CSCI 480/580 Lecture 10 - In-Class Problems

1. 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 $\beta = 0.6$.

   1. What is its $\gamma$ coordinate?
   2. Is it inside the triangle $\mathbf{abc}$?
   3. 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 $\mathbf{p}$?

2. Consider the triangle $\mathbf{abc}$ depicted below. Give the barycentric coordinates of each of the labeled points $\mathbf{w}$, $\mathbf{x}$, $\mathbf{y}$, $\mathbf{z}$.

   

3. 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.