2D Curves by Wassenaar J. PDF

By Wassenaar J.

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He did not use the name cissoid. The name of the curve, meaning 'ivy−shaped', is found for the first time in the writings of the Greek Geminus (about 50 BC). Because of Diocles' previous work on the curve his name has been added: cissoid of Diocles. Roberval and Fermat constructed the tangent of the cissoid (1634): from a given point there are either one ore three tangents. In 1658 Huygens and Wallis showed that the area between the curve and his asymptote is π/4. The cissoid of Diocles is a special case of the generalized cissoid, where line l and circle C have been substituted by arbitrary curves C1 and C2.

Euclid wrote about the parabola, and Apollonius (200 BC) gave the curve its name. Pascal saw the curve as the projection of a circle. In the parabola curve the parabola (with its vertex oriented downwards) is being repeated infinitely. Some properties of the parabola: • the parabola is the involute of the semi−cubic parabola • a path of a parabola is formed when the involute of a circle rolls over a line • the curve is a specimen of the sinusoidal spiral • it is the pedal of the Tschirnhausen's cubic Besides, the following curves can be derived from the parabola: kind of derived curve catacaustic rays perpendicular to the axis isoptic orthoptic curve Tschirnhausen's cubic hyperbola the parabola's directrix cardioid polar inverse focus as the center of inversion polar inverse cissoid vertex as the center of inversion roulette a parabola rolls over another (equal) parabola roulette a parabola rolls over a line cissoid: the path of the vertex catenary: the path of the focus Pedals of the parabola are given by: a pedal point pedal of the parabola 0 vertex cissoid (of Diocles) 1 foot of − intersection of (right) axis and − directrix strophoid 3 reflection of focus in trisectrix of directrix MacLaurin − on directrix oblique strophoid − focus line The Italian Luca Valerio determined the area of a parabola, in 1606; is was called the quadrature of the parabola.

The curve is a generalization of the Apollonian circle: Van Rees found some interesting properties of the curve, in 1829. In 1852 Steiner formulated the corresponding problem, unaware of the work of van Rees. It was Gomes Teixeira who remarked this equivalence (in 1915) between the work of Steiner and van Rees. g. Brocard, Chasles, Dandelin, Darboux and Salmon studied them. Nowadays, at the University of Crete, a group around Paris Pamfilos is working on these cubics. They gave the curve the name Apollonian cubic or isoptic cubic and they constructed a tool, named Isoptikon (796 kB) to draw an Apollonian cubic, given the two segments.

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2D Curves by Wassenaar J.


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