2.1. ASSUMED MATHSCore mathematical underpinnings

ASSUMED MATHS: COORDINATE SYSTEMSAssumed mathematical knowledge dealing with coordinate systems

Coordinate systems

The location of a point in space can be described in terms of a coordinate system, defined using an origin reference point and a number of coordinate axes.

A coordinate system may be given relative to a parent coordinate system.

The Cartesian (rectangular) coordinate system defines coordinate axes which are perpendicular to each other.

A given set of coordinate axes spanning a space is called the frame of reference, or basis, for the space. There are infinitely many frames of reference for a given coordinate space.

See links at end for reference material if

needed

ASSUMED MATHS: VECTORSAssumed mathematical knowledge dealing with vectors

VectorsThe following vector concepts should be familiar:

Vector structure (mostly restricted to 2, 3 or 4 components).

Vector addition, subtraction, scalar multiplication and length (including normalisation)

Common vector algebraic identities

Assume u, v and w are vectors and r and s are scalars

For addition and subtraction:

u + v = v + u(u + v) + w = u + (v + w)u − v = u + (−v)−(−v) = vv + (−v) = 0v + 0 = 0 + v = v

For scalar multiplication:r(s v) = (rs) v(r + s) v = r v + s vs(u + v) = s u + s v1 v = v

VectorsDot (scalar) product and common ● algebraic identities

Assume u, and v are vectors and r and s are scalars

u · v = u1v1 + u2v2 +· · ·+unvn

u · v = |u|| v| cos θu · u = |u|2

u · v = v · uu · (v ± w) = u · v ± u · wr u · s v = rs(u · v)

VectorsCross (vector) product and ×

algebraic identities and dependency upon coordinate system ‘handedness’. A right-handed system is assumed.

Assume u, v , w and x are vectors and r and s are scalars

u × v = −(v × u)u × u = 0u · (v × w) = (u × v) · wu × (v ± w) = u × v ± u × w(u ± v) × w = u × w ± v × w|u × v| = |u|| v| sin θ(u × v) · (w × x) = (u · w)(v · x) − (v · w)(u · x) (Lagrange’s identity)r u × s v = rs(u × v)

VectorsUnderstanding that the scalar triple product, i.e. (u × v) · w or [uvw] geometrically corresponds to the signed volume of the parallelepiped formed by vectors u, v and w.

ASSUMED MATHS: MATRICESAssumed mathematical knowledge dealing with matrices

MatricesThe following matrix concepts should be familiar:

Matrix structure (mostly restricted to 3x3 or 4x4), including identity, square, row and column matrices.

Transpose of a matrix.

MatricesMatrix addition, subtraction

and multiplication

Common matrix algebraic identities

If A is an m × n matrix and B an n × p matrix, then matrix multiplication (C = AB) is defined as:

Assume A, B and C are matrices and r and s are scalars

For addition and subtraction:A + B = B + AA + (B + C) = (A + B) + CA − B = A + (−B)−(−A) = As(A ± B) = sA ± sB(r ± s)A = r A ± sAr(sA) = s(r A) = (rs)A

For multiplication:AI = IA = AA(BC) = (AB)CA(B ± C) = AB ± AC(A ± B)C = AC ± BC(sA)B = s(AB) = A(sB)

For transposition:(A ± B)T = AT ± BT

(sA)T = sAT

(AB)T = BTAT

Matrices

Matrix determinants and inverse

The inverse of a 2x2 or 3x3 matrix is:

The determinant of a matrix A is denoted det(A) or |A|, is calculated as:1x1

2x2

3x3

ASSUMED MATHS: CALCULUSAssumed mathematical knowledge dealing with basic calculus

CalculusBasic calculus including: simple

differential calculus (rate of change over time of a variable) and integral calculus

ASSUMED MATHS: POLYHEDRAAssumed mathematical knowledge dealing with polygons and polyhedra

Polygons

Definition of a polygon, including edges and vertices, convex and concave, polygon mesh.

PolyhedraDefinition of polyhedra

including interior and exterior, polytope (bounded convex polyhedron).

ASSUMED MATHS: MISCELLANEOUSMiscellaneous mathematical aspects

Barycentric CoordinatesBarycentric coordinates parameterize the space

formed using a weighted combination of a set of reference points.

B

C

A

Consider two points A and B, any point on the line between A and B can be expressed as P = A + t(B − A) = (1 − t)A + tB or simply as P = uA + vB, where u + v = 1, i.e. P is on the segment AB if and only if 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1.

Expressions, as above, in terms of (u,v) are the barycentric coordinates of P with respect to A and B.

Line, Rays, Segments, Planes and HalfspacesDefinition of a line, ray and segment

Definition of a plane and half-space

Assume A, B and C are defined points and t, u and v are scalars, and n is a normal vector:

Minkowski Sum and DifferenceBasic understanding of the Minkowski sum and

Minkowski difference. Appreciate that two point sets intersect if, and only if, their Minkowski difference contains the origin.Assume A and B are two point sets, and a and b are position vectors of points in A and B. The Minkowski sum, A ⊕ B, is defined as the set

the Minkowski difference is obtained by adding A to the reflection of B about the origin; that is, A Ѳ B = A ⊕ (−B)

Voronoi regionsGiven a set S of points in the plane, the Voronoi region of a point P in S is defined as the set of points in the plane closer to (or as close to) P than to any other points in S.

Within a collision detection context, given a polyhedron P, let a feature of P be one of its vertices, edges, or faces. The Voronoi region of a feature of P is then the set of points in space closer to (or as close to) the feature than to any other feature of P.

DIRECTED READINGDirected mathematical reading

Directed

reading

Directed reading

• Read Chapter 3 (pp23-72) of Real Time Collision Detection• Read Section 4 (pp137-194) of Game Engine Architecutre.• Read Section 2 (pp15-42) and Section 9 (pp145-191) of Game Physics Engine Development

Directed

reading

• Consult the excellent Wolfram MathWorld http://mathworld.wolfram.com/

Summary

To do:Explore linked

mathematical resources.

Consider how you can best make use of a ‘just-in-time’ approach for mathematical concepts.

Today we explored:

Mathematical knowledge assumed within the module to cover collision detection and rigid body dynamics.