PhilonCubic

Philon's cubic

I use this name for a cubic curve intimately related to the problem of Philon:
Given two intersecting lines d , e and a point O not on these lines, consider all lines through O and their intersection points A, B with d, e. To find the line realizing the minimum segment AB (see Philon.html ).
The cubic results by considering to be fixed the line (d) and points O, Q. Then vary line e (turning about Q). For each position of the line e, the corresponding solution-point B of Philon's problem is a point on a cubic curve.
This is the cubic I mean above. It has an asymptote d' parallel to d. By identifying line OQ with the x-axis, as shown below, lines d and d' intersect the y-axis at points symmetric on O.
By the discussion in the aformentioned reference the cubic is defined as the geometric locus of points B, such that BD = AO, for A varying on line d. This is equivalent with the description of the cubic as the locus of points B, such that B is the symmetric of A w.r. to the projection F of E on OA. E being the middle of the fixed segment OQ.

If Q=(q,0) and C=(c,0), then the equation of the cubic is:
(x²+y²)(cx+qy-qc)-qx(cx+qy) = 0,
which is a "circular cubic" i.e. the circular points at infinity (1,i,0), (1,-i,0) are on the cubic.
The parametric equations of the cubic are:
x(t) = q*(1-t)*fac(t),
y(t) = c*t*fac(t),
where the factor fac(t) = q²(1-t)/((q(1-t))²+(ct)²) - 1.
The cubic has been drawn using this parametric representation. The corresponding script is to be found in the examples under the name PhilonCubic.txt.
The parameters q, c can be controlled by moving the green point with these coordinates. For this switch to the selection tool (press CTRL+1) catch and move this point.
The parameter d sets the interval [-d, d], where the t-parameter varies. N is the number of interpolation points, used to plot the curve.

Philon's cubic

See Also