Introduction to Computer Graphics CS 445 / 645 Lecture 10 Chapter 7: Transformations - PowerPoint PPT Presentation

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Introduction to Computer Graphics CS 445 / 645 Lecture 10 Chapter 7: Transformations

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  1. Introduction to Computer GraphicsCS 445 / 645Lecture 10Chapter 7: Transformations Gimbal

  2. Overview • Rotation representations • Euler • Axis-angle • Quaternion • Camera Transformations • Projections

  3. Basic 3D Transformations Rotate around Z axis: Rotate around Y axis: Rotate around X axis:

  4. 3-D Rotation • General rotations in 3-D require rotating about an arbitraryaxis of rotation • Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra • Standard approach: express general rotation as composition ofcanonical rotations • Rotations about X, Y, Z

  5. Composing Canonical Rotations • Goal: rotate about arbitrary vector A by  • Idea: we know how to rotate about X,Y,Z • So, rotate about Y by  until A lies in the YZ plane • Then rotate about X by  until A coincides with +Z • Then rotate about Z by  • Then reverse the rotation about X (by -) • Then reverse the rotation about Y (by -)

  6. Composing Canonical Rotations • First: rotating about Y by  until A lies in YZ • How exactly do we calculate ? • Project A onto XZ plane (Throw away y-coordinate) • Find angle  that rotates A to x-axis:  = -(90° - ) =  - 90 ° • Second: rotating about x-axis by  until A lies on z-axis • How do we calculate ?

  7. Composing Matrices • So we have the following matrices: • p: The point to be rotated about A by  • Ry : Rotate about Y by  • Rx : Rotate about X by  • Rz : Rotate about Z by  • Rx -1: Undo rotation about X by  • Ry-1: Undo rotation about Y by  • In what order should we multiply them?

  8. Compositing Matrices • Short answer: the transformations, in order, are written from right to left • In other words, the first matrix to affect the vector goes next to the vector, the second next to the first, etc. • So in our case: • p’ = Ry-1 Rx -1 Rz Rx Ry p

  9. Rotation Matrices • Notice these two matrices: • Rx : Rotate about X by  • Rx -1: Undo rotation about X by  • How can we calculate Rx  -1?

  10. Rotation Matrices • Notice these two matrices: • Rx : Rotate about X by  • Rx -1: Undo rotation about X by  • How can we calculate Rx  -1? • Obvious answer: calculate Rx (-) • Clever answer: exploit fact that rotation matrices are orthonormal

  11. Rotation Matrices • Notice these two matrices: • Rx : Rotate about X by  • Rx -1: Undo rotation about X by  • How can we calculate Rx  -1? • Obvious answer: calculate Rx (-) • Clever answer: exploit fact that rotation matrices are orthonormal • What is an orthonormal matrix? • What property are we talking about?

  12. Rotation Matrices • Orthonormal matrix: • orthogonal (columns/rows linearly independent) • normalized (columns/rows length of 1) • The inverse of an orthogonal matrix is just its transpose:

  13. Representing 3 Rotational DOFs • 3x3 Matrix (9 DOFs) • Rows of matrix define orthogonal axes • Euler Angles (3 DOFs) • Rot x + Rot y + Rot z • Axis-angle (4 DOFs) • Axis of rotation + Rotation amount • Quaternion (4 DOFs) • 4 dimensional complex numbers

  14. Rotation Matrices • What did we start with? • 3 rotations (roll, pitch, yaw) • Three numbers • Called Euler Angles • Axis-angle • Four numbers • And we end up with a 4x4 transformation matrix • Where inner 3x3 describes rotation

  15. Really, a whole matrix to represent three numbers? Underconstrained? • 9 DOFs must reduce to 3 • Rows must be unit length (-3 DOFs) • Rows must be orthogonal (-3 DOFs) • Drifting matrices is very bad • Numerical errors results when trying to gradually rotate matrix by adding derivatives • Resulting matrix may scale / shear • Gram-Schmidt algorithm will re-orthogonalize your matrix • Difficult to interpolate between matrices • Why would we do this?

  16. Euler Angles • (qx, qy, qz) = RzRyRx • Rotate qx degrees about x-axis • Rotate qy degrees about y-axis • Rotate qz degrees about z-axis • Axis order is not defined • (y, z, x), (x, z, y), (z, y, x)… are all legal • Pick one

  17. Euler Angles • Rotations not uniquely defined • ex: (z, x, y) = (90, 45, 45) = (45, 0, -45)takes positive x-axis to (1, 1, 1) • Cartesian coordinates are independent of one another, but Euler angles are not • Gimbal Lock • Term derived from mechanical problem that arises in gimbal mechanism that supports a compass or a gyro

  18. A Gimbal • Hardware implementation of Euler angles (used for mounting gyroscopes and globes)

  19. Gimbal Lock

  20. Gimbal Lock • Occurs when two axes are aligned • Second and third rotations have effect of transforming earlier rotations • ex: Rot x, Rot y, Rot z • If Rot y = 90 degrees, Rot z == -Rot x

  21. Gimbal Lock http://www.anticz.com/eularqua.htm

  22. Interpolation • Interpolation between two Euler angles is not unique • ex: (x, y, z) rotation • (0, 0, 0) to (180, 0, 0) vs. (0, 0, 0) to (0, 180, 180) • Interpolation about different axes are not independent

  23. Interpolation

  24. Axis-angle Notation • Define an axis of rotation (x, y, z) and a rotation about that axis, q: R(q, n) • 4 degrees of freedom specify 3 rotational degrees of freedom because axis of rotation is constrained to be a unit vector

  25. Axis-angle Rotation Given r – Vector in space to rotate n – Unit-length axis in space about which to rotate q – The amount about n to rotate Solve r’ – The rotated vector r’ r n

  26. Axis-angle Rotation • Step 1 • Compute rk an extended version of the rotation axis, n • rk = (n ¢ r) n rk r’ r

  27. Axis-angle Rotation • Compute r? • r? = r – (n ¢ r) n r? r’ r

  28. v q Axis-angle Rotation • Compute v, a vector perpendicular to rk and r? • v = rk£ r? • Use v and r? and q to compute r’ cos(q) r? + sin(q) v r?

