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CS361

Week 5 - Monday. CS361. Last time. What did we talk about last time? Trigonometry Transforms Affine transforms Rotation Scaling Shearing Concatenation of transforms. Questions?. Assignment 2. Project 2. Student Lecture: Normal Transforms and the Euler Transform . Normal transforms.

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CS361

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  1. Week 5 - Monday CS361

  2. Last time • What did we talk about last time? • Trigonometry • Transforms • Affine transforms • Rotation • Scaling • Shearing • Concatenation of transforms

  3. Questions?

  4. Assignment 2

  5. Project 2

  6. Student Lecture: Normal Transforms and the Euler Transform

  7. Normal transforms • The matrix used to transform points will not always work on surface normals • Rotation is fine • Uniform scaling can stretch the normal (which should be unit) • Non-uniform scaling distorts the normal • Transforming by the transpose of the adjoint always gives the correct answer • In practice, the transpose of the inverse is usually used

  8. Intuition about why it works • Because of the singular value theorem, we can write any square, real-valued matrix M with positive determinant as: • M = R1SR2 • where R1 and R2 are rotation matrices and S is a scaling matrix • (M-1)T = ((R1SR2)-1)T = (R2-1S-1R1-1)T = (R1-1)T(S-1)T(R2-1)T = R1S-1 R2 • Rotations are fine for the normals, but non-uniform scaling will distort them • The transpose of the inverse distorts the scale in the opposite direction

  9. Normal transform rules of thumb • With homogeneous notation, translations do not affect normals at all • If only using rotations, you can use the regular world transform for normals • If using rotations and uniform scaling, you can use the world transform for normals • However, you'll need to normalize your normals so they are unit • If using rotations and non-uniform scaling, use the transpose of the inverse or the transpose of the adjoint • They only differ by a factor of the determinant, and you'll have to normalize your normals anyway

  10. Inverses • For normals and other things, we need to be able to compute inverses • Rigid body inverses were given before • For a concatenation of simple transforms with known parameters, the inverse can be done by inverting the parameters and reversing the order: • If M = T(t)R() then M-1 = R(-)T(-t) • For orthogonal matrices, M-1 = MT • If nothing is known, use the adjoint method

  11. Euler transform • We can describe orientations from some default orientation using the Euler transform • The default is usually looking down the –z axis with "up" as positive y • The new orientation is: • E(h, p, r) = Rz(r)Rx(p)Ry(h) • h is head, like shaking your head "no" • Also called yaw • p is pitch, like nodding your head back and forth • r is roll… the third dimension

  12. Gimbal lock • One trouble with Euler angles is that they can exhibit gimbal lock • In gimbal lock, two axes become aligned, causing a degree of freedom to be lost • Euler angles can describe orientations in multiple ways, however the wrong choice of rotations may cause gimbal lock

  13. Rotation around an arbitrary axis • Sometimes you want to rotate around some arbitrary axis r • To do so, create an orthonormal basis r, s, and t as follows • Take the smallest component of r and set it to 0 • Swap the two remaining components and negate the first one • Divide the result by its norm, making it a normal vector s • Let t = r x s • This basis can be made into a matrix M • The final transform X transforms the r-axis to the x-axis, does the rotation by α and then transforms back to r • X = MTRx(α)M

  14. Quaternions • Quaternions are a compact way to represent orientations • Pros: • Compact (only four values needed) • Do not suffer from gimbal lock • Are easy to interpolate between • Cons: • Are confusing • Use three imaginary numbers • Have their own set of operations

  15. Definition • A quaternion has three imaginary parts and one real part • We use vector notation for quaternions but put a hat on them • Note that the three imaginary number dimensions do not behave the way you might expect

  16. Operations • Multiplication • Addition • Conjugate • Norm • Identity

  17. More operations • Inverse • One (useful) conjugate rule: • Note that scalar multiplication is just like scalar vector multiplication for any vector • Quaternion quaternion multiplication is associative but not commutative • For any unit vector u, note that the following is a unit quaternion:

  18. Quaternion transforms • Take a vector or point p and pretend its four coordinates make a quaternion • If you have a unit quaternion the result of is p rotated around the u axis by 2 • Note that, because it's a unit quaternion, • There are ways to convert between rotation matrices and quaternions • The details are in the book

  19. Slerp • Short for spherical linear interpolation • Using unit quaternions that represent orientations, we can slerp between them to find a new orientation at time t = [0,1], tracing the path on a unit sphere between the orientations • To find the angle  between the quaternions, you can use the fact that cos = qxrx + qyry + qzrz+ qwrw

  20. Upcoming

  21. Next time… • Vertex blending • Morphing • Projections

  22. Reminders • Keep reading Chapter 4 • Exam 1 next Wednesday

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