Lecture 3: Lines and Planes

Lines

Explicit equation:

$$y = m * x + b$$

Implicit equation:

Consider two known points of a triangle $(x_0, y_0)$ and $(x_1, y_1)$.

We can use these points to determine the slope and plug into the explicit equation:

$$y = \frac{(y_1 - y_0)}{(x_1 - x_0)} x + b$$ $$0 = (y_1 - y_0) x - (x_1 - x_0) y + c$$ $$f(x, y) = (y_1 - y_0) x - (x_1 - x_0) y + c$$

Here we have replaced the unknown y-intercept $b$ with another constant, $b * (x_1 - x_0) = c$. We don’t really care what $b$ was so we’ll just move on with $c$.

We know that $(x_0, y_0)$ is on the point, so let’s plug in an solve for $c$.

$$f(x_0, y_0) = (y_1 - y_0) x_0 - (x_1 - x_0) y_0 + c = 0$$ $$c = (x_1 - x_0) y_0 - (y_1 - y_0) x_0$$ $$c = x_1 y_0 - x_0 y_0 - y_1 x_0 + y_0 x_0$$ $$c = x_1 y_0 - y_1 x_0$$

Plug back in to our original equation with $c$

$$f(x, y) = (y_1 - y_0) * x - (x_1 - x_0) * y + x_1 * y_0 - y_1 * x_0 = 0$$

Planes

A plane is defined by a point $p_1$ and a normal $\hat n$. The plane consists of all points for which $(p-p_1)$ is perpendicular to $\hat n$.

Or, using the dot product:

$$f(p) = (p - p_1) \cdot \hat n = 0$$

And to get a more commonly seen equation:

$$f(x, y, z) = (x - p_x) * n_x + (y - p_y) * n_y + (z - p_z) * n_z = 0$$ $$= n_x * x + n_y * y + n_z * z - (n_x * p_x + n_y * p_y + n_z * p_z)$$ $$= A * x + B * y + C * z + D$$

where

$$A = n_x$$ $$B = n_y$$ $$C = n_z$$ $$D = - (n_x * p_x + n_y * p_y + n_z * p_z)$$