Points of inflexion at circular points at infinity.

Question

According to the Wikipedia article on Cassini Ovals, a Cassini oval has double-points, which are also inflexion points, at circular points I and J at infinity. I don't understand how to show that I and J are inflexion points. Wikipedia references a very old text by Basset which makes the same claim. Unfortunately, I was not able to find any "computational" support for this statement; only a general discussion about points of contact was given. I am looking for an explanation or a good reference that I can use to understand this problem. Thanks in advance!

Answer 1

The Wikipedia article on Circular points at infinity gives their homogeneous coordinates as $$(x_0,y_0,z_0) = (1,\pm i,0). \tag{1}$$ The homogeneous equation for Cassini ovals is $$ P(x,y,z) := (x^2+y^2)^2 - 2 a^2 (x^2-y^2) z^2 + (a^4-b^4) z^4 = 0. \tag{2}$$ Verify for the circular points in equation $(1)$ that $$ P(x_0,y_0,z_0) = 0. \tag{3}$$

As for inflection points, differentiate the equation $(2)$ taking partials to get the equation of the tangent line at $\,(x_0,y_0,z_0).\,$ Namely, $$ \frac14 dP(x,y,z) = x((x^2+y^2)-a^2z^2)dx + y((x^2+y^2)+a^2z^2)dy + z(a^2(y^2-x^2)+(a^4-b^4)z^2)dz.\tag{4}$$ If we evaluate this equation $(4)$ with $\,x^2+y^2=z=0\,$ as in equation $(1)$ it vanishes which verifies that there is a double point at infinity. Also equation $(4)$ is cubic in $\,x,y,z\,$ which verifies that the circular points are also inflection points of the curve.