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 C S 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 C S From 3D to 2D: Orthographic and Perspective Projection—Part 1
•History
• Geometrical Constructions 3D Viewing I • Types of Projection
• Projection in Computer Graphics
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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 C S 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 C S Drawing as Projection Early Examples of Projection • Plan view (orthographic projection) from Mesopotamia, 2150 BC: earliest known technical • Painting based on mythical tale as told by Pliny the drawing in existence Elder: Corinthian man traces shadow of departing lover
Carlbom Fig. 1-1 • Greek vases from late 6th century BC show perspective(!)
detail from The Invention of Drawing, 1830: Karl Friedrich • Roman architect Vitruvius published specifications of plan / elevation drawings, perspective. Illustrations Schinkle (Mitchell p.1) for these writings have been lost
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1 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 C S 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 C S Most Striking Features of Linear Early Perspective Perspective • Ways of invoking three dimensional space: shading • || lines converge (in 1, 2, or 3 axes) to vanishing point suggests rounded, volumetric forms; converging lines suggest spatial depth of room • Objects farther away are more foreshortened (i.e., smaller) than closer ones • Not systematic—lines do not converge to single vanishing point • Example: perspective cube
edges same size, with farther ones smaller
parallel edges converging
Giotto, Franciscan Rule Approved, Assisi, Upper Basilica, c.1295-1300
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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 C S 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 C S Setting for “Invention” of Brunelleschi and Vermeer Perspective Projection • Brunelleschi invented systematic method of • The Renaissance: new emphasis on importance of determining perspective projections (early 1400’s). individual viewpoint and world interpretation, power Evidence that he created demonstration panels, with of observation—particularly of nature (astronomy, specific viewing constraints for complete accuracy of anatomy, botany, etc.) reproduction. Note the perspective is accurate only from one POV (see Last Supper) – Massaccio – Donatello • Vermeer created perspective boxes where picture, when viewed through viewing hole, had correct – Leonardo perspective –Newton • Vermeer on the web: • Universe as clockwork: intellectual rebuilding of universe along mechanical lines – http://www.grand-illusions.com/vermeer/vermeer1.htm – http://essentialvermeer.20m.com/ – http://brightbytes.com/cosite/what.html
Ender, Tycho Brahe and Rudolph II in Prague (detail of clockwork), c. 1855 url: http://www.mhs.ox.ac.uk/tycho/catfm.htm?image10a Vermeer, The Music Lesson, c.1662-1665 (left) and reconstruction (center)
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2 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 C S 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 C S Hockney and Stork Alberti • An artist named David Hockney proposed that many • Published first treatise on perspective, Della Pittura, Renaissance artists, including Vermeer, might have been in 1435 aided by camera obscura while painting their masterpieces, • “A painting [the projection plane] is the intersection raising a big controversy of a visual pyramid [view volume] at a given distance, with a fixed center [center of projection] and a defined position of light, represented by art with lines and colors on a given surface [the rendering].” (Leono Battista Alberti (1404-1472), On Painting, pp. 32-33)
• David Stork, a Stanford optics expert, refuted Hockney’s claim in the heated 2001 debate about the subject among artists, museum curators and scientists. He also wrote the article “Optics and Realism in Renaissance Art”, using scientific techniques to disprove Hockney’s theory Hockney, D. (2001) Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters. New York: Viking Studio. Stork, D. (2004) Optics and Realism in Renaissance Art. Scientific American 12, 52-59.
