A Proposal for the Automatic Computation of Envelopes of Families of Plane Curves
The idea of envelope of a family of plane curves is an elementary notion in differential geometry. As such, its implementation in dynamic geometry environments is quite universal (Cabri, The Geometer’s Sketchpad, Cinderella, GeoGebra,...). Nevertheless, most of these programs return, when computing certain envelopes, both some spurious solutions and the curves that truly fit in the intuitive definition of envelope. The precise distinction between spurious and genuine parts has not been made before: This paper proposes such distinction in an algorithmic way, ready for its implementation in interactive geometry systems, allowing a finer classification of the different parts resulting from the current, advanced approach to envelope computation and, thus, yielding a more precise output, free from extraneous components.
KeywordsAutomated deduction in geometry dynamic geometry envelopes parametric polynomial systems symbolic computation
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With the occasion of ProfessorWen-tsunWu’s Centennial Birthday, we would like to honor, through this humble contribution, the decisive role of Professor Wentsun Wu’s work in the development of symbolic computation methods for the automatization of geometric reasoning.