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

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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)