From 3D to 2D: Orthographic and Perspective Projection—Part 1

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INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS From 3D to 2D: Orthographic and Perspective Projection—Part 1 Drawing as Projection • A painting based on a mythical tale as told by Pliny the Elder: a Corinthian maiden traces the shadow of her departing lover. • History

• Geometrical Constructions

• Types of Projection

• Projection in Computer Graphics

detail from The Invention of Drawing, 1830: Karl Friedrich Schinkle (Mitchell p.1)

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INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Early Examples of Projection Early Perspective

• Plan from Mesopotamia, 2150BC is earliest • Ways of invoking three dimensional space: known technical drawing in existence. rounded, volumetric forms suggested by shading, spatial depth of room suggested by • Greek vases from late 6th century BC show converging lines. perspective(!) • Not systematic—lines do not converge to a • Vitruvius, a Roman architect published single “vanishing” point. specifications of plan and elevation drawings, and perspective. Illustrations for these writings have been lost.

Giotto, Confirmation of the rule of Saint Francis, c.1325 (Kemp p.8)

Carlbom Fig. 1-1

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painting

mirror

Filippo Brunelleschi (1377-1446) Kemp pp.12,13

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INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Alberti The Visual Pyramid and Similar • Published first treatise on perspective, Della Triangles Pittura, in 1435. • “A painting is the intersection of a visual • Projected image is easy to calculate. Based on pyramid at a given distance, with a fixed – height of object (AB) – distance from eye to object (CB) center and a defined position of light, – distance from eye to picture plane (CD) represented by art with lines and colors on a – and using the relationship CB : CD as AB : ED given surface.” (Alberti, On Painting pp.32- 33) A’ A picture plane E

object projected object eye C D B

CB : CD as AB : ED

• AB is the component of A’ in the plane of the Leono Battista Alberti (1404-1472) projection

Andries van Dam September 17, 1998 3D Viewing I 7/31 Andries van Dam September 17, 1998 3D Viewing I 8/31 INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Las Meninas (1656) by Diego Dürer Velàzquez • Concept of similar triangles described both • Point of view influences content and meaning geometrically and mechanically in widely of what is seen. read treatise by Albrecht Dürer (1471-1528). • Are royal couple in mirror about to enter room? Or is their image a reflection of the painting on the far left? • Analysis through computer reconstruction of the painted space.

Albrecht Dürer, Artist Drawing a Lute, 1525

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INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Piero della Francesca Geometrical Construction of The Resurrection (1460) Projections • Perspective can be used in an unnatural way to control perception. • Use of two points of view concentrates viewer’s attention alternately on face of Christ and sarcophagus.

from Vredeman de Vries’s Perspective, Kemp p.117

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• Projectors are straight lines Geometric Projections • Projection surface is a plane( picture plane, a) Perspective: determined by Center of projection plane) Projection (COP) (in our diagrams the “eye”)

b) Parallel: determined by Direction of Projection (DOP) (projectors are parallel—do not converge to an “eye” or COP) projectors A A

Projectors A© Projectors A© B B B© B© eye, or Projection Projection plane Center of plane Center of Projection projection Center of projection (COP) projectors at infinity picture (a) (b) plane

This drawing itself is a perspective projection.

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INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Logical Relationship Between Types of Projection Types of Projections

Planar geometric projections

Parallel Perspective

Orthographic Oblique One-point

Top Cabinet Two-point (plan) Front Axonometric Cavalier Three-point elevation Side elevation Other Isometric Other

Andries van Dam September 17, 1998 3D Viewing I 15/31 Andries van Dam September 17, 1998 3D Viewing I 16/31 INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Multiview Orthographic Axonometric Projections • Used for: • Same method as multiview – engineering drawings of machines, machine parts orthographic projections, – working architectural drawings except projection plane • Pros: not parallel to any of the dimetric – accurate measurement possible coordinate planes. Parallel – all views are at same scale lines are equally • Cons: foreshortened. dimetric – does not provide “realistic” view or sense of 3D form • Isometric: Angles between • Usually need multiple views to get a three- all three principal axes are o dimensional feeling for object. equal (120 ). The same scale ratio applies along each isometric Projection axis. plane (top view) • Dimetric: Angles between Projectors for two of the principal axes are dimetric side view equal. Two scale ratios are Projectors for top view needed. • Trimetric: Angles different orthographic between the three principal Projection plane axes. Three scale ratios Projectors for (side view) required. front view Carlbom Fig.3-8 Projection • Note: different names for different views, but plane (front view) all part of a continuum of parallel projections of the cube.

