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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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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
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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
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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
VPN || a principal coordinateaxis
DOP || VPN
adjacent faces, none exact,uniformly foreshortened(function of angle betweenface normal and DOP)
3) Oblique
VPN || a principal coordinateaxis
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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Vanishing Points and
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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Vanishing Points and
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
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