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Game Mathematics

Game Mathematics. Essential Mathematics for Game Development. Matrices Vectors Fixed-point Real Numbers Triangle Mathematics Intersection Issues Euler Angles Angular Displacement Quaternion Differential Equation Basics. Matrices. a 11 .. a 1n . . . . a m1 .. a mn.

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Game Mathematics

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  1. Game Mathematics

  2. Essential Mathematics for Game Development • Matrices • Vectors • Fixed-point Real Numbers • Triangle Mathematics • Intersection Issues • Euler Angles • Angular Displacement • Quaternion • Differential Equation Basics

  3. Matrices a11 .. a1n . . . . am1 .. amn • Matrix basics • Definition • Transpose • Addition A = (aij) = C = A T cij = aji C = A + B cij = aij + bij

  4. Scalar-matrix multiplication • Matrix-matrix multiplication C = aA cij = aaij r C = A B cij = Saikbkj k = 1

  5. Transformations in Matrix form • A point or a vector is a row matrix (de facto convention) V = [x y z] • Using matrix notation, a point V is transformed under translation, scaling and rotation as : V’ = V + D V’ = VS V’ = VR where D is a translation vector and S and R are scaling and rotation matrices

  6. To make translation be a linear transformation, we introduce the homogeneous coordinate system V (x, y, z, w) where w is always 1 • Translation Transformation x’ = x + Tx y’ = y + Ty z’ = z + Tz V’ = VT 1 0 0 0 0 1 0 0 0 0 1 0 Tx Ty Tz 1 [x’ y’ z’ 1] = [x y z 1] = [x y z 1] T

  7. Scaling Transformation x’ = xSx y’ = ySy z’ = zSz V’ = VS Sx 0 0 0 0 Sy 0 0 0 0 Sz 0 0 0 0 1 [x’ y’ z’ 1] = [x y z 1] = [x y z 1] S Here Sx, Sy and Sz are scaling factors.

  8. Rotation Transformations 1 0 0 0 0 cosq sinq 0 0 -sinq cosq 0 0 0 0 1 Rx = Ry = Rz = cosq 0 -sinq 0 0 1 0 0 sinq 0 cosq 0 0 0 0 1 cosq sinq 0 0 -sinq cosq 0 0 0 0 1 0 0 0 0 1

  9. Net Transformation matrix • Matrix multiplication are not commutative [x’ y’ z’ 1] = [x y z 1] M1 and [x” y” z” 1] = [x’ y’ z’ 1] M2 then the transformation matrices can be concatenated M3 = M1 M2 and [x” y” z” 1] = [x y z 1] M3 M1 M2 = M2 M1

  10. Vectors • A vector is an entity that possesses magnitude and direction. • A 3D vector is a triple : • V = (v1, v2, v3), where each component vi is a scalar. • A ray (directed line segment), that possesses position, magnitude and direction. V = (x2-x1, y2-y1, z2-z1) (x1,y1,z1) (x2,y2,z2)

  11. X = V + W • = (x1, y1, z1) • = (v1 + w1, v2 + w2, v3 + w3) • Addition of vectors • Length of vectors W V + W W V + W V V |V| = (v12 + v22 + v32)1/2 U = V / |V|

  12. X = VXW • = (v2w3-v3w2)i + (v3w1-v1w3)j + (v1w2-v2w1)k • where i, j and k are standard unit vectors : • i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) • Cross product of vectors • Definition • Application • A normal vector to a polygon is calculated from 3 (non-collinear) vertices of the polygon. Np polygon defined by 4 points V2 • Np = V1XV2 V1

  13. N(X) = detJJ-1TN(x) where X = F(x) Jthe Jacobian matrix,Ji(x) = • Normal vector transformation dF(x) dxi "Global and Local Deformations of Solid Primitives" Alan H. Barr Computer Graphics Volume 18, Number 3 July 1984 • (take scaling as example)

