Ray-ray Intersection

$$\begin{aligned} c &= a + t q \\ c &= b + u r \end{aligned}$$

$$\begin{aligned} \begin{bmatrix}c_x \\ c_y\end{bmatrix} &= \begin{bmatrix}a_x \\ a_y\end{bmatrix} + t \begin{bmatrix}q_x \\ q_y\end{bmatrix} \\ \begin{bmatrix}c_x \\ c_y\end{bmatrix} &= \begin{bmatrix}b_x \\ b_y\end{bmatrix} + u \begin{bmatrix}r_x \\ r_y\end{bmatrix} \\ \end{aligned}$$

$$\begin{aligned} c_x &= a_x + t q_x \\ c_y &= a_y + t q_y \\ \end{aligned}$$

$$\begin{aligned} c_x &= b_x + u r_x \\ c_y &= b_y + u r_y \\ \end{aligned}$$

$$\begin{aligned} a_x + t q_x &= b_x + u r_x \\ a_y + t q_y &= b_y + u r_y \\ \end{aligned}$$

$$\begin{aligned} a_x r_y + t q_x r_y &= b_x r_y + u r_x r_y \\ a_y r_x + t q_y r_x &= b_y r_x + u r_x r_y \\ \end{aligned}$$

$$\begin{aligned} a_x r_y + t q_x r_y - a_y r_x - t q_y r_x &= b_x r_y - b_y r_x \\ \end{aligned}$$

$$\begin{aligned} t q_x r_y - t q_y r_x &= b_x r_y - a_x r_y - b_y r_x + a_y r_x \\ t (q_x r_y - q_y r_x) &= r_y (b_x - a_x) + -b_y r_x - -a_y r_x \\ t (q_x r_y - q_y r_x) &= r_y (b_x - a_x) - r_x (b_y - a_y) \\ t &= \frac{r_y (b_x - a_x) - r_x (b_y - a_y)}{q_x r_y - q_y r_x} \\ \end{aligned}$$

$$\begin{aligned} a_x q_y + t q_x q_y &= b_x q_y + u q_y r_x \\ a_y q_x + t q_x q_y &= b_y q_x + u q_x r_y \\ \end{aligned}$$

$$\begin{aligned} a_x q_y - a_y q_x &= b_x q_y + u q_y r_x - b_y q_x - u q_x r_y \\ \end{aligned}$$

$$\begin{aligned} u q_x r_y - u q_y r_x &= b_x q_y - a_x q_y - b_y q_x + a_y q_x \\ u (q_x r_y - q_y r_x) &= q_y (b_x - a_x) - b_y q_x - -a_y q_x \\ u (q_x r_y - q_y r_x) &= q_y (b_x - a_x) - q_x (b_y - a_y) \\ u &= \frac{q_y (b_x - a_x) - q_x (b_y - a_y)}{q_x r_y - q_y r_x} \\ \end{aligned}$$

$$\begin{aligned} d &= b - a \\ \mathit{det} &= q_x r_y - q_y r_x \\ t &= \frac{d_x r_y - d_y r_x}{\mathit{det}} \\ u &= \frac{d_x q_y - d_y q_x}{\mathit{det}} \\ \end{aligned}$$