Why Transforms? Want to animate objects and camera Translations Rotations Shears And more.. Want to be able to use projection transforms

Transcription

1 Why Transforms? Want to animate objects and camera Translations Rotations Shears And more.. Want to be able to use projection transforms ITCS 3050:Game Engine Programming 1 Geometric Transformations

2 Implementing Transformations We use affine (linear) transforms Why? Can be represented using matrices Can be composed to form complex transforms. Need to transform vertices only, and use (fast) scanconversion for rasterization. Can do a lot (not everything) with matrices Use 3 3 and 4 4 matrices ITCS 3050:Game Engine Programming 2 Geometric Transformations

3 Affine Transform: Definition Point P (P x, P y, P z ) is transformed into Q(Q x, Q y, Q z ) as follows: Q x Q y = ap x + dp y + gp z + T x = bp x + ep y + hp z + T y Q z = cp x + fp y + kp z + T z Q x Q y = a d g b e h P x P y + T x T y Q z c f k P z T z Q = M P + T ITCS 3050:Game Engine Programming 3 Geometric Transformations

4 Primitive Transformations: Translation Q = M P + T M = T = Q x = Q y Q z = I T x T y T z P x + T x P y + T y P z + T z (Identity) Note: Translation does not fit within a 3 3 matrix representation! ITCS 3050:Game Engine Programming 4 Geometric Transformations

5 Primitive Transformations: Scale M = Q x = Q y Q z S x S y 0, T = [ ] 0 0 S z P xs x P y S y P z S z S x = S y = S z : Uniform Scaling Else, Differential Scaling ITCS 3050:Game Engine Programming 5 Geometric Transformations

6 Primitive Transformations:Rotation M z = cosθ sinθ 0 Sinθ Cosθ 0 T = [ ] Q = M z P + T = P x Cosθ P y Sinθ = P x Sinθ + P y Cosθ Q x Q y Q z = P z Y Q y Q P y O θ R φ Exercise: Derive the above 2D rotation matrix. ITCS 3050:Game Engine Programming 6 Geometric Transformations R Q x P x P X

7 Example: Transforming a Circle Assume a circle with origin at (0,0,0) and unit radius gltranslatef(8,0,0); RenderCircle(); gltranslatef(3,2,0); glscalef(2,2,2); RenderCircle(); Y gltranslatef(3,2,0) glscalef (2,2,2) gltranslatef (8,0,0) X ITCS 3050:Game Engine Programming 7 Geometric Transformations

8 Homogeneous Coordinate Representation Motivation Translation: Q = P + T (Tx, T y, T z ) Rotation: Scale: Q = R(θ) P Q = S(Sx, S y, S z ) P Translation involves a vector addition instead of a vector-matrix multiplication. Better if all transforms use vector-matrix multiply. ITCS 3050:Game Engine Programming 8 Geometric Transformations

9 Example: Homogeneous Coordinates in 2D Add a third coordinate w: P 2d = (x, y) = P H = (x, y, w) To convert from P H to P 2D, project to w = 1 (divide by w coordinate and discard the 3rd coordinate). w = 0 represents points at infinity. W P (x, y, w) W = 1 Plane (x/w, y/w,1) X Y ITCS 3050:Game Engine Programming 9 Geometric Transformations

10 Homogeneous Form: Translation T H (T x, T y, T z ) = T x T y T z Thus T x T y T z P x P y P z 1 = P x + T x P y + T y P z + T z 1 ITCS 3050:Game Engine Programming 10 Geometric Transformations

11 Homogeneous Form: Scale S x S S H (S x, S y, S z ) = y S z ITCS 3050:Game Engine Programming 11 Geometric Transformations

12 Homogeneous Form: Rotations Rotation about X, Y, Z R z (θ) = R x (θ) = R y (θ) = Cosθ Sinθ 0 0 Sinθ Cosθ Cosθ Sinθ 0 0 Sinθ Cosθ Cosθ 0 Sinθ Sinθ 0 Cosθ ITCS 3050:Game Engine Programming 12 Geometric Transformations

