ofprojection of CA in the direction of CB

Travel from C in the direction of B for distance m to getB

AB:ED as CB:CD We can use this relationship to calculate the projection

of AB on ED

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Drer

Concept of similar triangles described bothgeometrically and mechanically in widelyread treatise by Albrecht Drer (1471-1528)

Albrecht Drer, Artist Drawing a Lute

Woodcut from Drers work about the Art of

Measurement.Underweysung der messung,Nurenberg, 1525

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Las Meninas (1656)

by Diego Velzquez Point of view influences content and meaning ofwhat is seen

Are royal couple in mirror about to enter room?Or is their image a reflection of painting on farleft?

Analysis through computer reconstruction of thepainted space

verdict: royal couple in mirror is reflectionfrom canvas in foreground, not reflection ofactual people (Kemp pp. 105-108)

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Robert Campin

The Annunciation Triptych(ca. 1425)

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Piero della Francesca The

Resurrection (1460) Perspective can be used in unnatural waysto control perception

Use of two viewpoints concentrates viewersattention alternately on Christ andsarcophagus

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Leonardo da Vinci The Last

Supper (1495) Perspective plays very

large role in this painting

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Geometrical Construction of

Projections

2 point perspectivetwo vanishing points

from Vredeman de Vriess Perspective,Kemp p.117

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Planar Geometric Projection

Projectors are straight lines, like the stringin Drers Artist Drawing a Lute.

Projection surface is plane (picture plane,projection plane)

This drawing itself is perspective projection

What other types of projections do youknow?

hint: maps

projectors

eye, or

Center ofProjection(COP)

projectors

pictureplane

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Main Classes of Planar

Geometrical Projectionsa) Perspective: determined by Center ofProjection (COP) (in our diagrams, theeye)

b) Parallel: determined by Direction ofProjection (DOP) (projectors are parallel

do not converge to eye or COP).Alternatively, COP is at

In general, a projection is determined bywhere you place the projection plane

relative to principal axes of object(relative angle and position), and whatangle the projectors make with theprojection plane

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Types of Projection

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Logical Relationship Between

Types of Projections

Parallel projections used for engineering andarchitecture because they can be used formeasurements

Perspective imitates eyes or camera andlooks more natural

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Multiview Orthographic

Used for: engineering drawings of machines, machineparts

working architectural drawings

Pros:

accurate measurement possible

all views are at same scale Cons:

does not provide realistic view or sense of3D form

Usually need multiple views to get a three-dimensional feeling for object

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Axonometric Projections

Same method as multivieworthographic projections,except projection plane notparallel to any ofcoordinate planes;parallel lines equallyforeshortened

Isometric: Angles betweenall three principal axesequal (120). Same scaleratio applies along eachaxis

Dimetric: Angles betweentwo of the principal axes

equal; need two scaleratios Trimetric: Angles different

between three principalaxes; need three scaleratios

Note: different names fordifferent views, but all partof a continuum of parallelprojections of cube; thesediffer in where projectionplane is relative to its cube

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Isometric Projection (1/2)

Used for:

catalogue illustrations

patent office records furniture design

structural design

3d Modeling in real time (Maya, AutoCad, etc.)

Pros:

dont need multiple views

illustrates 3D nature of object

measurements can be made to scale along principalaxes

Cons:

lack of foreshortening creates distorted appearance

more useful for rectangular than curved shapes

Construction of anisometric projection: projectionplane cuts each principal axis by45

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Isometric Projection (2/2)

Video games have been using isometric projection for ages.It all started in 1982 with Q*Bertand Zaxxon whichwere made possible by advances in raster graphicshardware

Still in use today when you want to see things indistance as well as things close up (e.g. strategy,simulation games)

Technicallysome games today arent isometric but instead are a generalaxonometric (trimetric) with arbitrary angles, but people still call themisometric to avoid learning a new word. Other inappropriate terms used foraxonometric views are 2.5D and three-quarter.http://simcity.ea.com/about/inside_scoop/3d1.php

SimCity IV (Trimetric) StarCraft II
http://simcity.ea.com/about/inside_scoop/3d1.phphttp://simcity.ea.com/about/inside_scoop/3d1.php

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Oblique Projections

Projectors at oblique angle to projectionplane; view cameras have accordionhousing, used for skyscrapers

