C++ Barycentric Coordinates: Calculate Point Weights Inside a Triangle

Barycentric coordinates are a way of representing a point inside a triangle using three weights, one for each vertex of the triangle. These weights are the ratios of the areas of the sub-triangles formed by the point and the edges of the triangle. In C++, we can calculate the barycentric coordinates of a point (x,y) inside a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) using the following code:

double alpha, beta, gamma;
double triangle_area = 0.5 * ((x2 - x1) * (y3 - y1) - (x3 - x1) * (y2 - y1));
alpha = 0.5 * ((x2 - x) * (y3 - y) - (x3 - x) * (y2 - y)) / triangle_area;
beta = 0.5 * ((x3 - x) * (y1 - y) - (x1 - x) * (y3 - y)) / triangle_area;
gamma = 0.5 * ((x1 - x) * (y2 - y) - (x2 - x) * (y1 - y)) / triangle_area;

The weights alpha, beta, and gamma represent the barycentric coordinates of the point (x,y) with respect to the vertices (x1,y1), (x2,y2), and (x3,y3) respectively. These weights satisfy the equation alpha + beta + gamma = 1.