1. Core mathematical underpinnings
2.1. Assumed Maths
Core mathematical underpinnings

2. Assumed Maths: Coordinate Systems
Assumed mathematical knowledge dealing with coordinate systems

3. See links at end for reference material if needed
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.

4. Assumed Maths: Vectors
Assumed mathematical knowledge dealing with vectors

5. Vectors The following vector concepts should be familiar:
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) = v
v + (−v) = 0
v + 0 = 0 + v = v
For scalar multiplication:
r(s v) = (rs) v
(r + s) v = r v + s v
s(u + v) = s u + s v
1 v = v
The 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

6. Vectors Dot (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 θ
u · u = |u|2
u · v = v · u
u · (v ± w) = u · v ± u · w
r u · s v = rs(u · v)
Dot (scalar) product and common ● algebraic identities

7. Vectors
Assume u, v , w and x are vectors and r and s are scalars
u × v = −(v × u)
u × u = 0
u · (v × w) = (u × v) · w
u × (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)
Cross (vector) product and × algebraic identities and dependency upon coordinate system ‘handedness’. A right-handed system is assumed.

8. Vectors
Understanding 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.

9. Assumed Maths: Matrices
Assumed mathematical knowledge dealing with matrices

10. Matrices The 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.

11. Matrices Matrix addition, subtraction and multiplication
Assume A, B and C are matrices and r and s are scalars
For addition and subtraction:
A + B = B + A
A + (B + C) = (A + B) + C
A − B = A + (−B)
−(−A) = A
s(A ± B) = sA ± sB
(r ± s)A = r A ± sA
r(sA) = s(r A) = (rs)A
For multiplication:
AI = IA = A
A(BC) = (AB)C
A(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 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:

12. Matrices Matrix determinants and inverse
The determinant of a matrix A is denoted det(A) or |A|, is calculated as:
1x1
2x2
3x3
The inverse of a 2x2 or 3x3 matrix is:

13. Assumed Maths: Calculus
Assumed mathematical knowledge dealing with basic calculus

14. Calculus
Basic calculus including: simple differential calculus (rate of change over time of a variable) and integral calculus

15. Assumed Maths: Polyhedra
Assumed mathematical knowledge dealing with polygons and polyhedra

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

17. Polyhedra
Definition of polyhedra including interior and exterior, polytope (bounded convex polyhedron).

18. Assumed Maths: Miscellaneous
Miscellaneous mathematical aspects

19. Barycentric Coordinates
Barycentric coordinates parameterize the space formed using a weighted combination of a set of reference points. C
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. B A

20. Line, Rays, Segments, Planes and Halfspaces
Definition 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:

21. Minkowski Sum and Difference
Basic 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)

22. Voronoi regions
Given 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.

23. Directed
reading
Directed Reading
Directed mathematical reading

24. Directed
reading
Directed reading
Read Chapter 3 (pp23-72) of Real Time Collision Detection
Read Section 4 (pp ) of Game Engine Architecutre.
Read Section 2 (pp15-42) and Section 9 (pp ) of Game Physics Engine Development
Consult the excellent Wolfram MathWorld

25. To do: Summary Explore linked mathematical resources.
Today we explored:
Mathematical knowledge assumed within the module to cover collision detection and rigid body dynamics.
To do:
Explore linked mathematical resources.
Consider how you can best make use of a ‘just-in-time’ approach for mathematical concepts.