Transformation Geometry — Math 331

February 9, 2004

Discussion

• Definition: By affine basis of R^{n} is meant a sequence P_{0}, P_{1}, …, P_{n} of n+1 barycentrically independent points of R^{n}.

• Proposition. Any point of R^{n} is uniquely representable as a barycentric combination of the points in a given affine basis of R^{n}.

Proof. Given P and an affine basis P_{0}, P_{1}, …, P_{n} use the fact from linear algebra that the vectors P_{1} − P_{0}, …, P_{n} − P_{0} form a linear basis of R^{n} and that P − P_{0} is uniquely a linear combination of those vectors.

• Terminology. The coefficients used to represent a point P as a barycentric combination of P_{0}, P_{1}, …, P_{n} are called barycentric coordinates or affine coordinates of P with respect to (or relative to) P_{0}, P_{1}, …, P_{n}.

• Definition. Any sequence of n + 1 numbers that is proportional to (a non-zero multiple of) a sequence of barycentric coordinates of P with respect to an affine basis P_{0}, P_{1}, …, P_{n} is a sequence of homogeneous coordinates of P with respect to (or relative to) P_{0}, P_{1}, …, P_{n}.

Example. (a, b, c) is a sequence of homogeneous coordinates for the point where the angle bisectors of Δ ABC meet relative to the vertices of the triangle since 1/(a + b + c) times that triple is the corresponding sequence of barycentric coordinates.

• Theorem. The point where the three altitudes of a triangle meet has homogeneous coordinates relative to the vertices of the triangle given by the areas of the three sub-triangles formed by that point and the three vertices when all of the angles in the triangle are acute.

Exercises due Wednesday, February 11

1. Let A, B, and C be the points

A = (0, −1) , B = (3, 4) , C = (−1, 1) .

(a) Find the point P where the three altitudes of Δ ABC meet.

(b) Find the areas of the three triangles: Δ BCP, Δ CAP, and Δ ABP.

(c) Find a triple of homogeneous coordinates for P relative to A, B, and C.

2. Show that three distinct points A, B, and C are collinear if there is a triple of numbers (u, v, w), not all zero, of weight 0, i.e., u + v + w = 0, such that uA + vB + wC = 0.

3. Let f(x) = Ax be the linear transformation of the plane where A is the matrix

A = \frac{1}{5} ( \begin{matrix}3 & 4 \\ − 4 & 3 \end{matrix} ) .

(a) What points x of the plane are “fixed” by f, i.e., satisfy f(x) = x?

(b) What lines in the plane are carried by f to other lines?

(c) What lines L in the plane are “stabilized” by f, i.e., satisfy the condition that f(x) is on L if x is on L?

4. Find homogeneous coordinates relative to the vertices of a given triangle for the point where the three perpendicular bisectors of the sides of the triangle meet.

Hint: Use the fact that the perpendicular bisectors are the altitudes of the triangle whose vertices are their feet (i.e., the midpoints of the three sides).