13 Shear Transform 2D: Y X X Shear (h > 0) (Q x Q y ) = (P x + hp y, gp x + P y ) = [ ] [ ] 1 h Px g 1 P y 3D: M sh = 1 h yx h zx 0 h xy 1 h zy 0 h xz h yz where h xy = shear along Y axis due to X, etc. ITCS 3050:Game Engine Programming 13 Geometric Transformations

14 Reflection Transform 2D: M y = [ ] 1 0, M 0 1 x = [ ] B Y B A A C C X 3D: (Reflection about a plane) M z = ITCS 3050:Game Engine Programming 14 Geometric Transformations

15 Affine Transform Inverses T 1 (T x, T y, T z ) = T ( T x, T y, T z ) S 1 (S x, S y, S z ) = S(1/S x, 1/S y, 1/S z ) R 1 z (θ) = R T = Cosθ Sinθ 0 0 Sinθ Cosθ In general, MM 1 = M 1 M = I ITCS 3050:Game Engine Programming 15 Geometric Transformations

16 Transformation About a Reference Point Y x y z r r r Y M X X Translate (x r, y r, z r ) to the origin. Transform. Invert translation M = T (x r, y r, z r ).S(S x, S y, S z ).T ( x r, y r, z r ) For rotation about a point, replace scale by a rotation transform. ITCS 3050:Game Engine Programming 16 Geometric Transformations

17 Composing Affine Transforms If M 1, M 2,..., M n are affine transforms to be applied in succession to a point P. Q = (M n.(m n 1...(M 2.(M 1.P )...) = (M n.m n 1...M 3.M 2.M 1 )P For transforming large numbers of points, the matrix product (M 1.M 2.M 3...M n 1.M n ) needs to be performed only once. Y Y M comp X X M comp = T (T x, T y, T z ).S(S x, S y, S z ).R(θ).T ( T x, T y ) ITCS 3050:Game Engine Programming 17 Geometric Transformations

18 Transforming Direction Vectors Motivation: Need to transform (vertex) normals, light direction Matrices used to transform points, lines cannot be used to transform direction vectors. Must use the transpose of the inverse of geometry transforming matrices. Translations do not affect normals, and rotations are orthogonal (R 1 = R T ) N = (M 1 ) T Y Original Y Incorrect Y Correct poly normal 0 X X X Scale by 0.5 along X ITCS 3050:Game Engine Programming 18 Geometric Transformations

19 The Euler Transform Euler Transform : a means to describe orientations Given a view down the negative Z axis, with up in the Y and X to the right, E(h, p, r) = R z (r)r x (p)r y (h) with (h, p, r) referring to the head, pitch, roll angles, and E 1 = E T ITCS 3050:Game Engine Programming 19 Geometric Transformations

20 The Euler Transform (contd) Gimbal lock, the loss of a degree of freedom may occur. For instance, h = 0, p = π/2radians, cos(r + h) 0 sin(r + h) E(h, π/2, r) = sin(r + h) 0 cos(r + h) 010 which is a function of only one angle. Useful to extract the Euler angles from a composite rotation matrix, F = f 00 f 01 f 02 f 10 f 11 f 12 = R z (r)r x (p)r y (h) f 20 f 21 f 22 Expand RHS and solve (details, Section 3.2.2) ITCS 3050:Game Engine Programming 20 Geometric Transformations

21 Quaternions Invented by Sir William Rowan Hamilton in 1843 Extension of complex numbers Introduced to computer graphics by Ken Shoemake in 1985 A compact, efficient means to representing and interpolating orientations. Superior to both Euler angles and matrices, especially for rotations. ITCS 3050:Game Engine Programming 21 Geometric Transformations

22 Quaternions: Definitions A quaternion ˆq has 4 components, ˆq = (q v, q w ) = iq x + jq y + kq z + q w = q v + q w i 2 = j 2 = k 2 = 1, jk = kj = i, ki = ik = j, ij = ji = k q w is the real part, and q v the imaginary part. All normal vector operations (addition, scale, dot and cross produces) can be performed on q v. Multiplication: ˆqˆr = (iq x + jq y + kq z + q w )(ir x + jr y + kr z + r w ) = (q v r v + r w q v + q w r v, q w r w q v r v ) ITCS 3050:Game Engine Programming 22 Geometric Transformations

