Mathematics for real-time graphics

Transcription

1 Mathematics for real-time graphics Josef Pelikán CGG MFF UK Praha Math 2018 Josef Pelikán, 1 / 65

2 Content homogeneous coordinates, matrix transformations coordinate-system conversions coordinate systems, projections, frustum perspective correct interpolation orientations Euler angles, quaternions orientation interpolation smooth interpolations and approximations spline functions, natural spline, B-spline, Catmull-Rom,... Math 2018 Josef Pelikán, 2 / 65

3 Geometric transformations in 3D Cartesian 3D coordinate vector [ x, y, z ] multiplying by a 3 3 matrix row vector multiplied from the right (DirectX) [x, y, z] [a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33] = [x ', y',z' ] column vector multiplied from the left (OpenGL) transform matrices 3 3 have serious drawback cannot do translations! Math 2018 Josef Pelikán, 3 / 65

4 Homogeneous coordinates homogeneous coordinate vector [ x, y, z, w ] transformation: multiplying by a 4 4 matrix [x, y, z, w] [a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 = [ x', y',z ', w' ] a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44] homogeneous matrix is able to translate and to do perspective projections Math 2018 Josef Pelikán, 4 / 65

5 Coordinate conversions from homogeneous coordinates [ x, y, z, w ] into Cartesian coordinates: by division (w 0) [ x/w, y/w, z/w ] coordinate vector [ x, y, z, 0 ] does not correspond to any real point in space can be interpreted as a directional vector (point in infinity) from Cartesian coordinates to homogeneous: trivial extension [ x, y, z ] [ x, y, z, 1 ] Math 2018 Josef Pelikán, 5 / 65

6 Elementary transformations Affine transformation: [a11 a12 a13 0 a 21 a 22 a 23 0 a 31 a 32 a 33 0 t 1 t 2 t 3 1 ] upper left submatrix [ a 11 to a 33 ] defines scaling, orientation and shear vector [ t 1, t 2, t 3 ] defines translation translation is performed after former transformations Math 2018 Josef Pelikán, 6 / 65

7 Normal vector transformation normal vectors must not be transformed by regular matrices (like point positions are) exception: M is rotational (orthonormal) normal-vector transformation matrix N: N = M 1 T x N x N' x N y y y Math 2018 Josef Pelikán, 7 / 65

8 Coordinate systems in OpenGL Object space (object modeling) Modeling transform (scene composition) World space (simulation) [x,y,z,w] -z View transform (camera position & view) Eye space (camera space) -z Projection transform (perspective orthographic) xy Clip space (frustum = homo cube) xy n f [x,y,z,w] -z xy -z xy Math 2018 Josef Pelikán, 8 / 65

9 Coordinate systems in OpenGL Clip space (frustum = homo cube) [x,y,z,w] Perspective divide (hardwired into HW) Viewport transform (3D scaling & translation) Normalized Device Space (frustum = cartesian cube) [x,y,z] -z Window space (actual pixel coords) [x,y,z] xy OpenGL: [-1,-1,-1] to [1,1,1] DirectX: [-1,-1, 0] to [1,1,1] [x,y] z actual screen coordinates (fragments) depth value compatible with actual depth-buffer Math 2018 Josef Pelikán, 9 / 65

10 Coordinate systems in OpenGL object space modeling of individual objects, modularity 3D modeling software (3DSMax, Blender, Rhino,..) world space absolute (real) coordinates in simulated virtual world object instantiation, collision detection, AI planning,.. camera space the whole virtual world transforms into coordinates relative to a camera center of projection: origin, view direction: -z (or z) Math 2018 Josef Pelikán, 10 / 65

11 Coordinate systems & transformations transformation model camera altogether modelview matrix world coordinates are not directly used in render. pipel. projection transformation defines visible volume = frustum [ l, r, b, t, n, f ] front & back clip distances: n, f result: homogeneous coordinate (before clipping) clip space mandatory output coordinate of vertex shader! Math 2018 Josef Pelikán, 11 / 65

12 Projection transformation far point f can be in infinity [ ] 2n r l 2n t b r l t b f n 1 r l t b f n 0 0 2fn 0 f n [ ] 2n r l 2n t b r l t b 1 1 r l t b 0 0 2n 0 xy n f -z Math 2018 Josef Pelikán, 12 / 65

