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Line of Intersection of Two Planes — result sheet
Find one finite point and a direction vector for the intersection line of two nonparallel planes.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For n1=(a1,b1,c1), n2=(a2,b2,c2), direction=n1 x n2. Return the closest-to-origin point on both planes using the Gram system and line point+t*direction.
Nonparallel plane normals have a nonzero cross product, which gives a direction parallel to both planes. The point convention is the unique intersection point closest to the origin; parallel or coincident planes are rejected.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Plane 1 coefficient a — plane coefficient units; x coefficient in plane 1: a1*x+b1*y+c1*z+d1=0.; minimum -1000000; maximum 1000000
- Plane 1 coefficient b — plane coefficient units; y coefficient in plane 1.; minimum -1000000; maximum 1000000
- Plane 1 coefficient c — plane coefficient units; z coefficient in plane 1.; minimum -1000000; maximum 1000000
- Plane 1 constant d — plane coefficient units; Constant term in plane 1.; minimum -1000000; maximum 1000000
- Plane 2 coefficient a — plane coefficient units; x coefficient in plane 2: a2*x+b2*y+c2*z+d2=0.; minimum -1000000; maximum 1000000
- Plane 2 coefficient b — plane coefficient units; y coefficient in plane 2.; minimum -1000000; maximum 1000000
- Plane 2 coefficient c — plane coefficient units; z coefficient in plane 2.; minimum -1000000; maximum 1000000
- Plane 2 constant d — plane coefficient units; Constant term in plane 2.; minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Plane 1 coefficient a | 1 |
| Plane 1 coefficient b | 0 |
| Plane 1 coefficient c | 0 |
| Plane 1 constant d | -1 |
| Plane 2 coefficient a | 0 |
| Plane 2 coefficient b | 1 |
| Plane 2 coefficient c | 0 |
| Plane 2 constant d | -2 |
The planes x=1 and y=2 intersect in the line (1,2,0)+t*(0,0,1); the returned point is the closest point to the origin.
Assumptions and limits
- Each plane is a finite real equation a*x+b*y+c*z+d=0 with a nonzero normal vector.
- The planes are nonparallel, so their intersection is one line; the direction returned is the raw n1 cross n2 vector and is not normalized.
- The selected point is the closest point to the origin on that line, not an arbitrary point and not a least-squares fit.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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