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BICORN
Curve studied by Sylvester
in 1864, Cayley in 1867 and G. de Longchamps in 1897.
Name given by Sylvester. Other name : cocked hat curve. 
Cartesian equation:
or .
Cartesian parametrization: or . Area: . Rational quartic, circular curve. 
First construction (Charlotte Scott,
Intermédiaire des mathématiciens, 1896, p. 250)
Let (C) and (C') be two tangent circles, of centres O and O' and of radius a; N be a point describing (C'). The bicorn is the locus of the intersection points between the line parallel to (OO') passing through N and the polar of N with respect to the circle (C). 

Second construction (G. de Longchamps, JMS,
1897)
Given two fixed points A(a, 0) and B(–a, 0), a circle (C) of centre C(b, c) and radius r, with a current point P, the parametrization of the locus of the orthocentre H of the triangle ABP is ; when , we get the bicorn. Note: when a = b = c = 0, we get the Kappa curve, when , we get the right strophoid., and when we get the Delanges trisectrix. 

Third construction (V. Jerabek, mathesis,
1912).
Given two fixed points A(a, 0) and B(–a, 0), a circle (C) of centre C(0, b) and radius c, and a variable diameter [PQ] of the circle (C), the parametrization of the locus of the intersection points of the lines (AP) and (BQ) (or (AQ) and (BP) ) is ; when , we get the bicorn. If M and N are the intersection points of the lines (AP) and (BQ) and the lines (AQ) and (BP), the line (MN) envelopes a right cissoid. It is possible to generalize by considering the two points P and Q as linked to a moving plane overlying a fixed plane containing A and B ; here, the moving plane is in rotational motion around C. 

With this construction we can get the bicorn as the orthogonal
projection of the intersection of an elliptic cone and a hyperbolic paraboloid
(biquadratic
curve).
The vertex of the cone is the point A_{1}(a,0, h) on the vertical line passing by A, and its directrix is the circle (C). The paraboloid is defined by the 3D quadrilateral (A_{1}B'BB_{1}) where B'(a, a, 0) and B are symmetric with respect to C, and B_{1}(–a, 0, h) is on the vertical line passing through B. 

Another characterization (Roland
Deaux, 1945): the bicorn is the locus of the centre of the incircle
of a triangle two vertices of which are fixed and the perimeter of which
equals the sum of the radiuses of its excircles.
A point M of a circle can be projected on two
points P and Q on two tangents to the circle perpendicular
to one another. When M describes the circle, the line (PQ)
envelopes a quartic with two cusps whose parametrization is, up to similarity,
given by .
But it is not a bicorn because the tangents at the cusps do not pass through the vertex. 



© Service audiovisuel École Polytechnique 
... Henri Lazennec applied Jerabek's construction and
replaced the circle by a cardioid.
He considers the points A(–4, 0), B(4, 0), C(0, –3), D(0, –2). The extremities of the line segment [PQ] of fixed length 4, passing through D and whose middle I describes the circle of centre C and radius 1, describe a cardioid with cusp D. When [PQ] turns on the cardioid, the intersection point M of (AP) and (BQ) describes a curve looking like the bicorn, with its tips upsidedown. Parametrization: . 
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© Robert FERRÉOL 2017