13 Coordinate systems & transformations perspective division just converts homogeneous coordinates into Cartesian normalized coordinates ( NDS ) standard-sized cube/cuboid OpenGL: [-1,-1,-1] to [1,1,1] DirectX: [-1,-1,0] to [1,1,1] window coordinates ( window space ) result of linear adjustment to window size in pixels used in rasterizer and all fragment processing Math 2018 Josef Pelikán, 13 / 65

14 Rigid body transformation preserves shapes, alters orientation & position translation and rotation conversion between coordinate systems (e.g. between world-space and camera-space) z u (up) r (right) v (view) y x left-handed = clockwise ( pravotočivý in Czech) Math 2018 Josef Pelikán, 14 / 65

15 Conversion between two orientations z u O s x y t [1, 0, 0 ] M stu xyz = s [0, 1, 0 ] M stu xyz = t [0, 0, 1 ] M stu xyz = u Coordinate system has an origin O and is defined by three unit vectors [ s, t, u ] [ M stu xyz = M xyz stu s x s y s z t x t y t z u x u y u z] T = M stu xyz Math 2018 Josef Pelikán, 15 / 65

16 Euler transformation y head roll arbitrary rotation decomposed into three components Leonard Euler ( ) x -z E(h,p,r) = R y (h) R x (p) R z (r) pitch h (head, yaw): plan view direction p (pitch): forward/backward pitching r (roll): rolling around the view vector Math 2018 Josef Pelikán, 16 / 65

17 Euler transformation II result matrix of rotation E= c r c h s r s p s h s r c h c r s p s h c p s h s r c p c r c p s p c r s h s r s p c h s r s h c r s p c h c p c h s(x).. sin(x), c(x).. cos(x) backward matrix angles computation h, p, r: p.. e 23 r.. e 21 /e 22 h.. e 13 /e 33 Math 2018 Josef Pelikán, 17 / 65

18 Rotations: different conventions main convention: 1. rotation around z by 2. rotation around x' by 3. rotation around z'' by x-convention: 1. rotation around z 2. rotation around original x 3. rotation around original z more systems (24): aeronautics, gyroscopes, physics,.. Math 2018 Josef Pelikán, 18 / 65

19 Quaternions Sir William Rowan Hamilton, 16 Oct 1843 (Dublin) i 2 = j 2 = k 2 = ijk = -1 usage in graphics since 1985 (Shoemake) generalization of complex numbers in 4D space q = (v, w) = i x + j y + k z + w = v + w sometimes (w, v)! imaginary part v = (x, y, z) = i x + j y + k z i 2 = j 2 = k 2 = -1, jk = -kj = i, ki = -ik = j, ij = -ji = k Math 2018 Josef Pelikán, 19 / 65

20 Quaternions: operations I addition (v 1, w 1 ) + (v 2, w 2 ) = (v 1 +v 2, w 1 +w 2 ) multiplication q r = ( v q v r + w r v q + w q v r, w q w r - v q v r ) i q y r z q z r y r w q x q w r x, j q z r x q x r z r w q y q w r y, k q x r y q y r x r w q z q w r z, q w r w q x r x q y r y q z r z Math 2018 Josef Pelikán, 20 / 65

21 Quaternions: operations II conjugation (v, w)* = (-v, w) norm (squared absolute value) q 2 = n(q) = q q* = x 2 + y 2 + z 2 + w 2 unit i = (0, 1) reciprocal q -1 = q* / n(q) multiplication by a scalar s q = (0, s) (v, w) = (s v, s w) Math 2018 Josef Pelikán, 21 / 65

22 Unit quaternions unit quaternion can be expressed by goniometry as q = (u q sin, cos ) for some unit 3D vector u q it represents a rotation (orientation) in 3D ambiguity: both q and -q represent the same rotation! identity (zero rotation): (0, 1) (f + p) power, exponential, logarithm: q = u q sin + cos = exp ( u q ), log q = u q q t = (u q sin + cos ) t = exp (t u q ) = u q sin t + cos t Math 2018 Josef Pelikán, 22 / 65