  14. Dot product of vectors • Definition • Application • |X| = V.W • = v1w1 + v2w2+v3w3 V q W V.W cosq = |V||W|

  15. Fixed Point Arithmetics (1/2) • Fixed Point Arithmetics : N bits (signed) Integer • Example : N = 16 gives range –32768  ă  32767 • We can use fixed scale to get the decimals a = ă / 28 8 integer bits 1 1 1 8 fractional bits ă = 315, a = 1.2305

  16. Fixed Point Arithmetics (2/2) e = a.c = ă / 28 . ĉ / 28  ĕ = (ă .ĉ) / 28 • Multiplication then Requires Rescaling • Addition just Like Normal e = a+c = ă / 28 +ĉ / 28  ĕ = ă + ĉ

  17. Fixed Point Arithmetics - Application • Compression for Floating-point Real Numbers • 4 bits reduced to 2 bits • Lost some accuracy but affordable • Network data transfer • Software 3D Rendering

  18. Triangular Coordinate System ha (xa,ya,za) Ac p Ab hb h (xb,yb,zb) Aa hc (xc,yc,zc) Aa Ab Ac h = ha + hb + hc where A = Aa + Ab + Ac If (Aa < 0 || Ab < 0 || Ac < 0) than the point is outside the triangle “Triangular Coordinate System” A A A

  19. Triangle Area – 2D Area of a triangle in 2D xa ya A = ½ xb yb xc yc xa ya = ½ (xa*yb + xb*yc + xc*ya – xb*ya – xc*yb – xa*yc) (xa,ya,za) (xb,yb,zb) (xc,yc,zc)

  20. Triangle Area – 3D Area of a triangle in 3D A = ½ (N. Sum(Pi1 cross Pi2)) where (i1, i2) = (a,b), (b,c), (c,a) float GmArea3(float *x0, float *x1, float *x2, float *n) { float area, len, sum1, sum2, sum0; len = (float) sqrt(n[0] * n[0] + n[1] * n[1] + n[2] * n[2]) * 2.0f; /* find sum of cross products */ sum0 = x1[1] * (-x0[2] + x2[2]) + x2[1] * (-x1[2] + x0[2]) + x0[1] * (-x2[2] + x1[2]); sum1 = x1[2] * (-x0[0] + x2[0]) + x2[2] * (-x1[0] + x0[0]) + x0[2] * (-x2[0] + x1[0]); sum2 = x1[0] * (-x0[1] + x2[1]) + x2[0] * (-x1[1] + x0[1]) + x0[0] * (-x2[1] + x1[1]); /* find the area */ return = (sum0 * n[0] + sum1 * n[1] + sum2 * n[2]) / len; }

  21. Triangular Coordinate System - Application • Terrain Following • Hit Test • Ray Cast • Collision Detection

  22. Intersection • Ray Cast • Containment Test

  23. Ray Cast – The Ray • Shot a Ray to Calculate the Intersection of the Ray with Models • Use Parametric Equation for a Ray x = x0 + (x1 – x0) t y = y0 + (y1 – y0) t, t = 0, z = z0 + (z1 – z0) t { 8 • When t = 0, the Ray is on the Start Point (x0,y0,z0) • Only the t  0 is the Answer Candidate • The Smallest Positive t is the Answer

  24. Ray Cast – The Plane • Each Triangle in the Models has its Plane Equation • Use ax + by + cz + d = 0 as the Plane Equation • (a, b, c) is the Plane Normal Vector • |d| is the Distance of the Plane to Origin • Substitute the Ray Equation into the Plane • Solve the t to Find the Intersect • Check the Intersect Point Within the Triangle or not by Using “Triangle Area Test” (p. 154)

  25. 2D Containment Test Intersection = 1, inside Intersection = 2, outside (x0, y0) Intersection = 0, outside Trick : Parametric equation for a ray which is parallel to the x-axis x = x0 + t y = y0 , t = 0, { 8 “if the No. of intersection is odd, the point is inside, otherwise, is outside”

