Describing a coordinate not from a origin, but from *more than one point* and using weighted averages. >[!definition] Barycentric Coordinates With respect to $\set{P, Q, R}$ > $\huge \ba{ a, b, c }_{PQR} = a\tilde P + b\tilde Q + c \tilde R$ > $ \huge \vector{PQ} \nparallel \vector{PR} $ For points, $a+b +c = 1$, > For Vectors, $a+b+c = 0$ ![[Pasted image 20231115175403.png|invert_Sepia]] $\ba{u, v, w}_{ABC}$ ![[Pasted image 20231116104345.png|invert_Sepia]] ### Standard Form >[!definition] Conversion to standard form > $ \huge \mathcal{C}_{PQR\to S} = \mat{\tilde P & \tilde Q & \tilde R}$ >[!info] Affine Transformations Barycentric coordinates will still be valid after an affine transformation, > >for ex. Barycentric coordinates for the centroid of a triangle will still be the centroid after any affine transformation >[!example] Converting to standard >$\huge \begin{align} P &= \pa{0, 1} \\ Q &= \pa{4, 0} \\ R &= \pa{3, 4} \\ \let M &= \ba{0.3, 0.1, 0.6} \\ \end{align}$ > >$\huge \begin{align} \tilde M &= 0.3 \tilde P + 0.1 \tilde Q + 0.6 ]\tilde R \\ &= 0.3\mat{0\\1\\1}_{S} + 0.1\mat{4\\0\\1}_{S} + 0.6\mat{3\\4\\0}_{S} \\ &= \pa{2.2, 2.7}_{S} \end{align}$ ## Interiors Any point is inside the triangle that determines the [[Coordinate System|Coordinate Space]] will have $a, b, c \ge 0$ in [[Barycentric Coordinates]]. $\huge \ba{a, b, c}_{PQR} \in \triangle_{PQR} \iff a, b, c \ge 0$ >[!example] >$\huge \begin{align} P&=\pa{0, 1} \\ Q&=\pa{4, 0} \\ R&=\pa{3, 4} \\ z&= (1, 3) \\ \text{Is } z &\in \triangle_{{PQR}}? \\ \\ \mathcal{C}_{S\to PQR} &= \mat{\tilde P & \tilde Q & \tilde R}^{-1} \\ &= \frac{1}{15} \mat{-4&-1&16\\3&-3&3\\1&4&-4} \\ \mathcal{C}_{S\to PQR} \tilde z &= \mat{ \frac{3}{5}, -\frac{1}{5}, \frac{3}{5} } \\ z &= \ba{\frac{3}{5}, -\frac{1}{5 }, \frac{3}{5}}_{{PQR}} \\ z_{y} < 0 &\therefore z \notin \triangle_{{PQR}} \\ \end{align}$ >[!example]- Problem Packet: Collisions If point V is traveling at some velocity $\vec v$, starting at point $P$ at $t=0$, when does it hit $\triangle_{Q_{a}, Q_{b}, Q_{c}}$? > >$\huge \begin{align} P = aQ_{a} + bQ_{b} + cQ_{c} + t\vec v \end{align} $