Chapter 4: Lines and Planes

Lines

Explicit equation:

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:

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 $c$.

Plug back in to our original equation with $c$

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:

And to get a more commonly seen equation:

where