  26. 3D Containment Test • Same as the 2D containment test “if the No. of intersection is odd, the point is inside, otherwise, is outside”

  27. Euler Angles • A rotation is described as a sequence of rotations about three mutually orthogonal coordinates axes fixed in space • X-roll, Y-roll, Z-roll • There are 6 possible ways to define a rotation • 3! R(q1, q2, q3) represents an x-roll, followed by y-roll, followed by z-roll R(q1, q2, q3) = c2c3 c2s3 -s2 0 s1s2c3-c1s3 s1s2s3+c1c3 s1c2 0 c1s2c3+s1s3 c1s2s3-s1c3 c1c2 0 0 0 0 1 where si = sinqi and ci = cosqi

  28. Euler Angles & Interpolation • Interpolation happening on each angle • Multiple routes for interpolation • More keys for constrains R z z y y x x R

  29. Angular Displacement • R(q, n), n is the rotation axis rh = (n.r)n rv = r - (n.r)n , rotate into position Rrv V = nxrv = nxr Rrv = (cosq)rv + (sinq)V -> Rr = Rrh + Rrv = rh + (cosq)rv + (sinq)V = (n.r)n + (cosq)(r - (n.r)n) + (sinq) nxr = (cosq)r + (1-cosq) n(n.r) + (sinq) nxr V rv q r r Rr rh n n V rv q Rrv

  30. Quaternion • Sir William Hamilton (1843) • From Complex numbers (a + ib), i 2 = -1 • 16,October, 1843, Broome Bridge in Dublin • 1 real + 3 imaginary = 1 quaternion • q = a + bi + cj + dk • i2 = j2 = k2 = -1 • ij = k & ji = -k, cyclic permutation i-j-k-i • q = (s, v), where (s, v) = s + vxi + vyj + vzk

  31. Quaternion Algebra q1=(s1, v1)and q2=(s2, v2) q3= q1q2 = (s1s2 - v1.v2 , s1v2 + s2v1 + v1xv2) Conjugate of q = (s, v), q = (s, -v) qq = s2 + |v|2 = |q|2 A unit quaternion q = (s, v), where qq = 1 A pure quaternion p = (0, v) Noncommutative

  32. Quaternion VS Angular Displacement Take a pure quaternion p = (0, r) and a unit quaternion q = (s, v) where qq = 1 and define Rq(p) = qpq-1 where q-1 = q for a unit quaternion Rq(p) = (0, (s2 - v.v)r + 2v(v.r) + 2svxr) Let q = (cosq, sinq n), |n| = 1 Rq(p) = (0, (cos2q - sin2q)r + 2sin2qn(n.r) + 2cosqsinqnxr) = (0, cos2qr + (1 - cos2q)n(n.r) + sin2qnxr) Conclusion : The act of rotating a vector r by an angular displacement (q, n) is the same as taking this displacement, ‘lifting’ it into quaternion space, by using a unit quaternion (cos(q/2), sin(q/2)n)

  33. Quaternion VS Rotation Matrix 1-2y2-2z2 2xy-2wz 2xz+2wy 0 2xy+2wz 1-2x2-2z2 2yz-2wx 0 2xz-2wy 2yz+2wx 1-2x2-2y2 0 0 0 0 1 q =(w,x,y,z)

