Plane Intersection

Topics: Geometry - 3D Space - Plane

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Given two planes in ${\mathbb{R}}^3$ we can find the line where they intersect by:

1. Finding their normal vectors and calculating the cross product between them
2. Finding a common point between them (we can find such a point by giving a variable a concrete value, then solving the resulting equation system)
3. Building a line with such a point and vector

In general, let:

$${\Pi }_1=Ax+By+Cz+D=0$$

$${\Pi }_2=Ex+Fx+Gx+H=0$$

…be the equations for two planes ${\Pi }_1$ and ${\Pi }_2$ in ${\mathbb{R}}^3$. Then, the intersection line for these planes is:

$$l=\{P+t(n_1\times n_2):t\in \mathbb{R}\}$$

…where $P$ is a common point between the planes, and $n_1$ and $n_2$ are the normal vectors for each respective plane.