Viewing Steve Marschner CS 4620 Cornell University Cornell CS4620 Fall 2020 Steve Marschner • 1
Viewing 1. Projection and perspective Cornell CS4620 Fall 2020 Steve Marschner • 2
Parallel projection • To render an image of a 3D scene, we project it onto a plane • Simplest kind of projection is parallel projection scene image projection plane Cornell CS4620 Fall 2020 Steve Marschner • 3
“Orthographic” projection (in graphics we normally don’t distinguish from axonometric) • Projection is along lines parallel to a fixed direction – projection plane is perpendicular to projection direction Cornell CS4620 Fall 2020 Steve Marschner • 4
“Orthographic” projection (in graphics we normally don’t distinguish from axonometric) • Projection is along lines parallel to a fixed direction – projection plane is perpendicular to projection direction – image height determines which objects appear in image Cornell CS4620 Fall 2020 Steve Marschner • 4
“Orthographic” projection (in graphics we normally don’t distinguish from axonometric) • Projection is along lines parallel to a fixed direction – projection plane is perpendicular to projection direction – image height determines which objects appear in image – size of projection does not change with distance Cornell CS4620 Fall 2020 Steve Marschner • 4
“Orthographic” projection (in graphics we normally don’t distinguish from axonometric) • Projection is along lines parallel to a fixed direction – projection plane is perpendicular to projection direction – image height determines which objects appear in image – size of projection does not change with distance Cornell CS4620 Fall 2020 Steve Marschner • 4
Classical projections—parallel • Emphasis on cube-like objects – traditional in mechanical and architectural drawing Planar Geometric Projections Parallel Perspective [after Carlbom & Paciorek 78] Oblique One-point Two-point Three-point Orthographic Multiview Axonometric Orthographic Cornell CS4620 Fall 2020 Steve Marschner • 5
Orthographic in architecture [Carlbom & Paciorek 78] Cornell CS4620 Fall 2020 Steve Marschner • 6
Orthographic in traditional drawing [Carlbom & Paciorek 78] – projection plane parallel to a coordinate plane – projection direction perpendicular to projection plane Cornell CS4620 Fall 2020 Steve Marschner • 7
Other parallel in traditional drawing [Carlbom & Paciorek 78] axonometric : projection plane perpendicular to projection direction but not parallel to coordinate planes Cornell CS4620 Fall 2020 Steve Marschner • 8
View volume: orthographic Cornell CS4620 Fall 2020 Steve Marschner • 9
History of projection • Ancient times: Greeks wrote about laws of perspective • Renaissance: perspective is adopted by artists Duccio c. 1308 Cornell CS4620 Fall 2020 Steve Marschner • 10
History of projection • Later Renaissance: perspective formalized precisely da Vinci c. 1498 Cornell CS4620 Fall 2020 Steve Marschner • 11
Plane projection in drawing Albrecht Dürer Cornell CS4620 Fall 2020 Steve Marschner • 12
Plane projection in drawing source unknown Cornell CS4620 Fall 2020 Steve Marschner • 13
Plane projection in photography • This is another model for what we are doing – applies more directly in realistic rendering [Source unknown] Cornell CS4620 Fall 2020 Steve Marschner • 14
Plane projection in photography [Richard Zakia] Cornell CS4620 Fall 2020 Steve Marschner • 15
Perspective projection • Projection is along lines through a fixed point – plane normally perpendicular to center projection line Cornell CS4620 Fall 2020 Steve Marschner • 16
Perspective projection • Projection is along lines through a fixed point – plane normally perpendicular to center projection line – angular field of view determines which objects appear in image α = 2 tan − 1 h 2 d Cornell CS4620 Fall 2020 Steve Marschner • 16
Perspective projection • Projection is along lines through a fixed point – plane normally perpendicular to center projection line – angular field of view determines which objects appear in image α = 2 tan − 1 h 2 d – size of projection is inversely proportional to distance y ′ = dy / z Cornell CS4620 Fall 2020 Steve Marschner • 16
Perspective projection • Projection is along lines through a fixed point – plane normally perpendicular to center projection line – angular field of view determines which objects appear in image α = 2 tan − 1 h 2 d – size of projection is inversely proportional to distance y ′ = dy / z Cornell CS4620 Fall 2020 Steve Marschner • 16
