Viewing Transformations CS5600 Computer Graphics by Rich Riesenfeld 5 March 2002 Lecture Set 11 CS5600

Homogeneous Coordinates An infinite number of points correspond to (x,y,1). They constitute the whole line (tx,ty,t). (tx,ty,t) w = 1 (x,y,1) CS5600

Illustration: Old Style, Simple Transformation Sequence for 3D Viewing CS5600 CS5600

Simple Viewing Transformation Example Points A B C D E F G H X -1 1 Y Z CS5600 CS5600

Simple Cube Viewed from (6,8,7.5) H=(-1,-1,1) E=(-1,1,1) F=(1,1,1) G=(1,-1,1) D=(-1,-1,-1) A=(-1,1,-1) C=(1,-1,-1) B=(1,1,-1) CS5600

Topology of Cube A B C D E F G H 1 B C E H D F G A CS5600 CS5600

Topology of Cube A: B D E B: A C F C: G D: H E: F: G: H: H E G F D A C CS5600 CS5600

Simple Example Give a Cube with corners View from Eye Position (6,8,7.5) Look at Origin (0,0,0) “Up” is in z-direction CS5600 CS5600

Translate Origin by (6,8,0) CS5600 CS5600

Simple Viewing Transformation Example CS5600 CS5600

Build LH Coord with (6,8,0) CS5600 CS5600

Build LH Coord with CS5600 CS5600

Rotate about y with 6 10 (6,8,0) 8 CS5600 CS5600

Simple Viewing Transformation Example CS5600 CS5600

Rotate about x-axis with 7.5 10 CS5600 CS5600

Look at the (3-4-5) Right Triangle 10 (4) 7.5 (3) (5) 12.5 CS5600 CS5600

Simple Viewing Transformation Examle CS5600 CS5600

View on 10x10 screen, 20 away 20 10 10 CS5600 CS5600

Map to canonical frustum 20 20 CS5600 CS5600

Scale x,y by 2 for normalization Will view a 20”x20” screen from 20” away. Scale to standard viewing frustum. CS5600 CS5600

Simple Viewing Transformation Example CS5600 CS5600

Clipping not needed, so project CS5600 CS5600

Transformation of Cube CS5600 CS5600

Cube Transformed for Viewing Pts A B C D E F G H X 2.8 -0.4 -2.8 0.4 Y -1.84 -3.28 -1.36 .08 1.36 -.08 1.84 3.28 Z 12.94 11.98 13.26 14.22 11.74 10.78 12.06 13.02 CS5600 CS5600

Transformed Cube G=(-2.8,1.84) E=(2.8,1.36) F=(-0.4,-.08) D=(0.4,.08) Pt X Y A 2.8 -1.84 B -0.4 -3.28 C -2.8 -1.36 D 0.4 08 E 1.36 F -.08 G 1.84 H 3.28 H=(0.4,3.28) G=(-2.8,1.84) E=(2.8,1.36) F=(-0.4,-.08) D=(0.4,.08) A: B D E B: A C F C: G D: H E: F: G: H: C=(-2.8,-1.36) A=(2.8,-1.84) B=(-0.4,-3.28) 25 CS5600

Recall mapping [a,b] [-1,1] Translate center of interval to origin Normalize interval to [-1,1] CS5600 CS5600

Recall mapping [a,b] [-1,1] Substitute x =a: x CS5600 CS5600

Recall mapping [a,b] [-1,1] Substitute x =b: x CS5600 CS5600

Map to the (1K x 1K) screen Assume screen origin (0,0) at lower left. This translates old (0,0) to center of screen (511,511). CS5600 CS5600

Map to the (1K x 1K) screen Proper scale factor for mapping: CS5600 CS5600

Combine Screen Transformation CS5600 CS5600

For General Screen: …… CS5600 CS5600

Transformation to Std Clipping Frustum CS5600 CS5600

Transforming to Std Frustum CS5600 CS5600

Transforming to Std Frustum CS5600 CS5600

Transforming to Std Frustum The right scale matrix to map to canonical form CS5600 CS5600

Transforming to Std Frustum CS5600 CS5600

Determining Rotation Matrix CS5600

Frame rotation, CS5600 CS5600

Inverse problem easy, CS5600 CS5600

In matrix representation of , Columns are simply images of CS5600 CS5600

Rotation matrix M columns given by frame’s pre-image Column i of is CS5600 CS5600

Inverse of rotation matrix M Recall, for rotation matrix R, So, CS5600 CS5600

Rotation matrix M Row i is simply Simply write M down! Thus, CS5600

Frame Rotation: CS5600 CS5600

46 CS5600

47 CS5600

48 CS5600

49 CS5600

50 CS5600

51 CS5600

52 CS5600

The End of Viewing Transformations Lecture Set 11 53 CS5600