Transformations

1
Transformations

• 10. March 2004

2
Linear Transformations

• In a vector space the important mappings are
  linear transformations
• L(l x m y) l L(x) m L(y). L-1 exists.
• L can be represented by a nonsingular square
  matrix.

3
Semi Linear Transformations

• A semi linear transformation is more general
• L(lx m y) f(l) L(x) f(m) L(y). L-1 exists,
  f K ! K is an automorphism of K.

4
Exercises

• Prove that in R, Q, Fp, (p- prime) there are no
  nontrivial automorphisms.
• Prove that z a z (conjugate) is an automorphism
  of C.
• Go to the library or internet and find a
  reference to the group of authomorphisms of
  complex numbers C and quaternions H.
• Determine the size of the group of authomorphisms
  of Fq, for q pk, a power of a prime.

5
Affine Transformations

• In an affine plane the important mappings are
  affine transformations (affinities).
• An affine transformation maps three collinear
  points to collinear points.
• Each affine transformation is of the form A(x)
  c, where A is a semilinear transformation.

6
Projective plane from R3.

• Consider the incidence structure, defined by
  1-dimensional and 2-dimensional subspaces of R3
  where the incidence is defined by inclusion.
• Call 1-dimensional subspaces points and
  2-dimensional subspaces lines.
• Exercise. Prove that this structure satisfies all
  three axioms for the projective plane.

7
Homogeneous Coordinates

• Let (a,b,c) ¹ (0,0,0) be a point in R3. There is
  exactly one line through the origin passing
  through (a,b,c). Hence a projective point can be
  represented by (a,b,c). However, for any l ¹ 0
  the same projective point can be represented by
  (l a, l b, l c).
• That is why (a,b,c) are called homogeneous
  coordinates.

8
Unit sphere model

• Take a unit sphere in R3.
• Let pairs of antipodal points be projective
  points.
• Let big circles be projective lines.
• Prove that this system is a model for a
  projective plane.

9
Stereographic Projection

• There is a homeomorphic mapping of a sphere
  without the north pole N to the Euclidean plane
  R2. It is called a stereographic projection.
• Take the unit sphere x2 y2 z2 1 and
  the plane z 0.
• The mapping p T0(x0,y0,z0) a
  T1(x1,y1) is shown on the left.

N
T0
T1
10
Stereographic Projection

• The mapping p T0(x0,y0,z0) a
  T1(x1,y1) is shown on the left.
• r1 r0/(1-z0)
• x1 x0/(1-z0)
• y1 y0/(1-z0)

N
T0
T1
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Example

• Take the Dodecahedron and a random point N on a
  sphere.
• Stereographic projection is depicted below.
• A better strategy is to take N to be a face
  center.

12
Example

• A better strategy is to take N to be a face
  center as shown on the left.