BESACE

 Curve studied and named by Cramer in 1750.

 Cartesian equation: or , with . Or . Cartesian parametrization: ,  of , where . Rational quartic. Total area: ac.

Given a diameter [AB] and a point O on a circle (C), the associated besace is the locus of the points M on a moving line (D) parallel to (OA) such that QM=OP where P is an intersection point between (D) and (C) and Q is the projection of O on (D) (here, A(a,0) and B(0,b)).

The third form of the Cartesian equation shows that besaces are the polyzomal curves median of the coaxial parabolas  and .

Besaces are the projections of Viviani's curve on the planes passing by the axis of the cylinder on which the curve is traced.

They are also the projections of the pancake curve on the planes passing by the axis of the associated cylinder.
The second parametrization above shows that besaces are examples of Lissajous curves.

When b = 0 (i.e. when the circle is tangent to Oy), we have the lemniscate of Gerono and when a = 0, a parabolic segment.

© Robert FERRÉOL 2017