23 Rotation using quaternion unit quaternion q = (u q sin, cos ) u q... axis of rotation,... angle u q vector (point) in 3D: p = [p x, p y, p z, 0] rotation of vector (point) p around u q by angle 2 p' = q p q -1 = q p q* Math 2018 Josef Pelikán, 23 / 65

24 Quaternion matrix conversions quaternion q converted to a matrix: M = 1 2 y 2 z 2 2 x y w z 2 x z w y 2 x y w z 1 2 x 2 z 2 2 y z w x 2 x z w y 2 y z w x 1 2 x 2 y 2 reverse conversion is based on equations ($): m 23 m 32 =4wx m 31 m 13 =4wy m 12 m 21 =4wz tr M 1 =4w 2 Math 2018 Josef Pelikán, 24 / 65

25 Matrix quaternion II 1. matrix_trace+1 has large enough absolute value: w= 1 2 tr M 1 x= m m w y= m 31 m 13 4w z= m 12 m 21 4w 2. otherwise compute a component with largest absolute value first and then apply $: 4x 2 =1 m 11 m 22 m 33 4y 2 =1 m 11 m 22 m 33 4z 2 =1 m 11 m 22 m 33 Math 2018 Josef Pelikán, 25 / 65

26 Spherical linear interpolation (slerp) two quaternions q and r real parameter 0 t 1 Interpolated quaternion (q r 0, else take -q) slerp(q,r,t) = q (q* r) t different formulation: slerp q, r,t = sin 1 t sin t q sin sin r cos =q x r x q y r y q z r z q w r w... the shortest spherical arc between q and r (quaternion splines will be explained later) Math 2018 Josef Pelikán, 26 / 65

27 Rotation between two vectors two vectors s and t: 1. normalization of s, t 2. unit rotation axis u = (s t) / s t 3. angle between s and t: e = s t = cos 2 s t = sin 2 4. final quaternion: q = ( u sin, cos ) q= q v, q w = e s t, 2 1 e 2 Math 2018 Josef Pelikán, 27 / 65

28 Slerp of rotational matrices two rotational matrices Q and R real parameter 0 t 1 Interpolated matrix slerp(q,r,t) = Q (Q T R) t technical problem:? power operation on matrices need to compute axis and angle Q T R not very efficient (for details see RotationIssues.pdf ) Math 2018 Josef Pelikán, 28 / 65

29 Rotation representation - summary rotational matrix + HW support, efficient point/vector transformation memory (float[9]), other operations are not so efficient rotational axis and angle + memory (float[4] or float[6]), similar to quaternion inefficient composition and interpolation quaternion + memory (float[4]), composition, interpolation inefficient point/vector transformation (for details see RotationIssues.pdf ) Math 2018 Josef Pelikán, 29 / 65

30 Interpolation in the screen space concerns clip-space and next spaces clip space: [ x, y, z, w ] NDS: [ x/w, y/w, z/w, w ] ( w could be preserved) window space (fragments): [ x i, y i, z i, w ] projective perspective transformation maps depth z to NDS non-linearly! nonuniform accuracy of z-buffer (distant parts could be less accurate: minimization of f/n ratio!) + interpolation of depth z can be done in the screen space linearly W-buffer instead of z-bufferu (not used any more) Math 2018 Josef Pelikán, 30 / 65

31 Perspective-correct interpolation [x 1,y 1,z 1,1/w 1,u 1 /w 1,v 1 /w 1 ] [x 2,y 2,z 2,1/w 2,u 2 /w 2,v 2 /w 2 ] w = 1 / (1/w) u = (u/w) w... [x 3,y 3,z 3,1/w 3,u 3 /w 3,v 3 /w 3 ] Linear interpolation + division: depth z is linear by itself for texture coordinates or w perspective-correct interpolation must be used: quantities 1/w, u/w, v/w,.. are linearly interpolated, [ u, v ] is then computed by division (hyperbolic interpolation) Math 2018 Josef Pelikán, 31 / 65

