2D computer graphics endowed with a self-rendering method—a procedure which assigns colors to the image pixels by an arbitrary algorithm. Complex models can be built by combining simpler objects, in the paradigms of object-oriented programming. In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion: other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. A translation operator is an operator formula_1 such that formula_2 If v
Hyperbolic geometry The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points (velocities). The hyperbolic plane is a plane where every point is a saddle point. There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature. By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative Gaussian curvature) in a three-dimensional Euclidean space. Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A