2 . 1. Assumed Maths

1. 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 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 Assume u, and v are vectors and r and s are scalars u · v = u1v1 + u2v2 +· · ·+unvn u · v = |u|| v| cosθ 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. 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 The determinant of a matrix A is denoted det(A) or |A|, is calculated as: Matrix determinants and inverse 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. 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. C 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 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 Consult the excellent Wolfram MathWorldhttp://mathworld.wolfram.com/ Directed reading

25. Summary 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.