Construction and exploration of particular curves and evolutes of theirs using GeoGebra dynamic software
Abstract
The article deals with the synthetic construction of specific planar curves which are defined by geometric motions and with the modeling of these constructions in GeoGebra dynamic system. We focus on the constructions of trajectories, envelopes, and centers of curvature and osculating circles of specific curves.
We present a par- ticular construction of all centers of curvature of these curves, i.e. the construction of the evolute of a curve. Our aim is to investigate the synthetic constructions without the use of coordinates and formulate the proofs of them.
We newly model selected types of curves in GeoGebra dynamic system which is an additional output to our theoretical conclusions and also a significant teaching aid at the same time. The con- structions are meant to be dynamic and the curves are formed gradually as the traces of points or curves.
All examples presented in this article are intended to be used in the undergraduate courses on kinematic geometry (mandatory courses for pre-service mathematics teachers who study teaching mathematics and descriptive geometry, i.e. the prospective secondary school teachers). The synthetic constructions together with their proofs demonstrated in GeoGebra dynamic system bring a new light into this area.
It allows students to imagine the main idea of the proofs of the constructions and to investigate the properties of the curves more easily based on the pure geometry and visual aspects. Dynamic constructions, i.e. the possibility of changing positions of points and curves play the significant role here.
GeoGebra offers also algebraic expressions which represent another tool for students how they can study and manip- ulate with those curves.