Viewing Transformations I Comp 535

Transcription

1 Viewing Transformations I Comp 535

2 Motivation Want to see our virtual 3-D worl on a 2-D screen 2

3 Graphics Pipeline Moel Space Moel Transformations Worl Space Viewing Transformation Ee/Camera Space Projection & Winow Transformation Screen Space 3

4 Viewing Transformations Projection: take a point from m imensions to n imensions where n < m There are essentiall two tpes of viewing transforms: Orthographic: parallel projection Points project irectl onto the view plane In ee/camera space (after viewing transformation): rop Perspective: convergent projection Points project through the origin onto the view plane In ee/camera space (after viewing transformation): ivie b 4

5 Parallel Projections We will eal onl with Orthographic Projection irection is parallel to projection plane normal Center of projection (COP) is at infinit Projectors from COP are parallel Parallel lines remain parallel All angles are preserve for faces parallel to the projection plane Center of projection at infinit 5 p p 2 Projection Ras p p 2

6 Orthographic Projection Points project orthogonall onto (i.e. normal to) the view plane: Projection lines are parallel 6

7 Projection Environment We will use a right-hane view sstem The ee or camera position is on the + ais a istance from the origin. The view irection is parallel to the ais The view plane is in the plane an passes through the origin 7

8 Parallel Projection A point in 3-space projects onto the viewing along ras parallel to the ais ( ) What is ( )? ( ) 8

9 Parallel Projection Looking own the ais: ( ) ( ) So = = 9

10 Parallel Projection Looking own the ais: ( ) ( ) So =

11 Parallel Projection Thus for parallel orthographic projections = = = So to perform a parallel projection on an object we can use matri multiplication p' Mp What is M? M i.e. we simpl rop the coorinate

12 Perspective in Art Before Perspective After Perspective 2

13 Samples Albrecht Dürer The Painter's Manual 525 3

14 Perspective Projection In the real-worl we see things in perspective: Parallel lines o not look parallel The converge at some point 4

15 Pinhole Camera Moel Image Plane How a real camera works. Wh on t the pictures appear upsie own? Virtual Image Plane To simplif things we ll put the image plane in front. This works fine but it isn t what s reall happening. 5

16 Perspective Projection Points project through the focal point (e.g. eepoint) onto the view plane: Projection lines converge 6

17 Perspective Projection Center of projection (COP) is no longer at infinit. Projection ras form a view frustum A prami with the tip at the COP ee view plane 7

18 Perspective Projection We will start with the projection plane parallel to the XY plane an perpenicular to the ais Lines parallel to the X or Y ais remain parallel X an Y istances become shorter as Z becomes more negative e.g. a cube viewe in perspective: 8

19 Perspective Projection Computation Assume the projection plane is normal to the Z ais locate at Z =. Assume the center of projection (COP eepoint) is locate at Z = What is p = ( )? projection plane p = ( ) p =( ) Center of Projection 9

20 Perspective Projection Computation Looking own the ais: - view plane p = ( ) p = ( ) B similar triangles: ' ' ' ee 2 '

21 Perspective Projection Computation Looking own the ais: p = ( ) B similar triangles: ' ee p = ( ) view plane ' ' - ' 2

22 Perspective Projection Computation So we have ' what is? ' ' sowe have ( ' ' ') can we put this into matri form? 22

23 23 Perspective Projection Computation We want: so c b a c b a so h g f e h g f e so l k j i l k j i so p o n m p o n m so p o n m l k j i h g f e c b a

24 24 Perspective Projection Computation So the matri that will give us the correct perspective is: M ba This works - what is the problem with it? Answer: The entries in the matri are point epenent! i.e. ever point will have to have a ifferent matri

25 25 Perspective Projection Computation How can we make the matri not epen on the points? The Cartesian point we want is which is equivalent to

26 Perspective Projection Computation Solution: use homogeneous points (remember them?) Our Cartesian point is: A homogeneous point that is equivalent to our esire Cartesian point is: can we come up with a matri that gives us what we nee but is point inepenent? 26

27 27 Perspective Projection Computation We want: p o n m l k j i h g f e c b a so c b a c b a so h g f e h g f e so l k j i l k j i so p o n m p o n m so

28 28 Perspective Projection Computation So the new matri we get is: M per This gives us correct results an is point inepenent

29 29 Focal Length is the focal length of the camera. What is going on here? What happens if we cut the focal length in half? Instea of we get What happens if we move all the objects twice as far awa? Instea of we get Decreasing the focal length is equivalent to making everthing more istant

30 3 Focal Length What woul happen if the focal length was reall reall big? M per What oes this look like? Orthographic Projection

31 Eample Project the triangle with vertices P = (35 7-3) P 2 = ( ) P 3 = ( ) onto the view plane LookFrom = ( 4) view plane P 3 P 2 P 3

32 Computing the Projection Point P = M per P with P = (357-3): ' P p 3 p 2 p (4) 7(4)

33 Computing the Projection Point P 2 = M per P 2 with P 2 = ( ): ' P (4) 35(4) p 3 p 2 p

34 Computing the Projection Point P 3 = M per P 3 with P 3 = ( ): ' P (4) 26(4) p 3 p 2 p

35 Final Projection The results of projecting the polgon onto the view plane: View Plane P 3 = (7 7 ) P 2 = (6 6 ) P = (2 4 ) 35

36 One More Problem What happens when Z >? The object is behin the camera. It ma get projecte onto to the screen. 36

37 Other Tpes of Projection 37

38 Pinhole Camera Moel What are we ignoring? The camera aperture is not eactl a pinhole. Large Aperture Small Aperture 38

39 Large Aperture Effects How oes a large aperture change things? Depth of Fiel We won t worr about creating it but it s not too har to o. Often we want pictures that are not blurre. 39

40 Depth of Fiel Image Plane 4

41 Photographing Lightning 4

42 Projecting Four-Dimensional Objects Just as we can project 3D objects into 2D. We can project 4D objects into 3D an then 2D. 42