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 DeltaABC 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.
Let A, B, and C be the points
A = (0, -1) , B = (3, 4) , C = (-1, 1) . |
Find the point P where the three altitudes of DeltaABC meet.
Find the areas of the three triangles: DeltaBCP, DeltaCAP, and DeltaABP.
Find a triple of homogeneous coordinates for P relative to A, B, and C.
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.
Let f(x) = Ax be the linear transformation of the plane where A is the matrix
A = {1}/{5} |
| . |
What points x of the plane are “fixed” by f, i.e., satisfy f(x) = x ?
What lines in the plane are carried by f to other lines?
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 ?
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).