In Euclidean space, the point on a plane
Contents
- Converting general problem to distance from origin problem
- Restatement using linear algebra
- Why this is the closest point
- Closest point and distance for a hyperplane and arbitrary point
- References
From this the distance from the origin to the plane can be found. If what is desired is the distance from a point not at the origin to the nearest point on a plane, this can be found by a change of variables that moves the origin to coincide with the given point.
Converting general problem to distance-from-origin problem
Suppose we wish to find the nearest point on a plane to the point (X0, Y0, Z0), where the plane is given by aX + bY + cZ = D. We define x = X - X0, y = Y - Y0, z = Z - Z0, and d = D - aX0 - bY0 - cZ0, to obtain ax + by + cz = d as the plane expressed in terms of the transformed variables. Now the problem has become one of finding the nearest point on this plane to the origin, and its distance from the origin. The point on the plane in terms of the original coordinates can be found from this point using the above relationships between x and X, between y and Y, and between z and Z; the distance in terms of the original coordinates is the same as the distance in terms of the revised coordinates.
Restatement using linear algebra
The formula for the closest point to the origin may be expressed more succinctly using notation from linear algebra. The expression
The Euclidean distance from the origin to the plane is the norm of this point,
Why this is the closest point
In either the coordinate or vector formulations, one may verify that the given point lies on the given plane by plugging the point into the equation of the plane.
To see that it is the closest point to the origin on the plane, observe that
Since
Alternatively, it is possible to rewrite the equation of the plane using dot products with
Closest point and distance for a hyperplane and arbitrary point
The vector equation for a hyperplane in
The closest point on this hyperplane to an arbitrary point
Written in Cartesian form, the closest point is given by
Thus in