  34. float tr, s; tr = m[0] + m[4] + m[8]; if (tr > 0.0f) { s = (float) sqrt(tr + 1.0f); q->w = s/2.0f; s = 0.5f/s; q->x = (m[7] - m[5])*s; q->y = (m[2] - m[6])*s; q->z = (m[3] - m[1])*s; } else { float qq[4]; int i, j, k; int nxt[3] = {1, 2, 0}; i = 0; if (m[4] > m[0]) i = 1; if (m[8] > m[i*3+i]) i = 2; j = nxt[i]; k = nxt[j]; s = (float) sqrt((m[i*3+i] - (m[j*3+j] + m[k*3+k])) + 1.0f); qq[i] = s*0.5f; if (s != 0.0f) s = 0.5f/s; qq[3] = (m[j+k*3] - m[k+j*3])*s; qq[j] = (m[i+j*3] + m[j+i*3])*s; qq[k] = (m[i+k*3] + m[k+i*3])*s; q->w = qq[3]; q->x = qq[0]; q->y = qq[1]; q->z = qq[2]; } M0 M1 M2 0 M3 M4 M5 0 M6 M7 M8 0 0 0 0 1

  35. Quaternion Interpolation • Spherical linear interpolation, slerp A P B t f sin((1 - t)f) sin(tf) slerp(q1, q2, t) = q1 + q2 sinf sinf

  36. Differential Equation Basics • Initial value problems • ODE • Ordinary differential equation • Numerical solutions • Euler’s method • The midpoint method

  37. Initial Value Problems • An ODE • Vector field • Solutions • Symbolic solution • Numerical solution . x = f (x, t) where fis a known function x is the state of the system, x is the x’s time derivative x & x are vectors x(t0) = x0, initial condition . . Follow the vectors … Start here

  38. Euler’s Method x(t + Dt) = x(t) + Dt f(x, t) • A numerical solution • A simplification from Tayler series • Discrete time steps starting with initial value • Simple but not accurate • Bigger steps, bigger errors • O(Dt2) errors • Can be unstable • Not even efficient

  39. The Midpoint Method Error term . .. Concept : x(t0 + h) = x(t0) + h x(t0) + h2/2 x(t0) + O(h3) Result : x(t0+h) = x(t0) + h(f(x0 + h/2 f(x0)) Method : a. Compute an Euler step Dx = Dt f(x, t) b. Evaluate f at the midpoint fmid = f((x+Dx)/2, (t+Dt)/2) c. Take a step using the midpoint x(t+Dt) = x(t) + Dt fmid c b a

  40. The Runge-Kutta Method • Midpoint = Runge-Kutta method of order 2 • Runge-Kutta method of order 4 • O(h5) k1 = h f(x0, t0) k2 = h f(x0 + k1/2, t0 + h/2) k3 = h f(x0 + k2/2, t0 + h/2) k4 = h f(x0 + k3, t0 + h) x(t0+h) = x0 + 1/6 k1 + 1/3 k2 + 1/3 k3 + 1/6 k4

  41. Initial Value Problems - Application • Dynamics • Particle system • Game FX System

  42. Game Geometry

  43. Game Models • Geometry • Position / vertex normals / texture coordinates • Topology • Primitive • Lines / triangles / surfaces / … • Property • Materials • Textures • Motion • Hierarchy

  44. Geometry Data • Vertex position • (x, y, z, w) • In model space or screen spane • Vertex normal • (nx, ny, nz) • Vertex color • (r, g, b) or (diffuse, specular) • Texture coordinates on vertex • (u1, v1), (u2, v2), … • Skin weights • (bone1, w1, bone2, w2, …)

  45. Topology Data • Lines • Line segments • Polyline • Open / closed • Indexed triangles • Triangle Strips / Fans • Surfaces • Non-uniform Rational BSpline (NURBS) • Subdivision

  46. Indexed Triangles • Geometric data • Vertex data • v0, v1, v2, v3, … • (x, y, z, nx, ny, nz, tu, tv) • or (x, y, z, vr, vg, vb, tu, tv) • Topology • Face v0 v3 v6 v7 • Edge table polygon normal vertex normal v0 v7 v3 v6 Right-hand rule for indexing

  47. Triangle Strips v0 v6 v2 v4 T0 T4 T2 T5 T3 T1 v5 v1 v7 v3 v0 , v1 ,v2 ,v3 ,v4 ,v5 ,v6 ,v7

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