How this calculator works
The perpendicular to a plane follows its normal. Projecting the point displacement onto a unit normal gives a signed distance; taking its absolute value gives the shortest distance. Subtracting that signed displacement gives the perpendicular foot. Reversing all plane coefficients reverses the signed distance but leaves the shortest distance and foot unchanged.
Formula
s = (Ax + By + Cz + D)/sqrt(A^2+B^2+C^2); distance = |s|; foot = P - s N/|N|.
Worked example
The plane z = 2 has coefficients (0, 0, 1, -2). For P = (1, 2, 5), the signed and shortest distances are both 3, and the foot is (1, 2, 2).
Advertisement
Frequently asked questions
What if the point lies on the plane?
Both distances are zero and the perpendicular foot equals the input point.
Is D the plane distance?
Only with a unit normal, and even then its sign follows the equation convention. In general divide by the normal magnitude.
What units are returned?
The distances and foot use the same coordinate unit as the points. Use a common unit on every axis.