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X(4), X(6), X(13), X(14), X(15), X(16), X(31412), X(42561)

X(43397) → X(43414)

other points below

Geometric properties :

K1228 is a cuspidal KHO-cubic, see K1191 for explanations and also CL075.

Its KHO-equation is : x^2 (4 y - 9 z) + 9 z^3 = 0 or, equivalently, 4 x^2 y - 9 z (x^2 - z^2) = 0.

H is a cusp with cuspidal tangent the Euler line.

K is a point of inflexion with inflexional tangent passing through the homothetic of H under h(G, 7/13).

Parametrization :

For any real number t or infinity (giving H), the KHO-point P(t) = (4, 9 t (1 - t^2), 4 t) lies on K1228. This gives a lot of simple points on the curve. Note that X(6), P(t), P(-t) are collinear.

The third point of the cubic on the line passing through P(t1), P(t2) is P(t3) where t3 = - (t1 + t2). Hence, three points P(t1), P(t2), P(t3) are collinear on the cubic if and only if t1 + t2 + t3 = 0.

In particular, the tangential of P(t) is P(- 2t).

K1228 meets the Kiepert hyperbola at H (twice), X(13), X(14) and two points K5, K6 = P(±√6 / 3) i.e. the KHO-points (± 2√6,3,4).

K1228 meets the Evans conic at X(13), X(14), X(15), X(16) and two points E5, E6 = P(±4√3 / 9) i.e. the KHO-points (± 9√3,11,12). These points are X(43409), X(43410).

KHO-points on the cubic :

(±3,-7,4) = P(± 4 /3) on the lines {14,16}, {13,15} respectively. These points are X(43401), X(43402).

(±6,5,4) = P(± 2 /3) on the lines {13,16}, {14,15} respectively. These are the tangentials of X(14), X(13). These points are X(43403), X(43404).

(±2,27,-4) = P(± 2), the tangentials of X(15), X(16). These points lie on the lines passing through H and X(11480), X(11481) respectively. These points are X(43397), X(43398).

Other points : (±3,20,-5), (±3,70,-7), (±1,54,-3), (±1,135,-4), (±8√6,15,8), (±√6,15,-4).

Other similar cuspidal cubics

All points and equations are given in KHO-coordinates unless otherwise specified.

Let m, n be two non-zero numbers The KHO-cubic K(m,n) with equation x^2(m y - n z) + n z^3 = 0 is the locus of point P(t) = (m, n t (1-t^2), m t) where t is a parameter or infinity (in which case P(t) is H).

K1228 is clearly K(4,9).

K(m,n) is a cuspidal cubic with cusp X(4) and cuspidal tangent the Euler line.

X(6) = P(0) is a point of inflexion with inflexional tangent m y - n z = 0 meeting the Euler line at (0,n,m).

K(m,n) passes through X(15) = (1,0,1) = P(1) and X(16) = (1,0,-1) = P(-1). The tangents at these points meet at (0,2n,-m) on the Euler line.

Three points P(t1), P(t2), P(t3) are collinear on K(m,n) if and only if t1 + t2 + t3 = 0. It follows that the tangential of P(t) is P(-2t).

In particular, P(-2) = (m,6n,-2m) and P(2) = (m,-6n,2m) are the respective tangentials of X(15), X(16). These points lie on the line passing through X(6) and (0,3n,-m).

The hessian of K(m,n) is the union of the Euler line counted twice and the van Aubel line (passing through H and K).

The polar conic of X(3) in K(m,n) is the union of the lines passing through H and X(371), X(372) respectively.

Note that these two latter curves are the same for every K(m,n).

Some remarkable examples

• K(1,-1) is the only K+ and its asymptotes (one only is real) meet at X, the homothetic image of X(6) under h(X4,1/3). The inflexional tangent at X(6) is parallel to the Euler line.

• K(2,3) and K(6,1) are the only cubics bitangent to the Kiepert hyperbola.

With K(2,3), the contacts are X(485), X(486) – with tangents passing through X(3) – and the cubic also contains X(42258), X(42259), X(42992) , X(42993).

With K(6,1), the contacts are imaginary namely (±3i,1,3) on the line X(6)X(140). It also contains X(42087), X(42088), X(42260), X(42261) and obviously X(4), X(6), X(15), X(16).

• K(3,2±2√3) are the only cubics bitangent to the Evans conic. The contacts are rather complicated.

• K(2,-3) meets the Evans conic at X(15), X(16), X(590), X(615), X(41973), X(41974). It also contains X(6560), X(6561) and obviously X(4), X(6).

• K(2,1) has its tangents at X(15), X(16) parallel to the Euler line and tangent at X(13), X(14) to the Kiepert hyperbola. The tangentials of X(15), X(16) are X(42109), X(42108) respectively.