3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture Steve Marschner (with previous instructors James/Bala)
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1 3D Tranformations CS 4620 Lecture 6 1
2 Translation 2
3 Translation 2
4 Translation 2
5 Translation 2
6 Scaling 3
7 Scaling 3
8 Scaling 3
9 Scaling 3
10 Rotation about z axis 4
11 Rotation about z axis 4
12 Rotation about x axis 5
13 Rotation about x axis 5
14 Rotation about y axis 6
15 Rotation about y axis 6
16 General Rotation Matrices A rotation in 2D is around a point A rotation in 3D is around an axis so 3D rotation is w.r.t a line, not just a point there are many more 3D rotations than 2D a 3D space around a given point, not just 1D 2D 3D 7
17 Properties of Rotation Matrices Columns of R are mutually orthonormal: RR T =R T R=I Right-handed coordinate systems: det(r)=1 8
18 Specifying rotations In 2D, a rotation just has an angle In 3D, specifying a rotation is more complex basic rotation about origin: unit vector (axis) and angle convention: positive rotation is CCW when vector is pointing at you Many ways to specify rotation Indirectly through frame transformations Directly through Euler angles: 3 angles about 3 axes (Axis, angle) rotation Quaternions 9
19 Euler angles An object can be oriented arbitrarily Euler angles: stack up three coord axis rotations ZYX case: Rz(thetaz)*Ry(thetay)*Rx(thetax) heading, attitude, bank (NASA standard airplane coordinates) pitch, yaw, roll 10
20 Roll, yaw, Pitch 11
21 3D rotations NASA standard Euler angles: stack up three coord axis rotations ZYX case: Rz(thetaz)*Ry(thetay)*Rx(thetax) 12
22 Specifying rotations: Euler rotations Euler angles 13
23 Gimbal Lock 14
24 Euler angles Gimbal lock removes one degree of freedom worth a look: (also 15
25 Matrices for axis-angle rotations Showed matrices for coordinate axis rotations but what if we want rotation about some random axis? Compute by composing elementary transforms transform rotation axis to align with x axis apply rotation inverse transform back into position Just as in 2D this can be interpreted as a similarity transform 16
26 Building general rotations Using elementary transforms you need three translate axis to pass through origin rotate about y to get into x-y plane rotate about z to align with x axis Alternative: construct frame and change coordinates choose p, u, v, w to be orthonormal frame with p and u matching the rotation axis apply similarity transform T = F R x (θ ) F 1 17
27 Orthonormal frames in 3D Useful tools for constructing transformations Recall rigid motions affine transforms with pure rotation columns (and rows) form right handed ONB that is, an orthonormal basis 18
28 Building 3D frames Given a vector a and a secondary vector b The u axis should be parallel to a; the u v plane should contain b u = u / u w = u x b; w = w / w v = w x u Given just a vector a The u axis should be parallel to a; don t care about orientation about that axis Same process but choose arbitrary b first Good choice is not near a: e.g. set smallest entry to 1 19
29 Building general rotations Alternative: construct frame and change coordinates choose p, u, v, w to be orthonormal frame with p and u matching the rotation axis apply similarity transform T = F R x (θ ) F 1 interpretation: move to x axis, rotate, move back interpretation: rewrite u-axis rotation in new coordinates (each is equally valid) (note above is linear transform; add affine coordinate) 20
30 Building general rotations Alternative: construct frame and change coordinates choose p, u, v, w to be orthonormal frame with p and u matching the rotation axis apply similarity transform T = F R x (θ ) F 1 interpretation: move to x axis, rotate, move back interpretation: rewrite u-axis rotation in new coordinates (each is equally valid) Sleeker alternative: Rodrigues formula 21
31 Specifying Rotations Many ways to specify rotation Indirectly through frame transformations Directly through Euler angles: 3 angles about 3 axes (Axis, angle) rotation: based on Euler s theorem Quaternions 22
32 Derivation of General Rotation Matrix Axis angle rotation 23
33 Axis-angle ONB 24
34 Axis-angle rotation 25
35 26
36 Rotation Matrix for Axis-Angle 27
37 Transforming normal vectors Transforming surface normals differences of points (and therefore tangents) transform OK normals do not --> use inverse transpose matrix 28
38 Transforming normal vectors Transforming surface normals differences of points (and therefore tangents) transform OK normals do not --> use inverse transpose matrix 28
39 Building transforms from points 2D affine transformation has 6 degrees of freedom (DOFs) this is the number of knobs we have to set to define one So, 6 constraints suffice to define the transformation handy kind of constraint: point p maps to point q (2 constraints at once) three point constraints add up to constrain all 6 DOFs (i.e. can map any triangle to any other triangle) 3D affine transformation has 12 degrees of freedom count them from the matrix entries we re allowed to change So, 12 constraints suffice to define the transformation in 3D, this is 4 point constraints (i.e. can map any tetrahedron to any other tetrahedron) 29
Essential Mathematics for Computer Graphics fast
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