lecture10

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

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

Overview: 

Overview Rotation representations Euler Axis-angle Quaternion Camera Transformations Projections

Basic 3D Transformations: 

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

3-D Rotation: 

3-D Rotation General rotations in 3-D require rotating about an arbitrary axis 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 of canonical rotations Rotations about X, Y, Z

Composing Canonical Rotations: 

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 -)

Composing Canonical Rotations: 

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 ?

Composing Matrices: 

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?

Compositing Matrices: 

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

Rotation Matrices: 

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

Rotation Matrices: 

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

Rotation Matrices: 

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?

Rotation Matrices: 

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:

Representing 3 Rotational DOFs: 

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

Rotation Matrices: 

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

Really, a whole matrix to represent three numbers? Underconstrained?: 

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?

Euler Angles: 

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

Euler Angles: 

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

A Gimbal: 

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

Gimbal Lock: 

Gimbal Lock

Gimbal Lock: 

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

Gimbal Lock: 

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

Interpolation: 

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

Interpolation: 

Interpolation

Axis-angle Notation: 

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

Axis-angle Rotation: 

Axis-angle Rotation r r’ n 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

Axis-angle Rotation: 

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

Axis-angle Rotation: 

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

Axis-angle Rotation: 

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?

Axis-angle Notation: 

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

Axis-angle Notation: 

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

Quaternion: 

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

Quaternion: 

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

Quaternion: 

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

Quaternion Example: 

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))

Quaternion Interpolation: 

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?

SLERP: 

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)

Quaternion Interpolation: 

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

Quaternion Code: 

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

3D Rendering Pipeline (for direct illumination): 

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

Reminder: Homogeneous Coords: 

Reminder: Homogeneous Coords What effect does the following matrix have? Conceptually, the fourth coordinate w is a bit like a scale factor

Homogenous Coordinates: 

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

Projection Matrix: 

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

Taxonomy of Projections: 

Taxonomy of Projections FVFHP Figure 6.10

Taxonomy of Projections: 

Taxonomy of Projections

Parallel Projection: 

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

Orthographic Projections: 

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

Oblique Projections: 

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

Orthographic Projection: 

Orthographic Projection Simple Orthographic Transformation Original world units are preserved Pixel units are preferred

Orthographic: Screen Space Transformation: 

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

Orthographic: Screen Space Transformation: 

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