The parabola is defined by these equations:
c(t) = [ 1/param* t^2 , t ]
The caustic of its normals is the cubic y^2 = x^3.
If t1+t2+t3 = 0, then the normals at c(t1), c(t2), c(t3)
all three intersect in
one point. Since the parabola
normals are tangents to the caustic, this property of
the caustic tangents is called a "geometric addition".
The construction: Connect the focal point F to an arbitrary point P
on the directrix (yellow line). The symmetry line between
F and P is the tangent of the parabola at a point Q such
that PQ is parallel to the axis.
Osculating circles at c(t) are computed using c(t), c'(t), c"(t).
They can also be visually identified: The circle is of course
tangential to the curve, but it also passes from one side of
the curve to the other (except at the vertex). Watch the demo!