Eye

Eye curve

Given the circle c and point A draw secants ABC and take D on AB such that DC/DB = k (constant).
The locus of D is the (red) eye curve shown.

From its definition follows that:
1) Every secant throught A intersects the curve at two points {D,F}.
2) The same secant intersects the circle at {B,C} and |DB|=|FC|.
3) The common to DF and BC middle E describes a circle with diameter AO.
4) The ratio ED/EB is constant and equal to m = (k+1)/(k-1).
5) The curve is symmetric on the axis OA.

6) Setting  OA = a, R for the radius of the circle the polar equation with center at A is:
                   (r-acos(fi))² = m²(R²-a²sin²(fi)).
7) In cartesian coordinates centered at A and x-axis AO:
                (x²+y²)(x²+y²-2ax-m²R²)+a²(x²+m²y²)=0.
8) Inverting this curve with respect to the circle of radius L=|AH|, centered at A we obtain an ellipse.
In fact, the inverted radius is r' = L²/r and from (6) we obtain:

The figure displays the ellipse obtained through the inversion with radius L = |AH|. It is characterized by its
properties:
a) To be tangent to {AH, AI} respectively at points {H, I} and,
b) to pass through K', which is the inverse of K,
c) its focal points are on the circle through {H,I,A}.
See Inversion.html , Ellipse.html .

Eye curve

See Also