Maps as objects: inversion, homothety, Moebius maps, with composition, inverse and the image of a set #2150

Chapter IV of Modenov's Problems in Geometry (#2131) is about inversion, and Chapters I–III use homotheties, rotations, reflections, similarities and Möbius maps throughout. The book composes them, inverts them, finds their fixed points, and maps circles and regions through them.

What AngouriMath has: apply(lambda(z, (a*z + b)/(c*z + d)), w) works, and lambdas compose numerically. What it lacks:

• a map with a domain, an inverse and fixed points;
• composition as an operation on maps that can be simplified;
• the image of a set under a map. image(f, z in S) parses, but membership in it is undecided even for 2z + 1 on the unit circle;
• the point at infinity, without which an inversion is not a bijection (Complex Infinity #217).

Wanted, for v3:

• maps as objects, with builders such as inversion(O, k) and homothety(O, k), that can be composed, inverted and applied to sets;
• an inversion that sends a circle to a circle as a decided result.

The builders could be tried in 2.8 as lambdas, but the object needs the node model. A lambda applied to a vector is wrong today (#2132).

Related: #495 (lambdas as entities), #330 (image sets), #217 (complex infinity).