20. Plane–Plane Distance
Two planes in are either:
- Identical — distance
- Parallel — separated by a constant perpendicular distance
- Intersecting — they cross along a line, distance at the intersection
20.1. Distance between parallel planes
Two planes are parallel iff their normal vectors are parallel:
are parallel iff for some .
Rewriting both with the same normal vector :
The perpendicular distance between them is:
Example
Same normal , .
20.2. Strategy: pick a point on , measure to
Equivalent computation: any point on has point–plane distance to equal to the inter-plane distance.
20.3. Non-parallel planes
If the normal vectors are not parallel, the planes intersect along a line in — distance . Any pair of non-parallel planes intersects (in 3-D).
The intersection line’s direction is (a vector orthogonal to both normals — see Cross Product).
20.4. Connections
- Planes
- Point–Plane Distance
- Cross Product — direction of intersection line