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November 26

Optics problem

I have a problem involving offset mirrors which will require some intricate 3D trigonometry to solve, unless it can be simplified with a bit of intuition. It is equivalent to what follows, which is easier to describe. There is a cuboidal room, with the only illumination a point source at the centre. An opaque planar object is parallel to one of the walls and casts a shadow wholly on it. Intuition says that the shadow will be of identical shape and inclination - specifically, a vertical rectangular object at some angle to the floor will produce a congruent rectangular shadow on the wall at the same angle. Further, the linear magnification of the shadow image can be got from the perpendicular distances from the wall of the source and object - if these are x and y respectively, the magnification is x/(x-y). Finally, there is a perfect analogy with parallel sections of a slant pyramid. Are all of these assumptions correct? 109.151.42.94 (talk) 10:35, 26 November 2013 (UTC)[reply]

Well, the word "congruent" is not used correctly there -- the correct word is "similar". But other than that it looks correct to me. Looie496 (talk) 17:01, 26 November 2013 (UTC)[reply]

Thanks. 109.151.42.94 (talk) 22:25, 26 November 2013 (UTC)[reply]

Actually we have a projection here, and if the planar object is parallel to a wall (and the shadow fits entirely on the wall, not crossing room's corners) the projection is just a scaling. --CiaPan (talk) 11:34, 27 November 2013 (UTC)[reply]

To do it the hard way, you only need to do the math for the 4 corners of the plate (and maybe the centers of each edge, if you question whether the edges of the shadow will be straight). StuRat (talk) 23:22, 26 November 2013 (UTC)[reply]