1. 1 Projection

2. 2 Model Transform Viewing Transform Modelview Matrix world coordinates Pipeline Review Focus of this lecture

3. 3 Review (Lines in R 2 )

4. 4 Parallel Projection Projection (R 2 ) viewpoint viewline

5. 5 Perspective Projection ~ ~

6. 6 Parallel Projection ~ ~

7. 7 Projection (R 3 ) See handout for proof!

8. 8 Example Vertices (0,0,0), (2,0,0), (2,3,0), (0,3,0) (1,1,1), (1,2,1) Parallel projection: onto z = 0 plane v = (0,0,1,0) T, n = (0,0,1,0) T

9. 9 Vertices (0,0,0), (2,0,0), (2,3,0), (0,3,0) (1,1,1), (1,2,1) Perspective projection: onto z = 0 plane from viewpoint (1,5,3) v = (1,5,3,1) T, n = (0,0,1,0) T

10. 10 p’ p”O Viewplane Coordinate Mapping

11. 11 Determine Viewplane Transform by Homogeneous Transformation K 4×3

12. 12 L L L: left inverse of K

13. 13 Example Viewplane origin (1,2,0) u-axis (3,4,0) v-axis (-4,3,0)

14. 14 Orthographic Projection Def: direction of projection  viewplane v n … is a parallel projection

15. 15 Definitions Direction cosine (ref)ref Foreshortening ratio = (length of projected segment)/(length of original segment)

16. 16 Theorem If the direction cosines of the plane normal (in world coordinate system) are n 1, n 2, and n 3, the foreshortening ratios in the x-, y-, and z- directions are (n 2 2 + n 3 2 ) 1/2, (n 1 2 + n 3 2 ) 1/2, and (n 1 2 + n 2 2 ) 1/2, respectively. Front, side, top views: n = (1,0,0,0), (0,1,0,0), or (0,0,1,0) as in engineering drawings

17. 17 Types of Orthographic Projections Axonometric projections: attempts to portray general 3D shape –Isometric projection: all foreshortening ratio are the same –Dimetric projection: exactly two are the same –Trimetric projection: all foreshortening ratio are different

18. 18 Axonometric Projections IsometricDimetricTrimetric f: foreshortening ratios

19. 19 Example (Dimetric)

20. 20

21. 21 Oblique Projection A particular parallel projection where direction of projection is not perpendicular to viewplane v n Oblique projection not available in OpenGL

22. 22 Cavalier Projection Lines  viewplane have f = 1 Planar faces  viewplane appear thicker v  /4 n Properties: viewplane

23. 23 Cabinet Projection To overcome ‘thickness’ problem, choose f  viewplane to be 1/2 Properties:  = arccot(2) v n

24. 24 Perspective Projection A perspective projection maps parallel lines in the space to parallel lines in the viewplane IFF the lines are parallel to the viewplane.

25. 25 Otherwise, they meet

26. 26 Vanishing Point Suppose (x i, y i, z i ) i =1,2,3 are a set of mutually perpendicular vectors. The viewplane normal (n 1, n 2, n 3 ) of a perspective projection can be perpendicular to (a) none (b) one (c) two of the vectors. (a) (b)(c) n n n

27. 27 Vanishing Point If a perspective projection maps a point-at- infinity (x,y,z,0) to a finite point (x’,y’,z’,1) on the viewplane, the lines in the direction (x,y,z) appear as lines converging to point on the (Cartesian) viewplane. The point (x’,y’,z’) is called the vanishing point in the direction (x,y,z).

28. 28 Three-point perspective Two-point perspective One-point perspective Vanishing point

29. 29 IMAGE FORMATION – Perspective Imaging Image courtesy of C. Taylor “The Scholar of Athens,” Raphael, 1518

30. 30 Example Determine (and verify it is indeed so) the vanishing point of an OpenGL setting. Eye = [15,0,0] Eye = [15,0,15]

31. 31 Numeric Example How about (1,0,1,0)? Viewpoint (15,0,15,1) Viewplane: x + z + 1 = 0

32. 32 Summary Projection –Parallel projection –Perspective projection Parallel projection –Orthographic Isometric Dimetric Trimetric –Oblique Cavalier Cabinet Perspective projection –Three-point perspective –Two-point perspective –One-point perspective Understand how they are differentiated

33. 33 Fig. 8. Constructing a perspective image of a house. (a) Drawing the floor plan and defining the viewing conditions (observer position and image plane). (b) Constructing a perspective view of the floor. (c) A reference height (in this case the height of an external wall) is drawn from the ground line and the first wall is constructed in perspective by joining the reference end points to the horizontal vanishing point v2. (d) All four external walls are constructed. (e) The elevations of all other objects (the door, windows and roofs) are first defined on the reference segment and then constructed in the rendered perspective view.

34. 34 Exercise Hand sketch a perspective drawing of a house Use Maxima to compute 2-point perspective projection, setting viewplane coordinate system

35. 35 Cross Ratio The cross-ratio of every set of four collinear points shown in this figure has the same value Cross ratio is preserved in projective geometry (ratio is NOT preserved) z1z1 z2z2 z3z3 z4z4