32 Interpolation example I Affine Correct Math 2018 Josef Pelikán, 32 / 65

33 Interpolation example II Affine Correct Math 2018 Josef Pelikán, 33 / 65

34 Approximation and interpolation approximation (e.g. B-spline) needs not to pass through control points interpolation (e.g. Catmull-Rom) curve passes through ctrl. points curve continuity G n geometric continuity of n th order (G 0 simple continuity, G 1 tangent, G 2 curvature, ) C n analytical continuity of n th order, n th derivative cont. (C 1 speed, C 2 acceleration), superior to geometric Math 2018 Josef Pelikán, 34 / 65

35 History curves in modeling industry Paul de Faget de Casteljau by Citroën (1959) Pierre Bèzier (by Renault , UNISURF) - late start, but his results were more popular application of spline function theory mostly in the USA (James Ferguson, 1964, Boeing, C 2 spline curves) spline function theory B-spline: Isaac Jacob Schoenberg, (ballistics, Aberdeen, MD, 1946) theory: Carl de Boor (also worked in General Motors) Gordon, Riesenfeld united Bèzier and B-spline curves (1972) Math 2018 Josef Pelikán, 35 / 65

36 Free-form curves I defined by a sequence of control points control polygon approximation or interpolation boundary conditions can be different controllability sometimes tangent vectors added in ctrl. pts. (Hermit) interpolation closer control locality change of single control point (one tangent vector) induces change in restricted neighborhood only Math 2018 Josef Pelikán, 36 / 65

37 Free-form curves II parametric expression (0 t 1) convex hull property curve lies in convex hull of its control polygon Cauchy condition for blending functions sufficient for convex hull property ensures affine transformation invariancy N 1 i =0 N 1 P t = i=0 w i t = 1 w i t P i P 1 P 0 P2 P 3 Math 2018 Josef Pelikán, 37 / 65

38 Splines Jay Greer Edson International Math 2018 Josef Pelikán, 38 / 65

39 Spline functions Named after elastic ruler used in ship design (pinned in several points by ducks ) definition: spline function of degree n: piece-wise polynomial (of degree n) maximum-smoothness connection: C n-1 continuity of n-1 th derivative (polyn. of degree n) global parametrization u, u 0 u u N [u 0, u 1,... u N ] individual parts are often uniformly parametrized uniform spline: t i = (u u i ) / (u i+1 u i ), 0 t i 1 Math 2018 Josef Pelikán, 39 / 65

40 Polynomial curve matrix notation: xn yn z n x P t = TC = [t n, t n 1,... t, 1 ] [ n 1 y n 1 z n 1 ] x 1 y 1 z 1 x 0 y 0 z 0 basis matrix M and vector of geometric conditions G: 1 0, k C = MG = [m ij ] i =n, j=1 [G.. k] G Math 2018 Josef Pelikán, 40 / 65 P t = TMG

41 Matrix notation of a curve P(t) = T C = T M G separation of a parameter vector (T) from polynomial basis (M) and geometric control conditions/points (G) differentiation (tangent, curvature) restricted to T control polynomial TM times geometry G cubic: n = 3, k = 4 11 m 12 m 13 m 14 Q(t )=[t 3,t 2 m,t,1] [m 21 m 22 m 23 m 24 G 2 m 31 m 32 m 33 m 34 G 3 m 41 m 42 m 43 m 44] [G1 4] G Math 2018 Josef Pelikán, 41 / 65

42 Hermite cubic curve Ferguson curve (cubic) geometry: endpoints and tangent vectors beginning (P 0 ) and end (P 1 ) of a curve tangents in beginning (T 0 ) and ending (T 1 ) points F (t)=[t 3,t 2,t, ] [ ] [P 0 P 1 1] T T Math 2018 Josef Pelikán, 42 / 65

43 Hermite cubic examples T 0 T 0 T 0 P 1 P 0 P 1 T 1 T1 P 0 P 0 P 1 T 1 Math 2018 Josef Pelikán, 43 / 65

44 More curves interpolating cubics derived from Hermite: general: Kochanek-Bartels (KB-spline, TCB cubic) special: cardinal spline, Catmull-Rom spline Akima interpolation ( Akima spline, not C 2 ) D-spline cubic another popular curves: Bèzier curves B-spline curve, Coons spline (approximation) natural spline (interpolation) Math 2018 Josef Pelikán, 44 / 65