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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 C S 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 C S The Visual Pyramid and Similar Dürer Triangles • Concept of similar triangles described both • Projected image is easy to calculate based on geometrically and mechanically in widely read treatise by Albrecht Dürer (1471-1528) – 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
object projected object
CB : CD as AB : ED Albrecht Dürer, Artist Drawing a Lute Woodcut from Dürer’s work about the Art of Measurement. • AB is component of A’ in the plane of projection ‘Underweysung der messung’, Nurenberg, 1525
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3 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 C S 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 C S Las Meninas (1656) by Diego Robert Campin Velàzquez The Annunciation Triptych • Point of view influences content and meaning of what is seen (ca. 1425) • Are royal couple in mirror about to enter room? Or is their image a reflection of painting on far left? • Analysis through computer reconstruction of the painted space – verdict: royal couple in mirror is reflection from canvas in foreground, not reflection of actual people (Kemp pp. 105-108)
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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 C S 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 C S Piero della Francesca Leonardo da Vinci The Resurrection (1460) The Last Supper (1495) • Perspective can be used in unnatural ways to control perception • Perspective plays very large role in this painting • Use of two viewpoints concentrates viewer’s attention alternately on Christ and sarcophagus
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4 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 C S 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 C S Geometrical Construction of Planar Geometric Projection Projections • Projectors are straight lines
• Projection surface is plane (picture plane, projection plane)
projectors
eye, or Center of Projection (COP) projectors
picture plane from Vredeman de Vries’s Perspective, Kemp p.117 • This drawing itself is perspective projection • 2 point perspective—two vanishing points • What other types of projections do you know? – hint: maps
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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 C S 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 C S Main Classes of Planar Types of Projection Geometrical Projections a) Perspective: determined by Center of Projection (COP) (in our diagrams, the “eye”)
b) Parallel: determined by Direction of Projection (DOP) (projectors are parallel—do not converge to “eye” or COP). Alternatively, COP is at ∞
• In general, a projection is determined by where you place the projection plane relative to principal axes of object (relative angle and position), and what angle the projectors make with the projection plane
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5 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 C S 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 C S Logical Relationship Between Multiview Orthographic Types of Projections • Used for: – engineering drawings of machines, machine parts – working architectural drawings •Pros: – accurate measurement possible – all views are at same scale • Cons: – does not provide “realistic” view or sense of 3D form • Usually need multiple views to get a three- dimensional feeling for object
• Parallel projections used for engineering and architecture because they can be used for measurements • Perspective imitates eyes or camera and looks more natural
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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 C S 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 C S Axonometric Projections Isometric Projection (1/2)
• Same method as multiview orthographic projections, except projection plane not parallel to any of coordinate planes; parallel lines equally foreshortened • Isometric: Angles between all Construction of an three principal axes equal isometric projection: projection (120º). Same scale ratio applies plane cuts each principal axis by 45° along each axis • Used for: • Dimetric: Angles between two of the principal axes equal; – catalogue illustrations need two scale ratios – patent office records • Trimetric: Angles different – furniture design between three principal axes; – structural design need three scale ratios •Pros: • Note: different names for – don’t need multiple views different views, but all part of a – illustrates 3D nature of object continuum of parallel projections of cube; these differ – measurements can be made to scale along principal in where projection plane is axes relative to its cube • Cons: – lack of foreshortening creates distorted appearance – more useful for rectangular than curved shapes
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6 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 C S 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 C S Isometric Projection (2/2) Oblique Projections • Video games have been using isometric projection for ages. • Projectors at oblique angle to projection plane; view It all started in 1982 with Q*Bert and Zaxxon which were cameras have accordion housing, used for skyscrapers made possible by advances in raster graphics hardware •Pros: – can present exact shape of one face of an object (can take accurate measurements): better for elliptical shapes than axonometric projections, better for “mechanical” viewing – lack of perspective foreshortening makes comparison of sizes easier – displays some of object’s 3D appearance • Cons: • Still in use today when you want to see things in distance as – objects can look distorted if careful choice not made well as things close up (e.g. strategy, simulation games) about position of projection plane (e.g., circles become ellipses) – lack of foreshortening (not realistic looking)
• Technically some games today aren’t isometric and are instead axonometric, but people still call them isometric to avoid learning a new word. Other inappropriate terms used for axonometric views are “2.5D” and “three- perspective oblique quarter.”
