Isoptics of Bézier curves

doi:10.1016/j.cagd.2012.05.002

Computer Aided Geometric Design

Volume 30, Issue 1, January 2013, Pages 78-84

https://doi.org/10.1016/j.cagd.2012.05.002 Get rights and content

Abstract

Given a planar curve $s (t)$ , the locus of those points from which the curve can be seen under a fixed angle is called isoptic curve of $s (t)$ .

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.

Highlights

► 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.

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An explicit characterization of isochordal-viewed multihedgehogs with circular isoptics
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A curve α is called $(ϕ, ℓ)$ -isochordal viewed if a straight segment of constant length ℓ can slide with its endpoints on α and such that their tangents to α at these endpoints make a constant angle ϕ. These tangents determine the so-called ϕ-isoptic curve of α. In this paper, an explicit characterization of all $(ϕ, ℓ)$ -isochordal-viewed multihedgehogs with circular ϕ-isoptics is provided by their support functions, which are obtained as the solutions of a differential equation. This allows to construct any example of these curves in a very simple way from some free parameters. In addition, it is shown that a regular polygon of side length ℓ can slide smoothly along these multihedgehogs.
Isoptic surfaces of polyhedra
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The theory of the isoptic curves is widely studied in the Euclidean plane $E^{2}$ (see Cieślak et al., 1991 and Wieleitner, 1908 and the references given there). The analogous question was investigated by the authors in the hyperbolic $H^{2}$ and elliptic $E^{2}$ planes (see Csima and Szirmai, 2010, Csima and Szirmai, 2012, Csima and Szirmai, submitted for publication), but in the higher dimensional spaces there are only few results in this topic.
In Csima and Szirmai (2013) we gave a natural extension of the notion of the isoptic curves to the n-dimensional Euclidean space $E^{n}$ $(n \geq 3)$ which is called isoptic hypersurface. Now we develope an algorithm to determine the isoptic surface $H_{P}$ of a 3-dimensional polyhedron $P$ .
We will determine the isoptic surfaces for Platonic solids and for some semi-regular Archimedean polytopes and visualize them with Wolfram Mathematica (Wolfram Research, Inc., 2015).
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Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in Nil Geometry
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Computer Aided Geometric Design

Abstract

Highlights

References (15)

Contributions to the geometry of cam mechanisms with oscillating followers

Journal of Mechanisms

Isoptic curves in the hyperbolic plane

Studies of the University of Zilina Mathematical Series

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra

Computer Algebra: Systems and Algorithms for Algebraic Computation

Handbook of Geometric Programming Using Open Geometry GL

A Catalog of Special Plane Curves

Cited by (16)

An explicit characterization of isochordal-viewed multihedgehogs with circular isoptics

Isoptic surfaces of polyhedra

Isoptic surfaces of segments in <sup>S2</sup>×R and <sup>H2</sup>×R geometries

Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in Nil Geometry

Isoptic curves of cycloids

Isoptic surfaces of segments in S<sup>2</sup>×R and H<sup>2</sup>×R geometries