45 Kochanek-Bartels cubic (KB-spline, TCB) derived from Hermite cubic (3DS Max, Lightwave) tangent vectors are derived from control points three additional scalar parameters (zero by default): - tension t: sharpness of a curve passing control point (absolute value of a tangent vector) - continuity c: in control points - bias b: tangent direction in control point Left and right tangent (T 0 and T 1 in local sense): L i = 1 t 1 c 1 b P 2 i P i 1 1 t 1 c 1 b P 2 i 1 P i 1 t 1 c 1 b R i = P 2 i P i 1 1 t 1 c 1 b P 2 i 1 P i Math 2018 Josef Pelikán, 45 / 65

46 Cardinal spline, Catmull-Rom spline Special cases of KB-spline: cardinal spline parameter a only (in fact relates to t, c = b = 0) T i = a P i 1 P i 1 0 a 1 Catmull-Rom spline a = t = 1/2 MG = 1 2 T i = 1 ] [ 2 P P i 1 i 1 [ Pi P i P i P i 2] Math 2018 Josef Pelikán, 46 / 65

47 Akima interpolation Alternative definition of tangent vectors for Hermite cubic: non-c 2! P i-2 Pi+2 P i-1 P i+1 P i C i T i = P i+1 P i 1 A i B i T i Math 2018 Josef Pelikán, 47 / 65

48 D-spline cubic one more variant of Hermite cubic tangent vector computed by the D-interpolation P i-1 b i P i c i P i+1 d i G =[ Pi P i 1 T i T i 1] T i b i Math 2018 Josef Pelikán, 48 / 65

49 Bèzier curves I polynomial curve of degree N N+1 control points - boundary control points define endpoints of a curve - boundary control-point pairs define tangent vectors parametric expression using Bernstein polynomials easy G 1 or C 1 connection spline-join is also possible, but much more complicated Bernstein polynomials: B n i t = n i ti 1 t n i 0 i n, 0 t 1 Math 2018 Josef Pelikán, 49 / 65

50 Bèzier curves II Cauchy condition convex combination of control points n i =0 B i n t = 1 P 1 P 2 for pro 0 t MG =[ ] [P 0 P 1 ] P 2 P 3 P 0 cubic Bèzier curve P 3 Math 2018 Josef Pelikán, 50 / 65

51 Joining Bèzier curves I G 1 connection (co-linear tangents) P 1 P 2 P 6 P 3 P 5 P 0 P 3 P 4 = k P 2 P 3 P 4 Math 2018 Josef Pelikán, 51 / 65

52 Joining Bèzier curves II C 1 connection (equal tangent vectors) P 1 P 6 P2 P 3 P 5 P 0 P 3 P 4 = P 2 P 3 P 4 Math 2018 Josef Pelikán, 52 / 65

53 Joining Bèzier curves III Quadratic spline from Bèzier segments P 1 P 4 P 5 P 8 P 2 P3 P 6 P 0 P 7 P 1 P 2 = P 2 P 3 P 3 P 4 = P 4 P 5... P 2k 1 P 2k = P 2k P 2k+1 Math 2018 Josef Pelikán, 53 / 65

54 Joining Bèzier curves IV Cubic spline from Bèzier segments P 7 P 1 P 2 P8 P 6 P 3 P 0 P 4 P 5 P 9 P 2 P 3 = P 3 P 4 P 5 P 6 = P 6 P 7... P 3k 1 P 3k = P 3k P 3k+1 Math 2018 Josef Pelikán, 54 / 65

55 De Casteljau (de Boor) algorithm geometric construction of Bèzier curve used as subdivision scheme or for computation of a specific point.. P 1 1 P 1 P 2 P 1 1 P 1 P 2 P 0 2 1/2 P 0 1 1/2 P 0 2 P 03 =L 3 =R 0 =B(1/2) P 1 2 P 2 1 1/4 1 P 0 3/4 P 0 3 =B(3/4) P 1 2 P 2 1 L 0 =P 0 P 3 =R 3 P 0 P 3 Math 2018 Josef Pelikán, 55 / 65

