BARYCENTRIC COORDINATES

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1 Computer Graphics Notes BARYCENTRIC COORDINATES Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview If we are given a frame in three-dimensional space we know how to define a local coordinate system with respect to the frame. However, given a set of points in three-dimensional space, we can also define a local coordinate system with respect to the these points. These coordinate systems are called barycentric coordinates and are discussed in these notes. What are Barycentric Coordinates? Consider a set of points P 0, P 1,..., P n and consider the set of all affine combinations taken from these points. That is all points P that can be written as α 0 P 0 + α 1 P α n P n + for some α 0 + α α n = 1 This set of points forms an affine space, and the coordinates (α 0, α 1,..., α n ) are called the barycentric coordinates of the points of the space. These coordinates system are frequently quite useful, and the interested student will notice that they are used extensively in working with triangles. In many cases (e.g. on a line, as shown below), this barycentric parameterization is exactly the parameterization that we usually use.

2 Example Point on a Line Segment To give a simple example of barycentric coordinates, consider two points P 1 and P 2 in the plane. If α 1 and α 2 are scalars such that α 1 + α 2 = 1, then the point P defined by P = α 1 P 1 + α 2 P 2 is a point on the line that passes through P 1 and P 2. If 0 α 1, α 2 1 then the point P is on the line segment joining P 1 and P 2. The following figure shows an example of a line and three points P, Q and R. These points were generated using the following αs: P : α 1 = 1 3, α 2 = 2 3 Q : α 1 = 3 4, α 2 = 1 4 R : α 1 = 4 3, α 2 = 1 3 P 2 P 1 Q p r Example Point in a Triangle To give a slightly more complex example of barycentric coordinates, consider three points P 1, P 2, P 3 in the plane. If α 1, α 2, α 3 are scalars such that α 1 + α 2 + α 3 = 1, then the point P defined by P = α 1 P 1 + α 2 P 2 + α 3 P 3 is a point on the plane of the triangle formed by P 1, P 2, P 3. The point is within the triangle P 1 P 2 P 3 if 0 α 1, α 2, α 3 1. If any of the α s is less than zero or greater than one, the point P is outside the triangle. If any of the α s is zero, we reduce to the example above and note that P is on one of the lines joining the vertices of the triangle. The following figure shows an example of such a triangle and three points P, Q and R, these points were calculated using the following α s: 2

3 P : α 1 = α 2 = 1 4, α 3 = 1 2. Q : α 1 = 1 2, α 2 = 3 4, α 3 = 1 4. R : α 1 = 0, α 2 = 3 4, α 3 = 1 4. P 2 Q r p P 1 p 3 Frames and Barycentric Coordinates There is a natural way to convert the local coordinates of a frame to barycentric coordinates for a certain set of points. Suppose we are given a frame F = ( v 1, v 2,..., v n, O) for an affine space A. Then we can write any point P in the space uniquely as P = p 1 v 1 + p 2 v p n v n + O where (p 1, p 2,..., p n ) are the local coordinates of the point P with respect to the frame F. If we define the points P i by P 0 = O P 1 = O + v 1 P 2 = O + v 2. P n = O + v n 3

4 (i.e., the origin of the frame and the points obtained by adding the coordinate vectors to the origin) and define p 0 to be p 0 = 1 (p 1 + p p n ) then we can see that P can be written as P = P 0 + p 1 (P 1 P 0 ) + p 2 (P 2 P 0 ) + + p n (P n P 0 ) or equivalently, in an affine way as, P = p 0 P 0 + p 1 P 1 + p 2 P p n P n where p 0 + p 1 + p p n = 1 In this form, the values (p 0, p 1, p 2,..., p n ) are barycentric coordinates of P relative to the points (P 0, P 1, P 2,..., P n ) How can Vectors be Represented? Following the above methods, we can also express the vectors of an affine space in terms of the points. In this case, if we are given the frame F = ( v 1, v 2,..., v n, O) then for any vector u, we can write u as u = u 1 v 1 + u 2 v u n v n for some constants u 1, u 2,..., u n (since the vectors of the frame are assumed to be linear independent). Now, if we define u 0 = (u 1 + u u n ) If we define the points P i by P 0 = O P 1 = O + v 1 P 2 = O + v 2. P n = O + v n 4

5 then u = u 1 (P 1 P 0 ) + u 2 (P 2 P 0 ) + + u n (P n P 0 ) or equivalently, in an affine way as, u = u 0 P 0 + u 1 P 1 + u 2 P u n P n where now we have that u 0 + u 1 + u u n = 0. References [1] DEROSE, T. Coordinate-free geometric programming. Technical Report , Department of Computer Science and Engineering, University of Washington, Seattle, Washington, Summary Barycentric coordinates are another important method of introducing coordinates into an affine space. If the coordinates sum to one, they represent a point ; if the coordinates sum to zero, they represent a vector. All contents copyright (c) Computer Science Department, University of California, Davis All rights reserved. 5