Abstract
The envelope of straight lines normal to a plane curve C is its evolute; the envelope of lines tangent to C is the original curve, together with the entire tangent line at each inflexion of C. We introduce some standard techniques of singularity theory and use them to explain how the first of these envelopes turns into the second, as the (constant) angle between the set of lines forming the envelope and the set of tangents to C changes from ½ to 0. In particular, we explain how cusps disappear and what happens at inflexions, where the evolute goes to infinity. We also study the family of “wavefronts” or “parallels” associated with these envelopes.
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