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Game MathematicsGame Mathematics

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MatricesMatrices VectorsVectors Fixed-point Real NumbersFixed-point Real Numbers Triangle MathematicsTriangle Mathematics Intersection IssuesIntersection Issues Euler AnglesEuler Angles Angular DisplacementAngular Displacement QuaternionQuaternion Differential Equation BasicsDifferential Equation Basics

Essential Mathematics for Game DevelopmentEssential Mathematics for Game Development

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Matrix basicsMatrix basics– DefinitionDefinition

– TransposeTranspose

– AdditionAddition

MatricesMatrices

A = (aij) =

a11 .. a1n

. . . .am1 .. amn

C = A T cij = aji

C = A + B cij = aij + bij

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– Scalar-matrix multiplicationScalar-matrix multiplication

– Matrix-matrix multiplicationMatrix-matrix multiplication

C = A cij = aij

C = A B cij = aikbkj

k = 1

r

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Transformations in Transformations in MatrixMatrix form form– A point or a vector is a row matrix (de facto convention)A point or a vector is a row matrix (de facto convention)

V = [x y z]

– Using matrix notation, a point Using matrix notation, a point VV is transformed under translation, is transformed under translation, scaling and rotation as :scaling and rotation as :

V’ = V + DV’ = VSV’ = VR

where D is a translation vector andS and R are scaling and rotation matrices

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– To make translation be a linear transformation, we introduce the To make translation be a linear transformation, we introduce the homogeneous coordinate systemhomogeneous coordinate system

V (x, y, z, w)

where w is always 1

– Translation TransformationTranslation Transformationx’ = x + Tx

y’ = y + Ty

z’ = z + Tz

V’ = VT

[x’ y’ z’ 1] = [x y z 1]

= [x y z 1] T

1 0 0 00 1 0 00 0 1 0Tx Ty Tz 1

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– Scaling TransformationScaling Transformation

x’ = xSx

y’ = ySy

z’ = zSz

V’ = VS

[x’ y’ z’ 1] = [x y z 1]

= [x y z 1] S

Sx 0 0 00 Sy 0 00 0 Sz 00 0 0 1

Here Sx, Sy and Sz are scaling factors.

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– Rotation TransformationsRotation Transformations

1 0 0 00 cos sin 00 -sin cos 00 0 0 1

Rx =

Ry =

Rz =

cos 0 -sin 0 0 1 0 0sin 0 cos 00 0 0 1

cos sin 0 0-sin cos 0 0 0 0 1 0 0 0 0 1

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– Net Transformation matrixNet Transformation matrix

– Matrix multiplication are Matrix multiplication are not commutativenot 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

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A vector is an entity that possesses A vector is an entity that possesses magnitudemagnitude an and d directiondirection..

A 3D vector is a triple :A 3D vector is a triple :– VV = (v = (v11, v, v22, v, v33)), where each component , where each component vvii is a scalar. is a scalar.

A ray (directed line segment), that possesses A ray (directed line segment), that possesses positionposition, , magnitudemagnitude and and directiondirection..

VectorsVectors

(x1,y1,z1)

(x2,y2,z2)

V = (x2-x1, y2-y1, z2-z1)

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AddAddiition of vectorstion of vectors

Length of vectorsLength of vectors

X = V + W = (x1, y1, z1) = (v1 + w1, v2 + w2, v3 + w3)

V

W

V + W

V

W

V + W

|V| = (v12 + v2

2 + v32)1/2

U = V / |V|

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Cross product of vectorsCross product of vectors– DefinitionDefinition

– ApplicationApplication» A normal vector to a polygon is calculated from 3 (non-collinear) A normal vector to a polygon is calculated from 3 (non-collinear)

vertices of the polygon.vertices of the polygon.

