A point \(\mathbf{p}\) in the plane of a triangle defined by 3D vertices \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) has barycentric coordinates \(\alpha = 0.8\) and \(\beta = 0.6\).
- What is its \(\gamma\) coordinate?
- Is it inside the triangle \(\mathbf{abc}\)?
- Suppose \(\mathbf{a}\) = \(\begin{bmatrix}1\\0\\0\\\end{bmatrix}\), \(\mathbf{b}\) = \(\begin{bmatrix}2\\1\\0\\\end{bmatrix}\), and \(\mathbf{c}\) = \(\begin{bmatrix}1\\1\\1\\\end{bmatrix}\). What are the cartesian coordinates of \(\mathbf{p}\)?
Consider the triangle \(\mathbf{abc}\) depicted below. Give the barycentric coordinates of each of the labeled points \(\mathbf{w}\), \(\mathbf{x}\), \(\mathbf{y}\), \(\mathbf{z}\).
The intersection between a ray and a triangle can be found by equating the parametric equation for the ray with the barycentric representation of the plane. Set these equal and derive a method to solve for the intersection point.