Mathematics for Realtime Animation part two

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Mathematics for Real-time Animation (part two)

• Ladislav Kavan

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Matrices

• A square m ? m matrix M is a table of real numbers

• Recall matrix-matrix multiplication,
  matrix-vector multiplication (vector ? one-column
  matrix)
• Properties
• M(NK) (MN)K (if defined)
• MN ? NM (in general)
• MI IM M (the identity matrix)

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Orthonormal Matrices

• A square matrix M is orthogonal (orthonormal) if
  its columns, interpreted as column vectors, are
  orthogonal (orthonormal).
• Properties of orthonormal matrices
• M-1 MT
• ?x,y?V ltMx, Mygt ltx, ygt
• ?x?V ?
• det(M) 1
• rows of M are orthonormal
• Especially for 3?3 orthonormal matrix, there
  always exists x?0 such that Mx x

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Rigid Body Motion
R - 3?3 matrix t - column vector

• R is a matrix of associated linear mapping, t is
  a translation vector.
• A rigid body motion is an affine transformation
  which preserves lengths (i.e. dot product).
• Matrix M represents a rigid body motion iff R is
  an orthonormal matrix.
• If R is orthonormal, then M transforms frames to
  frames (cartesian coordinate systems).

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Rigid Body Motion

• If det(R) 1, then R represents a rotation
• If det(R) -1, then R represents a reflexion
  (transforms right-hand basis to a left-hand one)
• A reflexion can not be obtained by a rigid body
  motion, therefore we require det(R) 1
• Identity (matrix I) is a rotation (with zero
  angle)
• Composition of two rotations is again a rotation
• Inverse rotation is also a rotation
• algebraically rotations form a group
• (Composition of two reflexions is not a
  reflexion).

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Rotation in R3

• In R3, there always exists a?0 such that Ra a
• What is a? Invariant direction the axis of
  rotation!
• Assume that a (1,0,0). Then R must look like

R preserves the first coordinate and in the other
two acts as a 2D rotation. The same is true for
any axis (although the matrix is not as
simple). Convention angle of rotation in a
right-hand sense
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Cross (Vector) Product

• The cross product of R3 vectors x (x1, x2, x3)
  and y (y1, y2, y3) is a vector

• Properties ?x, y, z ? R3 ?k?R
• neither commutative nor associative
• ltx ? y, xgt 0, ltx ? y, ygt 0 (orthogonality)
• (x y) ? z x ? z y ? z
• (kx) ? y k(x ? y)
• if x, y orthonormal, then x, y, x ? y is a
  right-hand orthonormal basis

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Cross Product to Matrix

• For each vector x (x1, x2, x3) ? R3 we can
  define a 3 ? 3 matrix M(x)

This matrix converts cross product to a matrix
multiplication, i.e. ?y?R3 x ? y M(x)
y Note that M(x) is anti-symmetric M(x)T -M(x)
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Rotation about a General Axis

• The rotation with axis a?R3 and angle ? can be
  expressed as a matrix
• R(a, ?) cos(?)I (1- cos(?))M(a)2 sin(?)M(a)
• where
• I is the identity matrix
• M(a) is the matrix representation of cross
  product
• known as the "Rodrigues formula"

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Combination of Points in A3

• Let x1, x2, ..., xn ? A3, and w1, w2, ..., wn ? R
• A point y ? A3 is an affine combination of points
  x1, x2, ..., xn if
• y w1 x1 w2 x2 ... wn xn, and
• w1w2...wn 1
• (Why 1? Recall affine coordinates!)
• A point y ? A3 is a convex combination of points
  x1, x2, ..., xn if
• it is an affine combination of x1, x2, ..., xn,
  and
• w1?0, w2?0, ..., wn?0
• Note If we require w1w2...wn 0, the result
  will be a vector (again recall affine
  coordinates)

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Affine and Convex Hull

• An affine (resp. convex) hull of points x1, x2,
  ..., xn ? A3 is the set of all possible affine
  (resp. convex) combinations of x1, x2, ..., xn
• (Compare with the linear hull!)
• Example What is the affine and convex hull in A3
  of
• two points?
• three points?
• four points?
• five points?

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Convex Sets

• A set of points S ? A3 is convex if
• ?x, y ? S ?t?0,1 (1-t)x ty ?S
• (i.e. for each two points in S, their connecting
  line segment is also in S)

Example
convex
non-convex