Wikipedia:Reference desk/Archives/Mathematics/2012 November 29

= November 29 =

Reflected image of a point in a plane, R^3
Hello. What is the quickest analytic method to find the reflected image P' of a point P(x0,y0,z0) across an arbitrary plane π: ax+by+cz+d=0 in R3? I solved the problem by picking an arbitrary point on π then considering vectors and projections but this is a lengthy and ugly process. Thanks. 24.92.74.238 (talk) 02:58, 29 November 2012 (UTC)


 * You want to find the point which is an equal distance from the plane, on the far side, on the normal vector to the plane, through the original point ? If so, once you find the normal projection of the point onto the plane, finding the point on the opposite side is almost trivial. If point 0 is the original, point 1 is the normal projection onto the plane, and point 2 is the reflected point behind the plane, find point 2 as follows:

x2 = x0 - 2(x0-x1) y2 = y0 - 2(y0-y1) z2 = z0 - 2(z0-z1)


 * StuRat (talk) 03:28, 29 November 2012 (UTC)


 * Yes; see http://www.9math.com/book/projection-point-plane for a more explicit answer. Looie496 (talk) 19:47, 29 November 2012 (UTC)