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Plane-Plane Intersection

Two non-parallel planes intersect in a line. The direction of the line is d = nโ‚ร—nโ‚‚ (cross product of normals). Find one point on both planes to get the parametric form r = rโ‚€ + td.

Concept Fundamentals
d = nโ‚ร—nโ‚‚
Direction
r = rโ‚€ + td
Parametric
nโ‚ร—nโ‚‚ = 0
Parallel
Same plane
Coincident

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Direction d is perpendicular to both normals. Parallel planes: no intersection line. Coincident planes: infinite intersection.

Key quantities
d = nโ‚ร—nโ‚‚
Direction
Key relation
r = rโ‚€ + td
Parametric
Key relation
nโ‚ร—nโ‚‚ = 0
Parallel
Key relation
Same plane
Coincident
Key relation

Ready to run the numbers?

Why: Plane-plane intersection is used in CSG (Constructive Solid Geometry), roof ridge design, CAD edge extraction, and game level design. The intersection line is perpendicular to both normals.

How: Compute d = nโ‚ร—nโ‚‚. If |d| โ‰ˆ 0, planes are parallel. Otherwise, set one coordinate (e.g. z=0) and solve the 2ร—2 system from both plane equations to find a point on the line.

Direction d is perpendicular to both normals.Parallel planes: no intersection line.
Sources:Khan Academy โ€” 3D GeometryWolfram MathWorld โ€” Plane-Plane Intersection

Run the calculator when you are ready.

Find Intersection LineEnter two plane equations

Sample Examples

Input

Plane 1

Plane 2

Results

Intersection Line

Point: (2, 3, 0)

Direction: (0, 0, 1)

Parametric: x = 2 + 0t, y = 3 + 0t, z = 0 + 1t

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

โˆฉ

Two planes intersect in a line or are parallel.

โ€” 3D Geometry

ร—

Cross product gives line direction.

โ€” Vector Algebra

Key Takeaways

  • Direction of intersection line: d = nโ‚ร—nโ‚‚ (cross product)
  • Parallel planes: nโ‚ร—nโ‚‚ = 0, no line
  • Find point by setting one coordinate (e.g. z=0) and solving 2ร—2 system
  • Parametric form: r = rโ‚€ + td
  • Coincident: parallel and same plane

Did You Know?

  • CSG (Constructive Solid Geometry) uses plane-plane intersection
  • Roof ridge lines are plane-plane intersections
  • Direction d is perpendicular to both normals
  • Two planes always intersect in a line (or are parallel)
  • CAD uses this for edge extraction
  • Level design in games uses plane intersections

Understanding

Two non-parallel planes intersect in a line. Direction d = nโ‚ร—nโ‚‚. Find one point on both planes.

dโƒ—=nโƒ—1ร—nโƒ—2\vec{d} = \vec{n}_1 \times \vec{n}_2
rโƒ—(t)=rโƒ—0+tdโƒ—\vec{r}(t) = \vec{r}_0 + t\vec{d}

Expert Tips

  • Set z=0 (or x/y) when |d_z| is smallest for numerical stability
  • Check |d| < ฮต for parallel
  • For coincident, check if a point on plane1 satisfies plane2
  • Normalize d for unit direction

FAQ

Q: When is there no intersection?
A: Parallel planes (nโ‚ร—nโ‚‚ = 0).
Q: Infinite intersections?
A: Coincident planes (same plane).
Q: How to find point on line?
A: Set one coordinate, solve 2ร—2 system from both plane equations.
Q: Applications?
A: CSG, roof design, CAD, level design.
Q: Why cross product for direction?
A: d must be perpendicular to both normals.
Q: Relation to line-plane intersection?
A: Intersection line can meet a third plane at a point.
Q: Can three planes meet at one point?
A: Yes, when not parallel and not sharing a line.

How to Use

  1. Enter plane 1: Aโ‚x+Bโ‚y+Cโ‚z+Dโ‚=0
  2. Enter plane 2: Aโ‚‚x+Bโ‚‚y+Cโ‚‚z+Dโ‚‚=0
  3. Get intersection line (point + direction) or special case

Disclaimer

Normals must be non-zero. Parallel planes have no intersection.

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