Maths & numbers
Intersection of two planes
Find a point and unit direction for the line where two 3D planes meet.
A clear answer. Here’s how.
Normalises both plane equations n·X+d=0. Their cross product gives line direction; cross-product identities give a point orthogonal to that direction. Parallel equations are classified as coincident or distinct using tolerances around 10⁻¹² for normal cross-products and 10⁻¹⁰ for offsets. Near-parallel planes can have distant, sensitive intersections. Direction sign is arbitrary.
See a worked example
Worked example
A fixed demonstration, separate from the values you enter above.
- Plane A normal X
- 0
- Plane A normal Y
- 0
- Plane A normal Z
- 1
- Plane constant d
- 0
- Plane B normal X
- 0
- Plane B normal Y
- 1
- Plane B normal Z
- 0
- Plane constant d
- 0
Example result
| Vector | X | Y | Z |
|---|---|---|---|
| Point on line | 0 | 0 | 0 |
| Unit line direction | -1 | 0 | 0 |
All intersection points are P + t·direction. Direction sign is arbitrary; coefficients are normalised internally.