Classical projections—perspective • Emphasis on cube-like objects – traditional in mechanical and architectural drawing Planar Geometric Projections Parallel Perspective [after Carlbom & Paciorek 78] Oblique One-point Two-point Three-point Orthographic Multiview Axonometric Orthographic Cornell CS4620 Fall 2020 Steve Marschner • 17
Perspective one-point : projection plane parallel to a coordinate plane (to two coordinate axes) two-point : projection plane parallel to one coordinate axis [Carlbom & Paciorek 78] three-point : projection plane not parallel to a coordinate axis Cornell CS4620 Fall 2020 Steve Marschner • 18
View volume: perspective Cornell CS4620 Fall 2020 Steve Marschner • 19
Field of view (or f.o.v.) • The angle between the rays corresponding to opposite edges of a perspective image – simpler to compute for “normal” perspective – have to decide to measure vert., horiz., or diag. • In cameras, determined by focal length – confusing because of many image sizes – for 35mm format (36mm by 24mm image) – 18mm = 67° v.f.o.v. — super-wide angle – 28mm = 46° v.f.o.v. — wide angle – 50mm = 27° v.f.o.v. — “normal” – 100mm = 14° v.f.o.v. — narrow angle (“telephoto”) Cornell CS4620 Fall 2020 Steve Marschner • 20
Field of view • Determines “strength” of perspective effects [Ansel Adams] close viewpoint far viewpoint wide angle narrow angle large scale differences small scale differences Cornell CS4620 Fall 2020 Steve Marschner • 21
Choice of field of view • In photography, wide angle lenses are specialty tools – “hard to work with” – easy to create weird-looking perspective effects • In graphics, you can type in whatever f.o.v. you want – and people often type in big numbers! [Ken Perlin] Cornell CS4620 Fall 2020 Steve Marschner • 22
Perspective “distortions” • Lengths, length ratios [Carlbom & Paciorek 78] Cornell CS4620 Fall 2020 Steve Marschner • 23
Oblique projection • View direction no longer coincides with projection plane normal (one more parameter) – objects at different distances still same size – objects are shifted in the image depending on their depth Cornell CS4620 Fall 2020 Steve Marschner • 24
Oblique projection • View direction no longer coincides with projection plane normal (one more parameter) – objects at different distances still same size – objects are shifted in the image depending on their depth Cornell CS4620 Fall 2020 Steve Marschner • 24
Oblique projection • View direction no longer coincides with projection plane normal (one more parameter) – objects at different distances still same size – objects are shifted in the image depending on their depth Cornell CS4620 Fall 2020 Steve Marschner • 24
Oblique projection • View direction no longer coincides with projection plane normal (one more parameter) – objects at different distances still same size – objects are shifted in the image depending on their depth Cornell CS4620 Fall 2020 Steve Marschner • 24
Off-axis parallel [Carlbom & Paciorek 78] axonometric : projection oblique : projection plane plane perpendicular to parallel to a coordinate plane projection direction but not but not perpendicular to parallel to coordinate planes projection direction. Cornell CS4620 Fall 2020 Steve Marschner • 25
Shifted perspective projection • Perspective but with projection plane not perpendicular to view direction – additional parameter: projection plane normal – exactly equivalent to cropping out an off-center rectangle from a larger “normal” perspective – corresponds to view camera in photography Cornell CS4620 Fall 2020 Steve Marschner • 26
Shifted perspective projection • Perspective but with projection plane not perpendicular to view direction – additional parameter: projection plane normal – exactly equivalent to cropping out an off-center rectangle from a larger “normal” perspective – corresponds to view camera in photography Cornell CS4620 Fall 2020 Steve Marschner • 26
Shifted perspective projection • Perspective but with projection plane not perpendicular to view direction – additional parameter: projection plane normal – exactly equivalent to cropping out an off-center rectangle from a larger “normal” perspective – corresponds to view camera in photography Cornell CS4620 Fall 2020 Steve Marschner • 26
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