Cusp — where it appears
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
Where the focus went
If the reflected rays do not meet at a point, they meet each other in pairs, and the curve they are all tangent to is what the mirror has instead of a focus. On a sphere of 1.6 m vertex radius it is 184.1 mm long and has a cusp; near the axis it sits at R/2, which is why a small enough spherical mirror passes for a good one. This is the bright shape in the bottom of a coffee cup, computed rather than admired.
The one shape that focuses
A paraboloid takes every ray parallel to its axis exactly through one point — 4.8e-9 mm of envelope across a 1.24 m aperture, which is arithmetic. A sphere with the same vertex curvature spreads the same bundle over 184.1 mm. So there is exactly one curved mirror that is a projection through a centre, it is a projection for exactly one bundle of rays, and every telescope in the world is built out of that sentence.
A wire with a corner in its shadow
A bent wire has no corner anywhere on it, and its shadow has one. The lamps that do it are not a coincidence — they are a surface in the room, two-dimensional, made of the wire's own tangent lines, and a lamp being carried across the room passes through it.
Named alongside it
The objects these essays reach for when they reach for this one.
ApertureCausticcentre of projectionConicEnvelopeFocusParaboloidRay tracingReflectionSpherical aberrationAstigmatismContour generator