X = V X W = (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)

NpV2

V1

polygon defined by 4 points

Np = V1 X V2

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» Normal vector transformationNormal vector transformation

N(X) = detJ J-1T N(x)

where X = F(x) J the Jacobian matrix, Ji(x) =

F(x)xi

(take scaling as example)

"Global and Local Deformations of Solid Primitives" Alan H. BarrComputer Graphics Volume 18, Number 3 July 1984

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Dot product of vectorsDot product of vectors– DefinitionDefinition

– ApplicationApplication

|X| = V . W = v1w1 + v2w2 + v3w3

V

W

cos =V . W

|V||W|

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Fixed Point Arithmetics : N bits (signed) IntegerFixed Point Arithmetics : N bits (signed) Integer– Example : N = 16 gives range –32768 Example : N = 16 gives range –32768 ă ă 32767 32767– We can use fixed scale to get the decimalsWe can use fixed scale to get the decimals

Fixed Point Arithmetics (1/2)Fixed Point Arithmetics (1/2)

a = ă / 28

1 1 18 integer bits

8 fractional bits

ă = 315, a = 1.2305

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Multiplication then Requires RescalingMultiplication then Requires Rescaling

Addition just Like Normal Addition just Like Normal

Fixed Point Arithmetics (2/2)Fixed Point Arithmetics (2/2)

e = a.c = ă / 28 . ĉ / 28

ĕ = (ă . ĉ) / 28

e = a+c = ă / 28 + ĉ / 28

ĕ = ă + ĉ

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Compression for Floating-point Real Compression for Floating-point Real NumbersNumbers– 4 bits reduced to 2 bits4 bits reduced to 2 bits– Lost some accuracy but affordableLost some accuracy but affordable– Network data transferNetwork data transfer

Software 3D RenderingSoftware 3D Rendering

Fixed Point Arithmetics - ApplicationFixed Point Arithmetics - Application

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h

ha

hb

hc

Aa

Ac

Ab

h = ha + hb + hc

where A = Aa + Ab + Ac

If (Aa < 0 || Ab < 0 || Ac < 0) thanthe point is outside the triangle

“Triangular Coordinate System”

Aa Ab Ac

A A A

p

(xa,ya,za)

(xb,yb,zb)

(xc,yc,zc)

Triangular Coordinate SystemTriangular Coordinate System

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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)

Triangle Area – 2DTriangle Area – 2D

(xa,ya,za)

(xb,yb,zb)

(xc,yc,zc)

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Area of a triangle in 3D A = ½ (N. Sum(Pi1 cross Pi2)) where (i1, i2) = (a,b), (b,c), (c,a)

Triangle Area – 3DTriangle Area – 3D

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;}

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Terrain FollowingTerrain Following Hit TestHit Test Ray CastRay Cast Collision DetectionCollision Detection

Triangular Coordinate System - ApplicationTriangular Coordinate System - Application

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Ray CastRay Cast Containment TestContainment Test

IntersectionIntersection

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Ray Cast – The RayRay Cast – The Ray

x = x0 + (x1 – x0) ty = y0 + (y1 – y0) t, t = 0,z = z0 + (z1 – z0) t

{

Shot a Ray to Calculate the Intersection of the Shot a Ray to Calculate the Intersection of the Ray with ModelsRay with Models

Use Parametric Equation for a RayUse Parametric Equation for a Ray

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When t = 0, the Ray is on the Start Point When t = 0, the Ray is on the Start Point (x(x00,y,y00,z,z00))

Only the t Only the t 0 is the Answer Candidate 0 is the Answer Candidate The Smallest Positive t is the AnswerThe Smallest Positive t is the Answer

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Ray Cast – The PlaneRay Cast – The Plane

Each Triangle in the Models has its Plane Each Triangle in the Models has its Plane EquationEquation

UUse se ax + by + cz + d = 0ax + by + cz + d = 0 as the Plane Equation as the Plane Equation ((a, b, c)a, b, c) is the Plane Normal Vector is the Plane Normal Vector |d||d| is the Distance of the Plane to Origin is the Distance of the Plane to Origin Substitute the Ray Equation into the PlaneSubstitute the Ray Equation into the Plane SSolve the olve the tt to Find the Intersect to Find the Intersect CCheck the Intersect Point Within the Triangle or heck the Intersect Point Within the Triangle or

not by Using “Triangle Area Test” (p. 154)not by Using “Triangle Area Test” (p. 154)

