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Given a plane
and a vector from the plane to the point is given by
Projecting onto gives the distance from the point to the plane as
Dropping the absolute value signs gives the signed distance,
which is positive if is on the same side of the plane as the normal vector and negative if it is on the opposite side. This can be expressed particularly conveniently for a plane specified in Hessian normal form by the simple equation
where is the unit normal vector. Therefore, the distance of the plane from the origin is simply given by (Gellert et al. 1989, p. 541). Given three points for , 2, 3, compute the unit normal
Then the distance from a point to the plane containing the three points is given by
where is any of the three points. Expanding out the coordinates shows that
as it must since all points are in the same plane, although this is far from obvious based on the above vector equation. |
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Tuesday, 31 January 2012
How to calculate the distance between a arbitrary point to a plane?
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