Abstract
We review the behavior of some popular dynamic geometry software when computing envelopes, relating the diverse methods implemented in these programs with the various definitions of envelope. Special attention is given to the new GeoGebra 5.0 version, that incorporates a mathematically rigorous approach for envelope computations. Furthermore, a discussion on the role, in this context, of the cooperation between GeoGebra and a recent parametric polynomial solving algorithm is detailed. This approach seems to yield accurate results, allowing for the first time sound computations of envelopes of families of plane curves in interactive environments.
Highlights
Loci determination is unanimously accepted as a key characteristic of dynamic geometry (DG) systems
From the developers side we observe the dominance of a simple strategy for envelope computation, consisting on tracing –for a sufficiently large number of instances– an element of the given family of curves, as the main tool for graphically suggesting the corresponding envelope
Through selected examples and cases, we review how the above mentioned DG systems perform when computing envelopes
Summary
Loci determination is unanimously accepted as a key characteristic of dynamic geometry (DG) systems. From the developers side we observe the dominance of a simple strategy for envelope computation, consisting on tracing –for a sufficiently large number of instances– an element of the given family of curves, as the main tool for graphically suggesting the corresponding envelope. The aim of this note is, first, to perform a systematic discussion of the construction of envelopes in the top four most widely distributed DG environments, namely Cabri, Cinderella, GeoGebra and The Geometer’s Sketchpad (GSP). To this end, we recall basic notions about envelopes and refer to different reputed sources, some classical and some very modern, for further details on the elusive concept of envelope. The algorithmic approach behind this command is presented, as well as some examples of its performance, along with a rough description of some essential limitations of the proposed method
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