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Intersection = 1, inside

Intersection = 2, outside

Intersection = 0, outside

Trick : Parametric equation for a ray which is parallel to the x-axis

x = x0 + t y = y0 , t = 0,

{ 8

(x0, y0)

2D Containment Test2D Containment Test

“ if the No. of intersection is odd, the point is inside, otherwise, is outside”

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3D Containment Test3D Containment Test

“ if the No. of intersection is odd, the point is inside, otherwise, is outside”

Same as the 2D containment testSame as the 2D containment test

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A A rotation is described as a sequence of rotation is described as a sequence of rotations about three mutually orthogonal rotations about three mutually orthogonal coordinates axes fixed in spacecoordinates axes fixed in space– X-roll, X-roll, Y-Y-roll, roll, Z-Z-rollroll

TThere are 6 possible ways to define a rotationhere are 6 possible ways to define a rotation– 3!3!

R(1, 2, ) represents an x-roll, followed by y-roll, followed by z-roll

R(1, 2, c2c3 c2s3 -s2 0 s1s2c3-c1s3 s1s2s3+c1c3 s1c2 0 c1s2c3+s1s3 c1s2s3-s1c3 c1c2 0 0 0 0 1 where si = sini and ci = cosi

Euler AnglesEuler Angles

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Interpolation happening on each angleInterpolation happening on each angle Multiple routes for interpolationMultiple routes for interpolation MMore keys for constrainsore keys for constrains

z

x

y

Rz

x

y

R

Euler Angles & InterpolationEuler Angles & Interpolation

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RR((, , nn), ), nn is the rotation axis is the rotation axis

n

r Rr

n

r

rv

rh

V

rv

V

Rrv

rh = (n.r)nrv = r - (n.r)n , rotate into position Rrv

V = nxrv = nxr

Rrv = (cos)rv + (sin)V-> Rr = Rrh + Rrv

= rh + (cos)rv + (sin)V = (n.r)n + (cos)r - (n.r)n) + (sin) nxr = (cos)r + (1-cos) n (n.r) + (sin) nxr

Angular DisplacementAngular Displacement

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Sir William Hamilton (1843)Sir William Hamilton (1843) FFrom Complex numbers (a + rom Complex numbers (a + iib), b), i i 22 = -1 = -1 16,16,October, 1843, October, 1843, Broome BridgeBroome Bridge in in DublinDublin 1 1 realreal + 3 + 3 imaginaryimaginary = 1 = 1 quaternionquaternion qq = a + b = a + bii + c + cjj + d + dkk ii22 = = jj22 = = kk22 = -1 = -1 ijij = = kk & & jiji = - = -kk, cyclic permutation , cyclic permutation ii--jj--kk--ii qq = ( = (ss, , vv), where (), where (ss, , vv) = ) = ss + + vvxxii + + vvyyjj + + vvzzkk

QuaternionQuaternion

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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

Quaternion AlgebraQuaternion Algebra

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Take a pure quaternion p = (0, r)and a unit quaternion q = (s, v) where qq = 1and 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 = (cos, sinn), |n| = 1

Rq(p) = (0, (cos2- sin2)r + 2sin2 n(n.r) + 2cossin nxr) = (0, cos2r + (1 - cos2)n(n.r) + sin2 nxr)

Conclusion :The act of rotating a vector r by an angular displacement (, n) is the same as taking this displacement, ‘lifting’ it into quaternion space, by using a unit quaternion (cos(/2), sin(/2)n)

Quaternion VS Angular DisplacementQuaternion VS Angular Displacement

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1-2y2-2z2 2xy-2wz 2xz+2wy 02xy+2wz 1-2x2-2z2 2yz-2wx 02xz-2wy 2yz+2wx 1-2x2-2y2 0 0 0 0 1

q = (w,x,y,z)