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INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Isometric Projection Oblique Projections • Used for: • Projectors are at an oblique angle to the – catalogue illustrations projection plane. View cameras have – patent office records accordion housing, used for skyscrapers. – furniture design – structural design • Pros: • Pros: – can present the exact shape of one face of an object (can take accurate measurements): better for elliptical – don’t need multiple views shapes than axonometric projections, better for – illustrates 3D nature of object “mechanical” viewing – measurements can be made to scale along principal – lack of perspective foreshortening makes comparison axes. of sizes easier • Cons: – displays some of bject’s three-dimensional appearance – lack of foreshortening creates distorted appearance. • Cons: – more useful for rectangular than curved shapes. – objects can look distorted if careful choice not made about position of projection plane (e.g., circles become Projection ellipses) plane – lack of foreshortening (not realistic looking)

y o o 120 120 Projection y Projector plane

Projection- plane x normal Projector x z z 120o Construction of an Example Carlbom Fig.2.2 isometric projection Projection-plane normal (Carlbom Fig. 2-6) Construction of an Example: plan oblique projection of a city oblique parallel projection

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o o 30 cavalier projection 45 of unit cube

• Cabinet: Angle between projectors and Projection plane parallel to circular face projection plane is arctan(2) = 63.4o. Perpendicular faces are projected at 50% scale.

1 1 1/2 1/2 1 1 Projection plane not parallel to circular face o o 45 30 cabinet projection of unit cube

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INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Examples of Orthographic and Summary of Parallel Projections Oblique Projections • Assume object face of interest lies in principal plane, i.e., parallel to xy, yz or zx planes. (DOP = Direction of Projection, VPN = View Plane Normal) x DOP 1) Multiview Orthographic • VPN || a principle coordinate axis • DOP|| VPN VPN • shows single face, exact y exact measurements

x 2) Axonometric • VPN || a principle coordi- multiview orthographic DOP VPN nate axis • DOP|| VPN • adjacent faces, none exact, not uniformly foreshortened(as exact a function of angle between y face normal and DOP)

x 3) Oblique || DOP • VPN a principle coordinate axis • DOP|| VPN cavalier cabinet • adjacent faces, one exact, VPN Carlbom Fig. 3-2 others uniformally exact y foreshortened Andries van Dam September 17, 1998 3D Viewing I 23/31 Andries van Dam September 17, 1998 3D Viewing I 24/31 INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS 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, the – presentation drawings for architecture, industrial number of vanishing points = number of principal design, engineering coordinate axes intersected by projection plane. – fine art y y • Pros: – gives a realistic view and feeling for three dimensional form of object x • Cons: z y – does not preserve shape of object or scale (except x where object intersects projection plane) z y One Point Perspective • Different from a parallel projection because x (z-axis vanishing point) – parallel lines not parallel to the projection plane z converge y – size of the object is diminished with distance – foreshortening is not uniform. y x x

z z

z x Three Point Perspective Two Point Perspective (z, x, and y-axis z, and x-axis vanishing points vanishing points) Andries van Dam September 17, 1998 3D Viewing I 25/31 Andries van Dam September 17, 1998 3D Viewing I 26/31

INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Vanishing Points (2/2) Vanishing Points and • What happens if same form is turned so that the View Point (1/3) its face normals are not perpendicular to the • We’ve seen two pyramid geometries for x, y, z coordinate axes? understanding perspective projection: y y

y v.p.

v.p. x eye (COP) z x New viewing situation: 1. perspective image 2. perspective image z cube is rotated, face normals z are no longer perpendicular is intersection of is result of foreshortening due to any of the principal axes. z a plane with light rays from object to eye (COP) to convergence of some v.p. Note: unprojected cube is depicted here parallel lines toward with a parallel projection vanishing points

Perspective drawing of the • How are these pictures related? rotated cube. y real cube is Although the projection plane somewhere back here only intersects one axis (z), projection plane three vanishing points were created. Note: can achieve ﬁnal results v.p. v.p. which are identical to x

previous situation in which z projection plane intersected all three axes. eye (COP)

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AB AB projection plane y in the XY plane

CD plane defined by x projectors to line AB A B A’ B A’ projectors eye plane defined by C (view point) projectors to line CD C’ z D eye (view point) x eye (view point) plane defined by projectors projectors to line CD z vanishing point eye (view point)

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INTRODUCTION TO COMPUTER GRAPHICS Projection in Computer Graphics

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