Isoptics of Bézier curves

Given a planar curve s(t)s(t), the locus of those points from which the curve can be seen under a fixed angle is called isoptic curve of s(t)s(t).Isoptics are well-known and widely studied, especially for some classical curves such as e.g. conics (Loria, 1911). They can theoretically be computed for a large class of parametric curves by the help of their support functions or by direct computation based on the definition, but unfortunately these computations are extremely complicated even for simple curves.Our purpose is to describe the isoptics of those curves which are still frequently used in geometric modeling – the Bézier curves. It turns out that for low degree Bézier curves the direct computation is possible, but already for degree 4 or 5 the formulas are getting too complicated even for computer algebra systems. Thus we provide a new way to solve the problem, proving some geometric relations of the curve and their isoptics, and computing the isoptics as the envelope of envelopes of families of isoptic circles over the chords of the curve.

► Isoptic curve of low degree Bézier curves are directly computed.
► Envelope properties of Bézier curves and their isoptic curves are proved.
► Isoptic curves of high degree Bézier curves are computed by these envelopes.