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2.1. A SSUMED M ATHS Core mathematical underpinnings

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Assumed mathematical knowledge dealing with coordinate systems

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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

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Assumed mathematical knowledge dealing with vectors

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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 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

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Dot (scalar) product and common ● algebraic identities Assume u, and v are vectors and r and s are scalars u · v = u 1 v 1 + u 2 v 2 +· · ·+u n v n 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)

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Cross (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 = 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)

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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.

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Assumed mathematical knowledge dealing with matrices

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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.

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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: 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 = A T ± B T (sA) T = sA T (AB) T = B T A T

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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

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Assumed mathematical knowledge dealing with basic calculus

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

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Assumed mathematical knowledge dealing with polygons and polyhedra

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

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

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Miscellaneous mathematical aspects

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Barycentric 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.

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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:

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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)

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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.

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Directed mathematical reading Directed reading

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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/

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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.

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