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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 C S 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 C S View Camera Examples of Oblique Projections
Plan oblique projection of city Construction of oblique parallel projection
source: http://www.usinternet.com/users/rniederman/star01.htm (Carlbom Fig. 2-4) Front oblique projection of radio
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7 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 C S 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 C S Example: Oblique View Main Types of Oblique • Rules for placing projection plane for oblique views: Projections projection plane should be chosen according to one or • Cavalier: Angle between projectors and projection several of following: plane is 45º. Perpendicular faces projected at full – parallel to most irregular of principal faces, or to one scale which contains circular or curved surfaces – parallel to longest principal face of object – parallel to face of interest
cavalier projection of unit cube
• Cabinet: Angle between projectors & projection plane: arctan(2) = 63.4º. Perpendicular faces Projection plane parallel to circular face projected at 50% scale
cabinet projection of unit cube Projection plane not parallel to circular face
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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 C S 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 C S Examples of Orthographic and Summary of Parallel Projections • Assume object face of interest lies in principal plane, Oblique Projections i.e., parallel to xy, yz, or zx planes. (DOP = Direction y y y of Projection, VPN = View Plane Normal)
y y y
y y y
y y y 1) Multiview Orthographic
y y y
y y y – VPN || a principal coordinate axis
y y y
y y y – DOP || VPN – shows single face, exact
yyyyyyyy y measurements
yyyyyyyy y
yyyyyyyy y 2) Axonometric – VPN || a principal coordinate axis multiview orthographic – DOP || VPN – adjacent faces, none exact, uniformly foreshortened (function of angle between face normal and DOP) 3) Oblique – VPN || a principal coordinate axis cavalier cabinet – DOP || VPN – adjacent faces, one exact, others uniformly foreshortened
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8 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 C S 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 C S Perspective Projections Vanishing Points (1/2) • Used for: • For right-angled forms whose face normals are –advertising perpendicular to the x, y, z coordinate axes, – presentation drawings for architecture, industrial number of vanishing points = number of design, engineering principal coordinate axes intersected by – fine art projection plane •Pros: y y – gives a realistic view and feeling for 3D form of object • Cons: y – does not preserve shape of object or scale (except x y where object intersects projection plane) z x x • Different from a parallel projection because z z – parallel lines not parallel to the projection plane One Point Perspective converge (z-axis vanishing point) y – size of object is diminished with distance x – foreshortening is not uniform y z x
z
Three Point Perspective (z, x, and y-axis vanishing z x Two Point Perspective points) (z, and x-axis vanishing points)
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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 C S 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 C S Vanishing Points (2/2) Vanishing Points and • What happens if same form is turned so its face normals are not perpendicular to x, y, z coordinate axes? the View Point (1/3) • We’ve seen two pyramid geometries for understanding perspective projection:
Unprojected cube depicted here with parallel projection
1. perspective image is • New viewing situation: cube intersection of a plane 2. perspective image is is rotated, face normals no result of foreshortening longer perpendicular to any with light rays from principal axes object to eye (COP) due to convergence of some parallel lines Perspective drawing • Combining these 2 views: toward vanishing points of the rotated cube • Although projection plane only intersects one axis (z), three vanishing points created
• But… can achieve final results identical to previous situation in which projection plane intersected all three axes
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9 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 C S 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 C S Vanishing Points and Vanishing Points and the View Point (2/3) the View Point (3/3) • Project parallel lines AB, CD on xy plane •Lines AB and CD (this time with A and C behind the • Projectors from eye to AB and CD define two planes, projection plane) projected on xy plane: A’B and C’D which meet in a line which contains the view point, or • Note: A’B not parallel to C’D eye • Projectors from eye to A’B and C’D define two planes • This line does not intersect projection plane (XY), which meet in a line which contains the view point because parallel to it. Therefore there is no vanishing • This line does intersect projection plane point • Point of intersection is vanishing point
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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 C S Next Time: Projection in Computer Graphics
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