23 Quaternions: Definitions (contd) Addition: ˆq + ˆr = (q v + r v, q w + r w ) Conjugate:ˆq = (q v, q w ) = ( q v, q w ) Norm: n(ˆq) = ˆqˆq = ˆq ˆq = q v q v + q 2 w = q 2 x + q 2 y + q 2 z + q 2 w Identity: î = (0, 1) Multiplicative Inverse: n(ˆq) 2 = ˆqˆq ˆqˆq n(ˆq) = 1 2 ˆq 1 1 = n(ˆq) 2 ˆq Other Properties: Conjugate, Norm rules, Linearity, Associativity, log, power. ITCS 3050:Game Engine Programming 23 Geometric Transformations

24 Unit Quaternions A quarternion ˆq, with n(ˆq) = 1. ˆq can be written as ˆq = (sinφu q, cosφ) = sinφu q + cosφ Given u q = 1, n(ˆq) = sin 2 φ(u q u q ) + cos 2 φ = 1 Quaternions are perfect for creating rotations and orientations. ITCS 3050:Game Engine Programming 24 Geometric Transformations

25 Rotation About Axis Unit quaternions can represent any 3D rotation Very compact, efficient rotation To rotate p about u q, put p = (p x p y p z p w ) T into a unit quaternion, sinφu q, cosφ Result: ˆq ˆpˆq 1 rotates ˆp around u q by the angle 2φ radians Note: q 1 = ˆq. ITCS 3050:Game Engine Programming 25 Geometric Transformations

26 Rotation About Axis: Traditional Method Goal: To rotate the vector r by θ about n, Rr = R(θ, n) θ r R r V r θ V r Rr r r n n Decompose r into r and r, parallel and perpendicular to n r r = (n r)n = r (n r)n ITCS 3050:Game Engine Programming 26 Geometric Transformations

27 Rotation About Axis: Traditional Method θ r R r V r θ V r Rr r r n n Determine Rr by computing V, orthogonal to r and n: Thus, V = n r Rr = (cosθ)r + (sinθ)v ITCS 3050:Game Engine Programming 27 Geometric Transformations

28 Rotation About Axis: Traditional Method θ r R r V r θ V r Rr r r n n Thus Rr = Rr + Rr = Rr + (cosθ)r + (sinθ)v = (n r)n + cosθ(r (n r)n) + (sinθ)n r = (cosθ)r + (1 cosθ)n(n r) + (sinθ)n r ITCS 3050:Game Engine Programming 28 Geometric Transformations

29 Rotation Using Quaternions T he same rotation, Rr = R(2θ, n) can be performed using the (unit) quaternion product ˆq ˆpˆq 1 Assume ˆq = (q v, q w ) ˆq is a unit quarternion ˆq = (sinθn, cosθ), n = 1 Compute the product ˆq ˆpˆq 1. Can be shown to be ˆq ˆpˆq 1 = (q 2 w v v)r + 2v(v r) + 2s(v r, 0) = (cos 2 θ sin 2 θ)r + 2sin 2 θn(n r) + 2cosθsinθ(n r)) = (rcos2θ + (1 cos2θ)n(n r) + sin2θ(n r), 0) ITCS 3050:Game Engine Programming 29 Geometric Transformations

30 Derivation: ˆq ˆpˆq 1 Given Compute pq 1 : Remember p = (0, r) q = (s, v) q 1 = ˆq = (s, v) ˆp(s, v)ˆq(s, v ) = s 2 vv, v v + sv + s v Thus, pq 1 = (0, r)(s, v) = (0 + r v), r ( v) + s.r ITCS 3050:Game Engine Programming 30 Geometric Transformations

