CS475/CS675 Computer Graphics

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1 CS475/CS675 Computer Graphics Viewing

2 Perspective Projection Projectors Centre of Projection Object Image Plane or Projection Plane 2

3 Parallel Projection Projectors Centre of Projection? Object Image Plane or Projection Plane 3

4 Parallel Projection Orthographic Projection Top View Multiviews (x=, y= or z= or principal planes). True size or shape for lines. For projection on the z= plane we get the projection matrix as Side View [ ] P z= = Front View 4

5 Parallel Projection Axonometric Projection Transform and then project using an orthographic projection such that at multiple adjacent faces are visible better representation of a 3D object using view. Face parallel to projection plane shows true shape and size. If U be the matrix formed by stacking up the unit vectors along the three axes, and T be the axonometric projection, then [ ][ T.U =T. xx ' y ' = x xy' yy' xz' yz ' ] 5

6 Parallel Projection Axonometric Projection If U be the matrix formed by stacking up the unit vectors along the three axes, and T be the axonometric projection, then [ ][ T.U =T. xx ' y ' = x xy' yy' xz' yz ' ] The foreshortening ratios for each projected principal axes are then given by: 2 f x= x x ' y x ' 2 2 f y= x y ' y y ' 2 2 f z= x z ' y z ' 2 6

7 Parallel Projection Axonometric Projection Depending on the kind of foreshortening they cause we can have three types of axonometric projections Trimetric (all foreshortenings are different) f x f y f z 7

8 Parallel Projection Axonometric Projection Depending on the kind of foreshortening they cause we can have three types of axonometric projections Trimetric (all foreshortenings are different) Dimetric (two foreshortenings are the same) f x f y f z f x= f z 8

9 Parallel Projection Axonometric Projection Depending on the kind of foreshortening they cause we can have three types of axonometric projections Trimetric (all foreshortenings are different) Dimetric (two foreshortenings are the same) Isometric (all foreshortenings are the same) f x f y f z f x= f z f x= f y= f z 9

10 Parallel Projection Axonometric Projection Assuming we rotate by R y and R x before we do the projection on the z= plane. T =P z=. R x. R y [ ][ [ = cos sin sin = cos. sin cos sin cos sin sin cos ][ cos. sin sin cos ] ]

11 Parallel Projection Axonometric Projection Now we apply this axonometric projection [ [ T to cos sin sin T.U = cos sin sin cos cos sin sin = cos sin sin cos U ][ ]. ]

12 Parallel Projection Axonometric Projection So the foreshortening ratios become f x =cos sin sin f y =cos f z =sin sin cos For Isometric projections, if we solve for ο θ=35.26 and =±45 f x = f y = f z then we get 2

13 Parallel Projection Oblique Projection The projectors are parallel to each other but they are not perpendicular to the plane of projection. Only planes parallel to plane of projection show true shape and size. 3

14 Parallel Projection Oblique Projection P ' ' x p, y p P ' x, y l x p = x l cos y p=y l sin z tan = or l=z cot l P x, y, z 4

15 Parallel Projection Oblique Projection P ' ' x p, y p P ' x, y l When =45 we get a Cavalier projection. Lines perpendicular to the projection plane are not foreshortened. When cot =/2 we get Cabinet projections. Lines perpendicular to the projection plane are foreshortened by half. is typically 3 or 45. P x, y, z 5

16 Perspective Projection Projectors Centre of Projection Object Image Plane or Projection Plane Projectors converge at a finite centre of projection. Parallel lines converge. We get non-uniform foreshortening. Shape is not preserved. We see in perspective so perspective viewing seems natural and helps in depth perception. 6

