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

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From 3D to 2D: Orthographic and Perspective Projection—Part 1 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) Andries van Dam September 17, 1998 3D Viewing I 1/31 Andries van Dam September 17, 1998 3D Viewing I 2/31 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 Andries van Dam September 17, 1998 3D Viewing I 3/31 Andries van Dam September 17, 1998 3D Viewing I 4/31 INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Setting for “Invention” of Brunelleschi Perspective Projection • Invented systematic method of determining perspective projections in early 1400’s. • The Renaissance: new emphasis on Evidence that he created demonstration importance of individual point of view and panels, with specific viewing constraints for interpretation of world, power of complete accuracy of reproduction. observation— particularly of nature (astronomy, anatomy, botany, etc.). – Massaccio – Donatello – Leonardo – Newton • Universe as clockworks: intellectual rebuilding of universe along mechanical lines. painting mirror Filippo Brunelleschi (1377-1446) Kemp pp.12,13 Andries van Dam September 17, 1998 3D Viewing I 5/31 Andries van Dam September 17, 1998 3D Viewing I 6/31 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 Andries van Dam September 17, 1998 3D Viewing I 9/31 Andries van Dam September 17, 1998 3D Viewing I 10/31 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 Andries van Dam September 17, 1998 3D Viewing I 11/31 Andries van Dam September 17, 1998 3D Viewing I 12/31 INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Planar Geometric Projection Main Classes of Planar • 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. Andries van Dam September 17, 1998 3D Viewing I 13/31 Andries van Dam September 17, 1998 3D Viewing I 14/31 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. Andries van Dam September 17, 1998 3D Viewing I 17/31 Andries van Dam September 17, 1998 3D Viewing I 18/31 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 Andries van Dam September 17, 1998 3D Viewing I 19/31 Andries van Dam September 17, 1998 3D Viewing I 20/31 INTRODUCTION TO COMPUTER GRAPHICS INTRODUCTION TO COMPUTER GRAPHICS Example: Oblique View Main Types of Oblique • Rules for placing projection plane for oblique Projections views: Projection plane should be chosen according to one or several of the followings: • Cavalier: Angle between projectors and – it is parallel to the most irregular of the principal faces, projection plane is 45o. Perpendicular faces or to the one which contains circular or curved are projected at full scale. surfaces – it is parallel to the longest principal face of the object – it is parallel to the face of interest 1 1 1 1 1 1 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 Andries van Dam September 17, 1998 3D Viewing I 21/31 Andries van Dam September 17, 1998 3D Viewing I 22/31 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
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