Projections Josef Pelikán & Alexander Wilkie CGG MFF UK Praha

Transcription

1 Projections Josef Pelikán & Aleander Wilkie CGG MFF UK Praha / 24

2 Basic Concepts plane of projection projection ras projection origin plane of projection 2 / 24

3 Classification of Linear Projections Parallel projections Projection ras are parallel to each other Orthogonal projections Projection ras are orthogonal to the projection plane Monge projection, floor plan, elevation, side view Aonometr (general orthogonal projection) Oblique projections Cabinet projection (the ais has ½ scale) Cavalier projection (same scale on all aes) 3 / 24

4 Monge projection elevation top view side view 4 / 24

5 Aonometr Isometric Projection a projection plane a a 5 / 24

6 Aonometr Dimetric Projection b projection plane a a 6 / 24

7 Aonometr Trimetric Projection c projection plane b a 7 / 24

8 Cabinet Projection projection plane = /2 / / 24

9 Cavalier Projection projection plane = / 24

10 Classification of Linear Projections (Central) perspective projections Projection ras form a beam that pass through a single point, the center of the projection Do not preserve parallelism (vanishing points!) One point perspective The plane of projection is parallel to two coordinate aes Lines parallel to the third coordinate ais meet in one vanishing point 0 / 24

11 One Point Perspective projection plane / 24

12 Classification of Linear Projections Two point perspective The plane of projection is parallel to one coordinate ais Lines parallel to the other aes meet in two vanishing points Three point perspective The plane of projection is in an arbitrar orientation Lines parallel to the coordinate aes meet in three vanishing points 2 / 24

13 Two Point Perspective perspective plane 3 / 24

14 Three Point Perspective projection plane intersects all aes 4 / 24

15 Orthogonal Projection Implementation [,] are usuall coordinates in the viewing plane, and depth (distance from the viewer) Fundamental views (top, front, side) These are just permutations of the, and aes (with possible sign change) General orthogonal projection (isometric) View direction (normal of the projection plane): N Orientation vector (up): u Transformation: Cs(S,u N,u,N) 5 / 24

16 Orthogonal Projection u N S u N N u u N Cs(S,u N,u,N) 6 / 24

17 Oblique Projection Implementation K perspective plane: foreshortening coefficient: K angle of the projection ais : [ K cos α K sin α ] 7 / 24

18 Central Projection Implementation General perspective projection: Center of the projection: S View direction (normal of the perspective plane): N Distance of the plane from the center of the projection: d Orientation vector (up): u Projection transformation: Use standard orientation (center at the origin, view direction along ): Cs(S,u N,u,N) Perspective projection: e.g. [ d/, d/, ] 8 / 24

19 Using the Standard Orientation u d N N S u N u u N Cs(S,u N,u,N) 9 / 24

20 Perspective Transform Does NOT conserve linearit! d projection plane, [ ] d / 24

21 Transformation of Linear Objects Perspective transform of lines Per: The following equation obviousl does not hold: Per(A + t [B - A]) = Per(A) + t [Per(B) - Per(A)] Using a difference algorithm (DDA) for visibilit calculations: Given point C(u) on the segment Per(A)Per(B): C(u), = Per(A), + u [Per(B), - Per(A), ] This also has to hold for depth : C(u) = Per(A) + u [Per(B) - Per(A) ] 2 / 24

22 Conservation of Linearit [-k,] [k,] d, projection plane [ ] d d d, 22 / 24

23 4D Clipping, = -k viewing area, = k, -k = 0 k limit hperplane: = -kw, = kw, = -kw, = kw, = 0 for w > 0: -kw < < kw, -kw < < kw, 0 < 23 / 24

24 End Further Information J. Fole, A. van Dam, S. Feiner, J. Hughes: Computer Graphics, Principles and Practice, Jiří Žára a kol.: Počítačová grafika, princip a algoritm, / 24