next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

DELANGES TRISECTRIX AND SECTRIX

Curve studied by Delanges in 1783.

 
The dotted curve is the Dürer folium, the inverse of which is the Delanges trisectrix.
Polar equation: .
Cartesian parametrization:  (t = q / 2).
Cartesian equation: .
Rational circular quartic.

 
Given a circle (C) (here, the circle with centre O and radius 2a) and a line (D0) passing through the centre of the circle (here Ox), the Delanges trisectrix is the locus of the point M on a variable line (D) passing through O such that the line parallel to D passing through M cuts (C) at N in such a way that (ON) is a bisector of (D0) and (D).
The Delanges trisectrix is the locus of the orthocentre of a triangle with a fixed side the opposite vertex of which describes a circle centred on the middle of the side, with radius the length of the side multiplied by .

See a similar construction for the bicorn, the right strophoid, and the Kappa.

The Delanges trisectrix is a special case of Cotes' spiral.
 
The construction opposite shows the property of trisection: the angle MOP is the third of AOP.

Its inverse curve with respect to O is the Dürer folium, which is, therefore, also a trisectrix.

Furthermore, the same construction shows that the curve with equation  is an n + 1 - sectrix, that can be called "Delanges sectrix".
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL  2017