×

Equioptic curves of conic sections. (English) Zbl 1202.51022

Let \(c\) be a given curve of the real Euclidean plane \(E_2\). The locus of points of \(E_2\) where \(c\) is seen under a given fixed angle \(\phi\in(0,2\pi)\) is called isoptic curve of \(c\); especially for \(\phi=\pi/2\) one speaks of the orthoptic curve of \(\,\,c\). The isoptic curve of an ellipse or a hyperbola is a quartic, the isoptic curve of a parabola is for \(\phi\not=\pi/2\) a hyperbola (and for \(\phi=\pi/2\) the doubly counted directrix).
The locus of points of \(E_2\) where two given curves \(c_1\) and \(c_2\) of \(E_2\) are seen under the same angle is called equioptic curve \(e(c_1,c_2)\) of \(c_1\) and \(c_2\). In the case that \(c_1\) and \(c_2\) are algebraic curves of respective degrees \(d_1\) and \(d_2\) the author gives an elimination procedure which applies resultants in order to get a description of \(e(c_1,c_2)\); but this algebraic equation describes an algebraic curve \(e_a(c_1,c_2)\) and \(e(c_1,c_2)\) is a proper subset of \(e_a(c_1,c_2)\) because the underlying structure is now the complex extended projective closure of \(E_2\) and \(e_a(c_1,c_2)\) contains the multiply counted ideal line as well as parasitic branches coming from points with \(|\cos\phi|>1\). On this way the author shows that the degree of \(e(c_1,c_2)\) is at most \(d_1^2d_2^2(d_1-1)^2(d_2-1)^2\).
In essential the author discusses the curve \(e_a(c_1,c_2)\) where \(c_1\) and \(c_2\) are irreducible conics. He proves for instance:
1. (Theorem 3.3, part 1) If \((c_1,c_2)\) is a pair confocal conics with center, then \(e_a(c_1,c_2)\) is the union of the two-fold ideal line, two further pairs of conjugate complex lines, and a circle containing the four common points of \(c_1\) and \(c_2\).
2. (Theorem 3.4, part 1) If \(c_1\) and \(c_2\) are irreducible conics of \(E_2\), then the absolute points of \(E_2\) are singular points on \(e_a(c_1,c_2)\).
The reviewer thinks that the formulation of the theorems is non-uniform: in Theorem 3.1 the author says, if \(c_1\) and \(c_2\) of \(E_2\) are irreducible conics with center, then the degree the equioptic \(e(c_1,c_2)\) is at most \(6\), only in the corresponding proof the reader finds that the ideal lines splits off with multiplicity \(2\).
Furthermore, the author studies the case where \(c_1\) and \(c_2\) are circles, and also the mixed case circle-conic. Finally, some light is shed on the problem of equioptic points of three irreducible conics.
The paper contains seven very aesthetical and illustrative figures.

MSC:

51N35 Questions of classical algebraic geometry
51M04 Elementary problems in Euclidean geometries
14N05 Projective techniques in algebraic geometry
78A05 Geometric optics
PDFBibTeX XMLCite
Full Text: Link