Transformation Geometry — Math 331

January 28, 2004

Discussion

• Terminology Revision. Any weight 1 linear combination of given points may be called a barycentric combination of those points, regardless of whether the coefficients are nonnegative.

• Definition. A sequence of r + 1 points p_{0}, p_{1}, …, p_{r} is called barycentrically independent if none of them is a barycentric combination of the others.

• Examples.

1. Any two distinct points P, Q are barycentrically independent. If P ≠ Q, the set of barycentric combinations of P and Q is the line through P and Q.

2. Three points A, B, C are barycentrically independent if and only if none lies on the line determined by the other two. Thus, the vertices of a triangle are barycentrically independent.

3. In R^{3} the four vertices of a tetrahedron are barycentrically independent.

• Proposition. A sequence of r + 1 points p_{0}, …, p_{r} is barycentrically independent if and only for given a_{0}, …, a_{r} and given b_{0}, …, b_{r} with a_{0} + … + a_{r} = 1 and b_{0} + … + b_{r} = 1 the following statement is true:

a_{0}p_{0} + … + a_{r}p_{r} = b_{0}p_{0} + … + b_{r}p_{r} if and only if a_{0} = b_{0}, …, a_{r} = b_{r} .

• Proof. Obtain this from corresponding facts about linear independence.

• Theorem. If p_{0}, p_{1}, …, p_{n} are barycentrically independent points of n-dimensional Euclidean space R^{n}, and q_{0}, q_{1}, …, q_{n} are any points of R^{m}, then there is one and only one affine map f from R^{n} to R^{m} for which f(p_{0}) = q_{0}, f(p_{1}) = q_{1}, …, f(p_{n}) = q_{n}.

Proof. Use the fact that there is a unique linear map taking prescribed values at the members of a basis of R^{n}.

• Theorem If a map f from R^{n} to R^{m} preserves barycentric combinations, then it must be an affine map.

Proof. Use two facts: (1) an affine map that carries 0 to 0 must be linear, and (2) a linear map is always given by a matrix.

Exercises due Friday, January 30

1. Let A, B, C, and D be four points in the plane R^{2}. Show that the polygonal path (sequence of line segments) from A to B, from there to C, then to D, and back to A is a parallelogram if and only if A − B + C − D = 0.

2. Show that an affine transformation of the plane carries a parallelogram to a parallelogram.

3. Show that there is one and only one affine transformation of the plane carrying a given parallelogram to another given parallelogram in a given vertex-matching way.

4. Show that any affine transformation of the plane carries the point where the diagonals of a given parallelogram meet to the point where the diagonals of the image parallelogram meet.

5. Explain why an affine transformation of the 3-dimensional space R^{3} must always carry a tetrahedron to a tetrahedron.