Circular_Cubics

Circular cubics

These are cubics created by intersection points of two families of circles (lines of circles or circle-bundles). The circular cubics above are created as geometric loci of such intersection-points. There are two bundles:
(I) Circles orthogonal to the two circles (EF) and (CD), whose radical axis is line (KL).
(II) Circles orthogonal to the two circles (AB) and (GH), whose radical axis is line (NO).
There is also the auxiliary line (IJ) and a point M moving on it.
The points of the locus (in dependence from M) are constructed as follows:
M is projected on the two radical axes, at points Q and P respectively. Then, with centers at these points, two circles are constructed, belonging, each, to the respective family:
Circle (QR) with center at Q, orthogonal to (EF) and (CD), hence belonging to family (I),
Circle (PS) with center at P, orthogonal to (GH) and (AB), hence belonging to family (II).
The two circles (QR) and (PS) intersect at points T and U. T and U describe the locus, as point M moves on the auxiliary line (IJ).
The locus depends on 5 objects (masters). First the four circles defining the families (I) and (II). Then the auxiliary line (IJ) realizing a linear relation between the members (circles) of the two families. Thus, modifying these 5 objects, we obtain a variety of shapes (circular cubics). Some of them need to increase the interpolation points in order to be drawn correctly (without big gaps). This can be done right-clicking on the corresponding component, selecting [Data] and then [InterPts]. Then press continuously the right arrow on the top of the [Data-dialog] to augment the number of interpolation points.

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