Pros:

can present exact shape of one face of anobject (can take accurate measurements):

better for elliptical shapes than axonometricprojections, better for mechanical viewing

lack of perspective foreshortening makescomparison of sizes easier

displays some of objects 3D appearance

Cons:

objects can look distorted if careful choice notmade about position of projection plane (e.g.,circles become ellipses)

lack of foreshortening (not realistic looking)

obliqueperspective

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source:http://www.usinternet.com/users/rniederman/star01.htm

View Camera

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Examples of Oblique

Projections

Construction ofoblique parallel projection

Plan oblique projection of city

Front oblique projection of radio

(Carlbom Fig. 2-4)

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Example: Oblique View

Rules for placing projection plane for obliqueviews: projection plane should be chosenaccording to one or several of following:

parallel to most irregular of principal faces, or to onewhich contains circular or curved surfaces

parallel to longest principal face of object

parallel to face of interest

Projection plane parallel to circular face

Projection plane not parallel to circular face

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Main Types of Oblique

Projections Cavalier: Angle between projectors andprojection plane is 45. Perpendicular facesprojected at full scale

Cabinet: Angle between projectors &

projection plane: arctan(2) = 63.4.Perpendicular faces projected at 50% scale

cavalierprojectionof unit cube

cabinetprojectionof unit cube

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Examples of Orthographic and

Oblique Projections

multiview

orthographic

cavalier cabinet

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Summary of Parallel Projections Assume object face of interest lies in principalplane, i.e., parallel toxy, yz, orzxplanes. (DOP

= Direction of Projection, VPN = View PlaneNormal)

1) Multiview Orthographic

VPN || a principal coordinateaxis

DOP || VPN

shows single face, exactmeasurements

2) Axonometric


DOP || VPN

adjacent faces, none exact,uniformly foreshortened(function of angle betweenface normal and DOP)

3) Oblique


DOP || VPN

adjacent faces, one exact,others uniformly foreshortene

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Used for: advertising presentation drawings for architecture,

industrial design, engineering

fine art

Pros:

gives a realistic view and feeling for 3D formof object

Cons:

does not preserve shape of object or scale(except where object intersects projectionplane)

Different from a parallel projection because parallel lines not parallel to the projectionplane converge

size of object is diminished with distance

foreshortening is not uniform

Perspective Projections

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Vanishing Points (1/2)

For right-angled forms whose facenormals are perpendicular to thex, y, zcoordinate axes, number of vanishingpoints = number of principal coordinateaxes intersected by projection plane

Three PointPerspective(z,x, and y-axisvanishing points)

Two Point Perspective(z, andx-axis vanishingpoints)

One Point Perspective

(z-axis vanishing point)

z

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Vanishing Points (2/2) What happens if same form is turned so its face

normals are notperpendicular tox, y,zcoordinateaxes?

Although projection plane only intersects oneaxis (z), three vanishing points created

But can achieve final results identical to

previous situation in which projection planeintersected all three axes Note: the projection plane still intersects all three of

the cubes edges, so if you pretend the cube isunrotated, and its edges the axes, then yourprojection plane is intersecting the three axes

New viewing situation:cube is rotated, facenormals no longerperpendicular to any

principal axesPerspective drawing

of the rotated cube

Unprojected cube depictedhere with parallel projection

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Vanishing Points and

the View Point (1/3) Weve seen two pyramid geometries forunderstanding perspective projection:

Combining these 2 views:

1. perspective image isintersection of aplane with light raysfrom object to eye

(COP)

2. perspective image isresult offoreshortening due toconvergence of someparallel lines towardvanishing points

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the View Point (2/3)

Project parallel linesAB, CD onxyplane Projectors from eye toAB and CD definetwo planes, which meet in a line whichcontains the view point, or eye

This line does not intersect projectionplane (XY), because parallel to it. Therefore

there is no vanishing point

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the View Point (3/3)

LinesAB and CD (this time withA and Cbehind the projection plane) projected onxyplane: AB and CD

Note: AB not parallel to CD

Projectors from eye to AB and CD definetwo planes which meet in a line which

contains the view point This line does intersect projection plane

Point of intersection is vanishing point

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Next Time: Projection in

Computer Graphics

Documents

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