56 Using [S]LERP operation linear interpolation LERP (SLERP for quaternions) LERP ( A, B, t ) = A (1 t) + B t P 1 Q 1 P 2 Cubic Bèzier: 1-t R 0 Q i = LERP( P i, P i+1, t ) Q 0 t S 0 = B(t) R 1 Q 2 R i = LERP( Q i, Q i+1, t ) S i = LERP( R i, R i+1, t ) P 0 P 3 Math 2018 Josef Pelikán, 56 / 65

57 [S]LERP for quadratic interpolation 1-t Q 0 P 1 Q 1 P 2 t R 0 = B(t) Quadratic Bèzier: Q i = LERP( P i, P i+1, t ) R i = LERP( Q i, Q i+1, t ) P 0 Math 2018 Josef Pelikán, 57 / 65

58 Cubic spline function assembled from cubic polynomials neighbor polynomials have C 2 joint elastic spline-ruler (see construction) interpolating cubic spline in knot points x 0, x 1,... x n function values y 0, y 1,... y n are prescribed S x = S k x =s k,0 s k,1 x x k s k,2 x x k 2 s k,3 x x k 3 x [x k, x k 1 ], k=0, 1,..., n 1 condition A: S x k = y k k=0, 1,..., n Math 2018 Josef Pelikán, 58 / 65

59 Interpolating cubic spline condition B (C 0 continuity): S k x k 1 = S k 1 x k 1 k=0, 1,..., n 2 condition C (C 1 continuity): S k ' ' x k 1 = S k 1 x k 1 k=0, 1,..., n 2 condition D (C 2 continuity): S '' '' k x k 1 = S k 1 x k 1 k=0, 1,..., n 2 natural cubic spline has additional condition E: S ' ' x 0 = S ' ' x n = 0 Math 2018 Josef Pelikán, 59 / 65

60 Natural cubic spline interpolating spline uniquely determined by the conditions (solution of linear system of equations s k,l ) has no local property (the whole curve changes after altering one control point) open spline: conditions A, B, C, D are not sufficient, 2 more DoF additional condition E (second derivatives in endpoints) closed (cyclic) spline: x 0 = x n C and D give us missing conditions for x 0 Math 2018 Josef Pelikán, 60 / 65

61 B-spline (basis spline) free-form curve shape is defined by a sequence of control points parametric form using basis/blending functions (dependency of a curve point on control polygon) local property (only local change after altering one CP) uniform cubic B-spline (Coons curve) unified set of basis functions (cubic polynomials) nonuniform B-spline more complicated definition using knot vector 0 t i 1 Math 2018 Josef Pelikán, 61 / 65

62 Coons B-spline P 1 P 5 P 3,1 P 6 P 3,2 P 2 P 0 P 7 P 3,3 P 4 continuity C 2 sharing 3 CP between neighb. altering one CP induces change in 4 segments MG = 1 6 [ ] [Pi 1 P i P i 1 P i 2] Math 2018 Josef Pelikán, 62 / 65

63 Spline interpolation of quaternions subsequent interpolation by a sequence of oriantations q 0, q 1,... q n slerp(q i,q i+1,t) not sufficient continuity (C 0 only) a slerp(a,b,t) b squad p,a,b, q,t = slerp slerp p,q,t,slerp a,b,t, 2t 1 t p squad(p,a,b,q,t) q s i t = squad q i,a i,b i,q i 1,t slerp(p,q,t) a i = b i 1 = q i exp[ 1 log q i q i 1 log q 1 i q i 1 ] 4 Math 2018 Josef Pelikán, 63 / 65

64 Vertex blending ( skinning ) vertex p of each triangle is connected to some neighbor bones (coordinate systems) quiescent position: t = 0 bones are animated by matrices B i (t) (object world) initial transformation M i = B i (0) n p t = i= 1 w i p M i 1 B i t, n i= 1 w i = 1 B 1 (t) w 1 p(t) w 2 B 2 (t) Math 2018 Josef Pelikán, 64 / 65

65 Sources Tomas Akenine-Möller, Eric Haines: Real-time rendering, 2 nd edition, A K Peters, 2002, ISBN: J. Žára, B. Beneš, J. Sochor, P. Felkel: Moderní počítačová grafika, 2. vydání, Computer Press, 2005, ISBN: (Dave Eberly) Math 2018 Josef Pelikán, 65 / 65