Line of intersection of two planes

Two distinct planes in three-dimensional space either stay parallel and never meet, or cut each other along a unique straight line. Finding that line of intersection of two planes is a standard analytic-geometry task: the direction is the cross product of the two normal vectors, and any particular point that satisfies both Cartesian equations pins the parametric form r of t equals point plus t times direction. Students, CAD scripters, and graphics engineers meet the same eight coefficients over and over: a1, b1, c1, d1 for the first plane and a2, b2, c2, d2 for the second.

What the line of intersection is and how the formulas lock it

A plane in Cartesian form is the set of points r equals (x, y, z) that satisfy n · r plus d equals zero, where n is a non-zero normal vector. Write the first plane as a1 x plus b1 y plus c1 z plus d1 equals zero and the second as a2 x plus b2 y plus c2 z plus d2 equals zero.

Every vector that lies in the first plane is orthogonal to n1 equals (a1, b1, c1); every vector that lies in the second is orthogonal to n2 equals (a2, b2, c2). Therefore any direction that lies in both planes is orthogonal to both normals, which is exactly the geometric meaning of the cross product: the direction of the line of intersection is v equals n1 cross n2.

If that cross product is the zero vector, the normals are linearly dependent, the planes are parallel — including the coincident case where they are the same plane written twice — and there is no unique line to return. When the cross product is non-zero the two planes meet along one line, and the remaining task is to name a particular point on that line. This solver chooses the unique point on the line closest to the origin, which always lies in the span of n1 and n2.

Solving the two-by-two Gram system n1 · r0 equals minus d1 and n2 · r0 equals minus d2 for coefficients alpha and beta yields r0 equals alpha n1 plus beta n2. The Gram determinant equals the squared length of n1 cross n2 by the Lagrange identity, so a non-zero direction guarantees a unique closest point. The parametric line is then r of t equals r0 plus t times (n1 cross n2).

A classroom check is immediate: the planes x plus y plus z equals one and x minus y plus z equals one meet along r of t equals (one half, zero, one half) plus t times (two, zero, minus two). The xy-plane z equals zero and the xz-plane y equals zero meet along the x-axis. Every step is elementary vector algebra on eight scalars, so the result is deterministic, has no iteration, and never depends on the clock or a random seed.

Where a two-plane intersection calculator earns its keep

Classrooms and tutoring platforms use a line of intersection of two planes calculator so learners can check the parametric form for arbitrary coefficients without a sign error in the cross product or a missed case when the largest direction component is not z. Computer-aided design and solid-modeling scripts recover an edge from two adjacent faces given in implicit form, then feed the parametric line into a clipping or extrusion step.

Graphics and computational-geometry pipelines meet the same eight numbers when two clipping planes, two mirror planes, or two constraint planes are known and the shared hinge must be sampled. Failed validation does not consume balance, which keeps exploratory scripts cheap when users type parallel coefficients while learning the contract. Keep requests small, pin the rounded direction and point in tests, and you get a reproducible line for homework, LMS plugins, CAD helpers, and edge workers without standing up a computer-algebra system of your own.

Prefer this contract when both surfaces are already written as Cartesian planes; if you only have three points on each plane, fit the two implicit equations first, then call this solver. If you need the angle between the planes rather than their common line, that is a different search and a different calculator.