Technical Background Display pipeline Transformations and roundoff errors

Technical Background Display pipeline Transformations and round-off errors Orientation representation Quaternions Rick Parent - CIS 681

Viewing Pipeline Transformation between spaces transformations Map to eye space Perspective divide Object space World space Eye space transformations perspective Clipping space Image space Viewport mapping Screen space Rick Parent - CIS 681

Ray Tracing transformations Map to eye space Object space World space Eye space transformations Trace rays Screen space Rick Parent - CIS 681

Transformations d Px h i Py a b c e f j k l m Pz n o q 1 p Rick Parent - CIS 681

Transformations 1 0 0 Tx Sx 0 0 1 0 Ty 0 Sy 0 0 1 Tz 0 0 Sz 0 0 0 1 Rick Parent - CIS 681

Rotation We know how to rotate about the global axes cos(q) 0 sin(q) 0 1 0 -sin(q) 0 cos(q) 1 0 0 0 cos(q) -sin(q) 0 sin(q) cos(q) 0 0 0 1 Rick Parent - CIS 681

Transformation round-off errors Rotate 5 degrees every frame M = 5 degree Rotation matrix Apply M to data in world space Rick Parent - CIS 681

Transformation round-off errors Rotate 5 degrees every frame m = 5 degree c = 0 degrees c=c+m M = rotation matrix Of c degrees Apply M to data in world space Rick Parent - CIS 681

Transformation round-off errors Rotate 5 degrees every frame M = 5 degree Rotation matrix C = Identity matrix C=C*M Apply C to data Rick Parent - CIS 681

Transformation round-off errors Rotate 5 degrees every frame M = 5 degree Rotation matrix m = 5 degree Apply M to data in world space C = Identity matrix c=c+m C=C*M M = rotation matrix Of c degrees Apply C to data c = 0 degrees Apply M to data in world space Rick Parent - CIS 681

Orientation Rotation about principle axes - fixed angles Rotation about object’s axes - Euler angles Axis-angle rotation Quaternion Rick Parent - CIS 681

Orientation We know how to rotate about the global axes cos(q) 0 sin(q) 0 1 0 -sin(q) 0 cos(q) 1 0 0 0 cos(q) -sin(q) 0 sin(q) cos(q) 0 0 0 1 Rick Parent - CIS 681

Fixed angles Rotate about global axes in a fixed order Rotating about global axes is what the rotation matrices do y Can use most any triple of axes Rotate about x, then y, then z x (10, 90, -45) z Rick Parent - CIS 681

Gimbal lock From some orientations, can’t do some rotations y (0, 90, 0) x Can’t rotate around x-axis z Rick Parent - CIS 681

Euler angles Rotate about axes of object Can use most any triple of axes y Roll, Pitch, Yaw (10, 90, -45) x z Rick Parent - CIS 681

Equivalence of Fixed angles and Euler angles Ru(a) = Rx (a) Rv (b)Ru (a) = Rx(a)Ry (b)Rx (- a)Rx (a) = Rx (a)Ry (b) Rw (g) Rv (b)Ru (a) = Rx (a)Ry (b) Rz (g) Rick Parent - CIS 681

Quaternions Keep axis-angle orientation as 4 -tuple Q = (s, x, y, z) = (s, v) Q 1*Q 2 = (s 1, v 1)*(s 2, v 2) = (s 1*s 2+v 1*v 2 , s 1*v 2 + s 2*v 1 + v 1 xv 2) Q 1+Q 2 = (s 1, v 1)+(s 2, v 2) = (s 1+s 2, v 1+v 2) Rick Parent - CIS 681

Quaternions Keep axis-angle orientation as 4 -tuple (sin(t/2), cos(t/2)*(x, y, z)) Rick Parent - CIS 681