31 Derivation: ˆq ˆpˆq 1 (contd) Compute q(pq 1 ): q(pq 1 ) = (s, v)(r v, r ( v) + sr) = (s(r v) v ( r v) s(v r)), (v ( r v)) + s(v r) + s( r v) + s 2 r + (r v) v = (0, (v v) ( r) + (v r) v + s(v r) + s( r v) + s 2 r + (r v) v = (s 2 v v)r + 2s(v r) + 2(v r)v For unit quaternions, s = cosθ, v = nsinθ, thus q(pq 1 ) = (cos 2 θ sin 2 θ)r + 2nsinθ(rnsinθ) + 2cosθ(nsinθ r) = rcos(2θ) + n(1 cos(2θ)) + sin(2θ)(n v) Identical to the traditional method, except for a factor of 2, easily adjusted by redefining q = (nsin θ 2, cosθ 2 ) ITCS 3050:Game Engine Programming 31 Geometric Transformations

32 Traditional Rotation vs Quaternion Rotation: Summary Expression obtained using traditional method: Rr = (cosθ)r + (1 cosθ)n(n r) + (sinθ)n r Expression obtained using quaternion rotation: Rr = (rcos(2θ)r + (1 cos(2θ))n(n r) + sin(2θ)(n r)) ITCS 3050:Game Engine Programming 32 Geometric Transformations

33 Quaternions in Matrix Form M q = 1 s(qy 2 + qz) 2 s(q x q y q w q z ) s(q x q z + q w q y ) 0 s(q x q y + q w q z ) 1 s(qx 2 + qz) 2 s(q y q z q w q x ) 0 s(q x q z q w q y ) s(q y q z + q w q x ) 1 s(qx 2 + qy) For unit quaternions, s = 2/n(ˆq), 1 2(qy 2 + qz) 2 2(q x q y q w q z ) 2(q x q z + q w q y ) 0 M q = 2(q x q y + q w q z ) 1 2(qx 2 + qz) 2 2(q y q z q w q x ) 0 2(q x q z q w q y ) 2(q y q z + q w q x ) 1 2(qx 2 + qy) ITCS 3050:Game Engine Programming 33 Geometric Transformations

34 Rotation from One Vector to Another To rotate from vector s to vector t Method: Normalize s and t. Compute unit rotation axis, u = s t s t e = s t = cos(2θ), s t = sin(2θ), 2φ, the angle between the vectors Quaternion ˆq = (sinφu, cosφ) With optimizations (refer text) e + hv 2 x hv x v y v z hv x v z + v y 0 hv R(s, t) = x v y + v z e + hvy 2 hv y v z v x 0 hv x v z v y hv y v z + v x e + hvz where h = 1 1 e ITCS 3050:Game Engine Programming 34 Geometric Transformations

35 Spherical Linear Interpolation Smooth interpolation of quaternions, from ˆq to ˆr. Useful in animating objects This is done by the following: ŝ(ˆq, ˆr, t) = (ˆrˆq 1 ) tˆq, Algebraic Form sin(φ(1 t)) ŝ(ˆq, ˆr, t) = slerp(ˆq, ˆr, t) = sinφ ˆq + sin(φt) sinφ ˆr The second form is more useful, effectively interpolating over a 4D sphere at fixed constant speed (geodesic interpolation). ITCS 3050:Game Engine Programming 35 Geometric Transformations

36 Application: Camera Positioning and Orientation Given camera positioned at (0, 0, 0) T and looking down v = (0, 0, 1) T Goal: To move lookat direction to w and move camera to a new position p Accomplished by X = T(p)R(v, w) Usually, will need to rotate the camera up direction to something more desirable. ITCS 3050:Game Engine Programming 36 Geometric Transformations

37 Rotation about an Arbitrary Axis s y r x M s y r M T x s y r x z t z t z t Sometimes, useful to rotate about an arbitrary axis, r Need two additional axes to form a basis, then change bases. Change from standard basis to new basis, rotate by given angle, and transform back ITCS 3050:Game Engine Programming 37 Geometric Transformations

38 Rotation about an Arbitrary Axis:Methodology Find orthonormal axes of basis: Determining s: Set smallest component of r to zero, swap the remaining two terms and negate the first, (0, r x, r y ) r x < r z, r x < r z s = ( r z, 0, r x ) r y < r x, r y < r z ( r y, r x, 0) r z < r x, r z < r y s = s/ s t = r s Rotation to standard basis: M = Final Transform: X = M T R x (α)m rt s T t T ITCS 3050:Game Engine Programming 38 Geometric Transformations