The segment may intersect the plane formed by with the normal in any (or all) generic point on the segment.

The line equation in parametric form can be expressed as:

where is the intersection between the segment and the plane and values of will yield points on that segment.

The equation of the plane can be expressed as:

where can be any known point in the plane .

We plug the line equation into the equation of the plane:

and distribute the normal vector :

then collect for :

then group on the right-hand side of the equation:

and collect again for we obtain the equation for :

Now, based on the equation, we can make the following judgments:

- We know from the vector properties that if then the vector and the segment are perpendicular meaning that the segment is either parallel to the plane formed by or that the segment is contained entirely in the plane described by .
- If any point on the segment is contained in the plane described by then the entire segment is contained within the plane.
- If any point on the segment is contained in the plane described by then the segment is parallel with the plane described by .

- If then the segment interesects the plane.
- If then the intersection falls on the first end-point.
- If then the intersection falls on the second end-point.

- If or then the segment does not intersect the plane.
- If then intersection occurs beyond the second end-point.
- If then intersection occurs before the first end-point.

fuss/mathematics/geometry/intersections/plane_and_segment.txt ยท Last modified: 2017/02/22 18:30 (external edit)

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