17 A digression into art 3th century, Arezzo by Giotto 7

18 A digression into art Early 5th century, The Little Garden of Paradise 8

19 A digression into art 5th century, The Baptistry in Florence, Filippo Brunelleschi 9

20 A digression into art 5th century, Fresco of Holy Trinity, Masaccio 2

21 A digression into art 5th century, School of Athens, Raphael 2

22 Perspective Projection y l2 l y' P' x ',y' P x, y, z zc y' y = l 2 l 2 l y'= z c z c z = l 2 l 2 l y z zc 22

23 Perspective Projection l2 l x' x zc P' x ',y' P x, y, z zc x' x = l 2 l 2 l x '= z z c z c z = l 2 l 2 l x z zc 23

24 Perspective Projection First we apply a perspective transform to a point X that takes it to X ' X '=P r. X [ ] [ ][ ] [ ] x x' x y' =. y = y z z' z w' r rz x '= x, rz y'= y, rz z'= z rz 24

25 Perspective Projection First we apply a perspective transform to a point X that takes it to X ' Now we add projection on the z= plane. X '=P z=. P r. X [ ] [ ][ ][ ] [ ] x' y' = z' w' x x '=, rz If r= zc x x y.. = y z r rz y y'=, rz then we get x '= x z zc z ' =, y '= y z zc 25

26 Perspective Projection VP z Vanishing point in the z direction. Set of lines not parallel to the projection plane converge at a vanishing point. 26

27 Perspective Projection To find the vanishing point along the z direction we apply the perspective transformation to the point at infinity along the z direction. X '=P r. X [ ] [ ][ ] [ ] x' y' =. = z' w' r r x '=, y '=, z'= r If r= / z c then we get z '= z c, i.e., the vanishing point lies an equal distance on the opposite side of the projection plane as the center of projection. 27

28 Perspective Projection Single point perspective Centre of projection (CoP) on x axis X '=P p. X [ ] [ ][ ] [ ] x' y' = z' w' p x x '=, px x x y. = y z z px y y'=, px z z '= px CoP is at / p,,,, VP is at / p,,, 28

29 Perspective Projection Single point perspective Centre of projection (CoP) on y axis X '=P q. X [ ] [ ][ ] [ ] x' y' = z' w' q x x '=, qy x x y. = y z z qy y y '=, qy z z '= qy CoP is at, /q,,, VP is at,/ q,, 29

30 Perspective Projection Two point perspective [ ][ ] [ ] P pq =P p. P q = p x '= x, px qy. q y'= y, px qy Two vanishing points Two CoPs? = p z '= q z px qy From 3

31 Perspective Projection Three point perspective [ ] P pqr =P p. P q. P r = p q r x x '=, px qy rz y y'= px qy rz z z '= px qy rz Three vanishing points Three CoPs? M. C. Esher, Ascending and Descending 3

32 Perspective Projection Generation of perspective views Transform and then apply single point persepective. Let us try to translate, apply a perspective and project to z= [ ][ ] [ T =P r z=. T l, m, n = r. l m = n l m r rn ] 32

33 Perspective Projection Generation of perspective views Translation along x=y line. 33

34 Perspective Projection Generation of perspective views y z z y Translation along the z axis causes change in scaling. 34

35 Perspective Projection Generation of perspective views Rotate about y axis and then apply single point persepective projection. [ ][ T =P r z=. R y = r cos. sin sin cos ] 35

36 Perspective Projection Generation of perspective views Rotate about y axis and then apply single point persepective projection. [ ][ ] T =P r z=. R y = [ cos = r sin sin r cos r cos. sin sin cos ] We get a two point perspective. 36

37 Perspective Projection Generation of perspective views Rotate about y axis, x axis and then apply single point persepective projection. T =P r z=. R x. R y [ ][ = cos sin. sin cos r ][ cos. sin sin cos ] 37

38 Perspective Projection Generation of perspective views Rotate about x axis, y axis and then apply single point persepective projection. T =P r z=. R y, R x [ cos sin sin sin cos sin cos = r cos sin r sin r cos cos ] We get a three point perspective. 38

39 Taxonomy Planar Projections Perspective Parallel Orthographic Axonometric Oblique Front Top Side One Point Trimetric Cavalier Dimetric Cabinet Isometric Two Point Three Point 39

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