[alogo] Strophoid

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]
[2_0] [2_1] [2_2]

Let z = (JKL) be a fixed right angle, M a fixed point and N a point moving on line KJ. Consider the circle (KNM) passing through K, N and M. Let P be the other intersection point of this circle with line KL. Project the fixed point K on the variable line NP. The locus of the projection Q is a strophoid. The strophoid has a cusp at K and the tangents there coincide with lines KN and KL. The asymptotic line is line (UV), passing through the projections of T, on the two orthogonal lines. T is the symmetric of the fixed point M with respect to K. The intersection-point W of the strophoid with the asymptotic line is such that WK is orthogonal to KM.
The equation of the strophoid is:
(x + y)(bx +ay) - (a + b)xy = 0.
(a, b) being the coordinates of the fixed point M, with respect to the two orthogonal axes (KJ) and (KL).
(Aubert, Papelier, t1, p. 132)

You can produce automatically the construction-recipe of the above figure by selecting the [Describes Schemes] tool (1st button/6th menu-item).

Produced with EucliDraw©