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X(2), X(6), X(13), X(14), X(485), X(486), X(3070), X(3071)

X(42498) → X(42509)

Geometric properties :

K1195 is a cuspidal KHO-cubic. See K1191 for explanations and also CL075.

Its KHO-equation is : x^2 (8y - 7z) - 3 (2y - z)^3 = 0.

K1195 has a cusp at G with cuspidal tangent the Euler line and a point of inflexion at K with inflexional tangent passing through X(5071). The Hessian of K1195 splits into the line GK and the Euler line counted twice.

K1195 is bitangent at X(13), X(14) to the Evans conic with tangents passing through X(549). The remaining common points are X(3070), X(3071).

K1195 meets the Kiepert hyperbola at G (twice), X(13), X(14), X(485), X(486). The tangentials of X(486), X(485) are X(3070), X(3071) respectively.


More generally, a cubic with KHO-equation x^2(v y + w z) - (2y - z)^3 = 0 and v y + w z ≠ 2y - z is a similar cuspidal cubic. See K1204 for instance.

If x : y : z are the KHO-coordinates of a point T (different from G and K) on the cubic, then the point

T(n) = ((-1)^n (v + 2w) x^3 , 2^n (2y - z) (w x^2 + 2^(2n) (2y - z)^2) , 2^n (2y - z) (- v x^2 + 2^(1 + 2n) (2y - z)^2))

is its nth tangential. Obviously, T(0) = T and T(n + 1) is the tangential of T(n). Conversely, T(n - 1) is the pretangential of T(n).

The limit points are T(+∞) = G and T(-∞) = K.

Peter Moses has computed a large selection of these cuspidal cubics. See Table 3 in CL075.