  29. rperp = r – (n.r) n q V = nx (r – (n.r) n) = nxr Rr rpar = (n.r) n r n Axis-angle Notation Rr = Rrpar + Rrperp = Rrpar + (cos q) rperp + (sin q) V =(n.r) n + cos q(r – (n.r)n) + (sin q) n x r = (cos q)r + (1 – cos q) n (n.r) + (sin q) n x r

  30. Axis-angle Notation • No easy way to determine how to concatenate many axis-angle rotations that result in final desired axis-angle rotation • No simple way to interpolate rotations

  31. Quaternion • Remember complex numbers: a + ib • Where i2 = -1 • Invented by Sir William Hamilton (1843) • Remember Hamiltonian path from Discrete II? • Quaternion: • Q = a + bi + cj + dk • Where i2 = j2 = k2 = -1 and ij = k and ji = -k • Represented as: q = (s, v) = s + vxi + vyj + vzk

  32. Quaternion • A quaternion is a 4-D unit vector q = [x y z w] • It lies on the unit hypersphere x2 + y2 + z2 + w2 = 1 • For rotation about (unit) axis v by angle q • vector part = (sin q/2) v = [x y z] • scalar part = (cos q/2) = w • (sin(q/2) vx, sin(q/2) vy, sin(q/2) vz, cos (q/2)) • Only a unit quaternion encodes a rotation - normalize

  33. Quaternion • Rotation matrix corresponding to a quaternion: • [x y z w] = • Quaternion Multiplication • q1 * q2 = [v1, w1] * [v2, w2] = [(w1v2+w2v1+ (v1 x v2)), w1w2-v1.v2] • quaternion * quaternion = quaternion • this satisfies requirements for mathematical group • Rotating object twice according to two different quaternions is equivalent to one rotation according to product of two quaternions

  34. Quaternion Example • X-roll of p • (cos (p/2), sin (p/2) (1, 0, 0)) = (0, (1, 0, 0)) • Y-roll 0f p • (0, (0, 1, 0)) • Z-roll of p • (0, (0, 0, 1)) • Ry (p) followed by Rz (p) • (0, (0, 1, 0) times (0, (0, 0, 1)) = (0, (0, 1, 0) x (0, 0, 1) = (0, (1, 0, 0))

  35. Quaternion Interpolation • Biggest advantage of quaternions • Interpolation • Cannot linearly interpolate between two quaternions because it would speed up in middle • Instead, Spherical Linear Interpolation, slerp() • Used by modern video games for third-person perspective • Why?

  36. SLERP • Quaternion is a point on the 4-D unit sphere • interpolating rotations requires a unit quaternion at each step • another point on the 4-D unit sphere • move with constant angular velocity along the great circle between two points • A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). • Any rotation is defined by 2 quaternions, so pick the shortest SLERP • To interpolate more than two points, solve a non-linear variational constrained optimization • Ken Shoemake in SIGGRAPH ’85 (www.acm.org/dl)

  37. Quaternion Interpolation • Quaternion (white) vs. Euler (black) interpolation • Left images are linear interpolation • Right images are cubic interpolation

  38. Quaternion Code • http://www.gamasutra.com/features/programming/19980703/quaternions_01.htm • Registration required • Camera control code • http://www.xmission.com/~nate/smooth.html • File, gltb.c • gltbMatrix and gltbMotion

  39. 3D Rendering Pipeline (for direct illumination) 3D Geometric Primitives Modeling Transformation Transform into 3D world coordinate system Lighting Illuminate according to lighting and reflectance Viewing Transformation Transform into 3D camera coordinate system Projection Transformation Transform into 2D screen coordinate system Clipping Clip primitives outside camera’s view Scan Conversion Draw pixels (includes texturing, hidden surface, ...) Image

  40. Reminder: Homogeneous Coords • What effect does the following matrix have? • Conceptually, the fourth coordinate wis a bit like a scale factor

  41. Homogenous Coordinates • Increasing w makes things smaller • We think of homogenous coordinates as defining a projective space • Increasing w “getting further away” • Will come in handy for projection matrices

  42. Projection Matrix • We talked about geometric transforms, focusing on modeling transforms • Ex: translation, rotation, scale, gluLookAt() • These are encapsulated in the OpenGL modelview matrix • Projection is also represented as a matrix • Next few slides: representing orthographic and perspective projection with the projection matrix

  43. Taxonomy of Projections FVFHP Figure 6.10

  44. Taxonomy of Projections

  45. Parallel Projection • Center of projection is at infinity • Direction of projection (DOP) same for all points View Plane DOP Angel Figure 5.4

  46. Orthographic Projections • DOP perpendicular to view plane Front Angel Figure 5.5 Top Side

  47. Oblique Projections • DOP not perpendicular to view plane Cavalier (DOP  = 45o) Cabinet (DOP  = 63.4o) H&B Figure 12.24

  48. Orthographic Projection • Simple OrthographicTransformation • Original world units are preserved • Pixel units are preferred

  49. Orthographic: Screen Space Transformation left =10 m right = 20 m top=20 m (max pixx, max pixy) (height in pixels) (0, 0) bottom=10 m (width in pixels)

  50. Orthographic: Screen Space Transformation • left, right, top, bottom refer to the viewing frustum in modeling coordinates • width and height are in pixel units • This matrix scales and translates to accomplish the transition in units