A parametric approach to 3D dynamic geometry

Abstract

Dynamic geometry systems are computer applications allowing the exact onscreen drawing of geometric diagrams and their interactive manipulation by mouse dragging. Whereas there exists an extensive list of 2D dynamic geometry environments, the number of 3D systems is reduced. Most of them, both in 2D and 3D, share a common approach for the management of geometric constructions, using numerical data for the present diagram instance and elementary methods for computing derived objects. This paper deals with a parametric approach for automatic management of 3D Euclidean constructions. An open source library, implementing the core functions in a 3D dynamic geometry system, is described here. The library deals with constructions by using symbolic parameters, so allowing a full algebraic knowledge about objects such as loci and envelopes. This parametric approach is also a prerequisite for performing automatic proof, and basic functions are defined for symbolically checking the truth of statements. Using recent results coming from the theory of parametric polynomial systems solving, the bottleneck in the automatic determination of geometric loci and envelopes is solved. As far as we know, there is no comparable library in the 3D case, except the paramGeo3D library (designed for computing equations of simple 3D geometric objects, but it lacks specific functions for finding loci and envelopes).

Geometrik Cisimler Konusunun Dinamik Geometri Yazılımı ile Öğretimi

“…It contributes to students' critical thinking and problem solving skills (Pesen, 2006). In many studies conducted in Turkey, it was emphasized that students' coincide with those in the literature which state that technology-supported geometry teaching is an intriguing (Şimşek & Yücekaya, 2014) and effective method; it increases student achievement (Bintaş & Bağcıvan, 2007;Forsythe, 2007;Yıldız, 2009); and the use of geometry software in teaching geometry provides a meaningful difference in student achievement (Botana, 2014;Erbaş & Yenmez, 2011). As expressed by the teacher, the positive aspects of the implementation are that the activities increase student motivation by attracting their attention; they help teachers save time during the teaching process in class; and they can also be actively repeated home.…”

A constructive approach to the quadrics of revolution using 3D dynamic geometry systems with algebraic capabilities

“…(2) construct a slider, k, in the "Graphics View," related to the slope of the generatrices of the cone, (3) construct the auxiliary sphere b of center C = (0,0,0) and radius w, (4) let H be one of the intersection points of the auxiliary sphere b and the z-axis, (5) construct sphere c of center H and radius r, (6) construct plane p, parallel to plane xy through H, (7) construct point E = (1,0,k), (8) construct line f (a generatrix of the cone) through C = (0,0,0) (apex of the cone) and E, (9) intersect line f and plane p, and denote J the intersection point, (10) construct sphere a of center H passing through J, Figure 25 Drawing a right circular cone as a trace with GeoGebra 5.…”

“…Let us underline that complex studies and computer packages on automatic 3D loci determination and theorem proving exist [7][8][9]. There are also specific studies on automatic theorem proving in the 2D case using GeoGebra [10], some focused on educational applications [11].…”

A Singular web service for geometric computations

“…An obvious work will consist of extending the described GeoGebra abilities to 3D loci and envelopes. Preliminary work on such computations in spatial DG has been discussed in [20], and the ongoing development of GeoGebra3D should allow such extension.…”