Wikipedia:Reference desk/Archives/Mathematics/2023 August 16

= August 16 =

What are these shapes called?
Empty hollow ones where a point exists such that the first bounce of any ray from that point is the same non-retroreflective angle no matter what (no infinitely many rings of Fresnel lens-like cheating). I drew one as a kid it seemed like this should be possible exactly but I wouldn't know how to prove it. Sagittarian Milky Way (talk) 00:42, 16 August 2023 (UTC)
 * A paraboloid with its focus? (See Parabolic reflector.) --Lambiam 07:13, 16 August 2023 (UTC)
 * No not a collimator, the reflection angle of a paraboloid varies wildly from retroreflection at the apex to the opposite of retroreflection (very glancing deflection) at parts of the surface very far from the apex. Sagittarian Milky Way (talk) 11:59, 16 August 2023 (UTC)
 * Your question is not abundantly clear. Bounces are not angles, and being non-retroreflective is a property of the material of which a surface is made, not of angles. In a parabolic reflector, all reflected rays make the same angle with any given straight line or flat surface. If that is not the desired answer, than with what is it that they should make the same angle? With the incident ray? In 2D, this means a segment of a logarithmic spiral. In 3D, a surface of revolution of such a segment will do. --Lambiam 14:13, 16 August 2023 (UTC)
 * A (positive albedo) sphere is a (shitty if too dark) retroreflector of light from the center regardless of what it's made of as nothing has an albedo of exactly 0. What if the angle between incident and reflected rays was the same positive number everywhere instead of 0 like rays from the sphere's special point? Sagittarian Milky Way (talk) 15:29, 16 August 2023 (UTC)
 * A surface of revolution of a segment of a logarithmic spiral, with the axis of revolution lying in the plane of the spiral and passing through its centre. Perhaps this also works when the segment shifts along the axis and expands while rotating, forming a surface not unlike (the interior of) some snail shells. --Lambiam 16:21, 16 August 2023 (UTC)
 * I see what you mean now, cut on a polar coordinate line, take any of the pieces and add a mirrored copy, spin on the bilateral symmetry line if 3D. Does either the 2D or 3D version have a name? Cause it's not strictly a logarithmic spiral anymore. Sagittarian Milky Way (talk) 16:54, 16 August 2023 (UTC)
 * In 2D you can take any collection of segments of logarithmic spirals that share their centre and have the same absolute value for their polar slopes. Adding a mirrored copy of a segment is a special case. I don't think this has a name. For the conic snail shell with an extra z-axis, use $$r=e^{k\varphi+qz},$$ where $$k$$ and $$q$$ are constant. By a symmetry argument, this should work. The surface of revolution is obtained by setting $$q=0.$$ I can imagine someone named this, but it is not found in our list of surfaces. The name "cochleoid" (after Ancient Greek ) is taken. --Lambiam 18:01, 16 August 2023 (UTC)
 * Yes it seems many 2D shapes are possible for each angle, even just 1 piece can intercept all of the plane's central point rays. Sagittarian Milky Way (talk) 18:28, 16 August 2023 (UTC)