UW EXTENSION CERTIFICATE PROGRAM IN GAME DEVELOPMENT 2 ND QUARTER: ADVANCED GRAPHICS Advanced Math

Goals 1. Learn more advanced math concepts 2. Review the basic physics formulas

lerp (Linear intERPolation)  Very common operation, appears everywhere  V a = V 0 *(1-a) + V 1 *a  More complex interpolations often expressed using lerps  For example, Bezier curves are composition of lerps  The problem: lerp doesn’t work with matrices  Resulting matrix is not a rotation  It works, sort of, with quaternions  Need to renormalize afterwards  Speed is not constant

Logarithmic space  What’s halfway between a scale of 1 and a scale of 100?  Translation composes by addition: T ab = T a + T b  Standard lerp works (arithmetic weighted average)  But scaling composes by multiplication: S ab = S a * S b  Lerp should be: S a = S 0 (1-a) * S 1 a (geometric weighted average)  Logarithms to the rescue:  S a = exp( log(S 0 ) * (1-a) + log(S 1 ) * a )  Exp and log can be costly, but so can arbitrary powers

Quaternions  4D vectors used to represent rotations  Always normalized! Q = [Q X Q Y Q Z Q W ]  Two quaternions Q and –Q represent the same rotation  Addition, subtraction: per-component  Scalar product: multiply all components with scalar  Used for linear interpolation, renormalize afterwards  Quaternion product  Follows a pattern similar to complex number multiplication

Quaternions  Often represented as vector and scalar: Q = [Q V Q W ] Q V = [Q X Q Y Q Z ]  Concatenate quaternions: Q 12 = Q 1 * Q 2  Inverse: Q -1 = [-Q V Q W ] ≈ [Q V -Q W ]  Rotate a vertex: V = [V X V Y V Z 0]  Q * V * Q -1  Interpretation: Q W = cos( θ /2) Q V = A * sin( θ /2)  Rotation of θ around axis A  Used because they help make animations smoother

slerp (Spherical LERP)  For rotations, it is done using quaternions  That’s the main reason for their existence, besides the compactness  It is a complex operation:  Extract angle Ω between 4D quaternions  Standard slerp:  Be careful with quaternion duplicity (Q ≈ -Q)  Watch out for math singularities

Quaternion logarithms  Rotations also compose by multiplying  And we do have quaternion logarithms!  Q = [V*sin( θ /2) cos( θ /2)]  log(Q) = [V * θ /2 0]  It can also encode multiple loops  It’s like a 3D rotation “angle”  It can be tricky to use, like angles but worse  log(Q) ≈ log(Q) + K * π  Finding shortest route can be very tricky

Spherical harmonics  It’s like a polynomial, but instead of giving values for values, it gives values for directions  Arbitrary number of coefficients  More coefficients  more detail  Coefficients must be a square: 1, 4, 9, 16, 25, …  Can be rotated by operating on the coefficients  Integrate (average) a function: choose 1 st coefficient  Integrate a product: dot-product of coefficients  Used for lighting

Basic physics  The Newtonian world is continuous  Ballistics: P = P 0 + V 0 * t + A/2 * t 2  We use discrete approximations  Ballistics: P += V * Δ t, V += A * Δ t  We call this “Euler integration”  of the differential equations:  V = P ′, A = P ″  The acceleration changes between frames  A(P, V, t), works well with Euler integration P = position P 0 = initial position V = velocity V 0 = initial velocity A = acceleration t = elapsed time Δ t = time step

Force fields  Acceleration basically same as force  F(P, V, t) = m * A(P, V, t), but the mass is typically constant  Constant force (gravity): A = 9.8 m/s 2, F = m * A  Position-dependent force (spring): F = K * (C - P)  C = spring’s resting position  Velocity-dependent force (drag): F = -K * V 2  Time-dependent force (wind): F = F(t)

Midpoint integration  Euler integration isn’t very precise  It can diverge and misbehave  Midpoint is simple and can behave better:  P += V * Δ t + A * Δ t/2, V += A * Δ t  Perfect results for ballistic trajectories P = position P 0 = initial position V = velocity V 0 = initial velocity A = acceleration t = elapsed time Δ t = time step midpointEuler

Runge-Kutta integration  It’s more complex but better behaved  P += V k * Δ t, V += A k * Δ t  V k = (V a + 2 * V b + 2 * V c + V d ) / 6  A k = (A a + 2 * A b + 2 * A c + A d ) / 6  Where:  P a = P, V a = V, A a = A(P a, V a, t)  P b = P + V a * Δ t/2, V b = V + A a * Δ t/2, A b = A(P b, V b, t + Δ t/2)  P c = P + V b * Δ t/2, V c = V + A b * Δ t/2, A c = A(P c, V c, t + Δ t/2)  P d = P + V c * Δ t, V d = V + A c * Δ t, A d = A(P d, V d, t + Δ t) P = position P 0 = initial position V = velocity V 0 = initial velocity A = acceleration t = elapsed time Δ t = time step