The research within the area of the curve theory has been moving in at least two directions, one being constructive method, and the other analytic. Both ways are scientifically sufficient and have the same base -- synthetic method. The phrase curve necessarily includes its visualization, these days obtained by both mentioned research methods. In my opinion, the synthetic-constructive method is more convenient to many geometers since it elegantly leads to the desired results, without use of coordinates and systems of equations. Without claiming to show something new and original here we give an overview of a large set of the plane rational curves and envelopes associated with the conics in different ways. The aim is to emphasize the beauty of geometry on one hand, and the simplicity and elegance of the synthetic method leading to the desired results on the other. It is necessary to have only one of many offered dynamic computer programs, sufficient knowledge of the synthetic geometry, and a little bit of imagination. The walk starts through the ocean of the curves associated with the conics in the Euclidean plane, and it continues in the pseudo-Euclidean and hyperbolic plane by using the Cayley-Klein models. It is easy to perceive the behavior of the curves in the real absolute points, which in the Euclidean case, because of the imaginarity of the absolute points, is often difficult even to visualize. Geometry is a science. But at the end, I expect you to agree with me that it is an art as well. Once when you become enough possessed by it, it becomes an intellectual game.

Mathematics