Quaternion VS Rotation MatrixQuaternion VS Rotation Matrix

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M0 M1 M2 0M3 M4 M5 0M6 M7 M8 0 0 0 0 1

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]; }

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Spherical linear interpolation, Spherical linear interpolation, slerpslerp

A

B

P

t

slerp(q1, q2, t) = q1 + q2

sin((1 - t))

sin sinsin(t)

Quaternion InterpolationQuaternion Interpolation

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Initial value problemsInitial value problems OODEDE

– Ordinary differential equationOrdinary differential equation

NNumerical solutionsumerical solutions– EEuler’s methoduler’s method– TThe midpoint methodhe midpoint method

Differential Equation BasicsDifferential Equation Basics

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An ODEAn ODE

VVector fieldector field SSolutionsolutions

– SSymbolic solutionymbolic solution– NNumerical solutionumerical solution

x = f (x, t)

where f is a known functionx is the state of the system, x is the x’s time derivative

x & x are vectorsx(t0) = x0, initial condition

.

Start here Follow the vectors …

Initial Value ProblemsInitial Value Problems

.

.

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A numerical solutionA numerical solution– A simplification from A simplification from Tayler seriesTayler series

DDiscrete time steps starting with initial valueiscrete time steps starting with initial value Simple but not accurateSimple but not accurate

– Bigger steps, bigger errorsBigger steps, bigger errors– OO((tt22) ) errorserrors

Can be unstableCan be unstable Not even efficientNot even efficient

x(t + t) = x(t) + t f(x, t)

Euler’s MethodEuler’s Method

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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 stepx = t f(x, t)

b. Evaluate f at the midpointfmid = f((x+x)/2, (t+t)/2)

c. Take a step using the midpointx(t+t) = x(t) + t fmid

. ..

a

b

c

Error term

The Midpoint MethodThe Midpoint Method

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Midpoint = Midpoint = Runge-KuttaRunge-Kutta method of order 2 method of order 2 Runge-KuttaRunge-Kutta method of order 4 method of order 4

– OO(h(h55))

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

The The Runge-KuttaRunge-Kutta Method Method

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DynamicsDynamics– Particle systemParticle system

Game FX SystemGame FX System

Initial Value Problems - ApplicationInitial Value Problems - Application

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Game GeometryGame Geometry

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GeometryGeometry– Position / vertex normals / texture Position / vertex normals / texture

coordinatescoordinates TopologyTopology

– PrimitivePrimitive» Lines / triangles / surfaces / …Lines / triangles / surfaces / …

PropertyProperty– MaterialsMaterials– TexturesTextures

MotionMotion HierarchyHierarchy

Game ModelsGame Models

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Vertex positionVertex position– (x, y, z(x, y, z, w, w))– In model space or screen spaneIn model space or screen spane

Vertex normalVertex normal– (n(nxx, n, nyy, n, nzz))

Vertex colorVertex color– (r, g, b) or (diffuse, specular)(r, g, b) or (diffuse, specular)

Texture coordinates on vertexTexture coordinates on vertex– (u(u11, v, v11), (u), (u22, v, v22), …), …

Skin weightsSkin weights– ((bonebone11, w, w11, bone, bone22, w, w22, …), …)

Geometry DataGeometry Data

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LinesLines– Line segmentsLine segments– PolylinePolyline

» Open / closedOpen / closed

Indexed trianglesIndexed triangles TTriangle Strips / Fansriangle Strips / Fans SSurfacesurfaces

– NNon-on-uuniform niform RRational ational BB SSpline (pline (NURBSNURBS)) SSubdivisionubdivision

Topology DataTopology Data

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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

v0

v3

v6

v7

Right-hand rule for indexing

polygon normal

vertex normal

Indexed TrianglesIndexed Triangles

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v0

v1

v2

v3

v4

v5

v6

v7

T0

T1

T2 T3

T4

T5

v0 , v1 , v2 , v3 , v4 , v5 , v6 , v7

Triangle StripsTriangle Strips