QL-Cu2: Eckart’s Cubic


QL-Cu2 is the Cubic that passes through the 27 centers of Morley’s Multiple Cardioids. See QL-27Qu1. It was discovered by Eckart Schmidt
(April, 2013). See Ref-34 Quadri-Figures Group, message # 37 (first notification), #52, #56 (asymptotes // axes Deltoid), #57 (QL-P1), #58 (27 Cardioid Centers).
It’s three asymptotes meet in one point QL-P1 (Miquel Center) at angles of 60°.
The asymptotes are parallel to the three axes of the Kantor-Hervey Deltoid QL-Qu2.
The construction of QL-Cu2 can be found at Ref-34, QFG message # 558.
Bernard Gibert made a detailed description at Ref-17d in a paper he wrote on the basis of this special cubic.
More information about the cubic also can be found at Ref-43.

QL-Cu2-EckartsCubic-01 
Equations:
Equation in CT-notation:
     b2 c2 l (a2 ( l m + l n - m n) - b2 (m - l) (m - n) - c2 (n - l) (n - m)) x3
+ a2 c2 m (b2 ( l m - l n + m n) - c2 (n - l) (n - m) - a2 ( l - m) ( l - n)) y3
+ a2 b2 n (c2 (-l m + l n + m n) - a2 (l - m) ( l - n) - b2 (m - l) (m - n)) z3
+ 3 a2 b2 c2 (l x + m y) (m y+ n z) (n z + l x) = 0
Equation in DT-notation:
   (m4 Sa3 - 2 m2 n2 Sa3 + n4 Sa3 + 2 l2 m2 Sa2 Sb + 2 l2 n2 Sa2 Sb + 2 m2 n2 Sa2 Sb + 2 n4 Sa2 Sb + l4 Sa Sb2 + 6 l2 n2 Sa Sb2 + n4 Sa Sb2 + 2 l2 m2 Sa2 Sc + 2 m4 Sa2 Sc + 2 l2 n2 Sa2 Sc + 2 m2 n2 Sa2 Sc + 2 l4 Sa Sb Sc + 6 l2 m2 Sa Sb Sc + 6 l2 n2 Sa Sb Sc + 2 m2 n2 Sa Sb Sc + 4 l4 Sb2 Sc + 4 l2 n2 Sb2 Sc + l4 Sa Sc2 + 6 l2 m2 Sa Sc2 + m4 Sa Sc2 + 4 l4 Sb Sc2 + 4 l2 m2 Sb Sc2) (l2 x3 + 3 m2 x y2 + 3 n2 x z2)
+ (m4 Sa2 Sb + 6 m2 n2 Sa2 Sb + n4 Sa2 Sb + 2 l2 m2 Sa Sb2 + 2 l2 n2 Sa Sb2 + 2 m2 n2 Sa Sb2 + 2 n4 Sa Sb2 + l4 Sb3 - 2 l2 n2 Sb3 + n4 Sb3 + 4 m4 Sa2 Sc + 4 m2 n2 Sa2 Sc + 6 l2 m2 Sa Sb Sc + 2 m4 Sa Sb Sc + 2 l2 n2 Sa Sb Sc + 6 m2 n2 Sa Sb Sc + 2 l4 Sb2 Sc + 2 l2 m2 Sb2 Sc + 2 l2 n2 Sb2 Sc + 2 m2 n2 Sb2 Sc + 4 l2 m2 Sa Sc2 + 4 m4 Sa Sc2 + l4 Sb Sc2 + 6 l2 m2 Sb Sc2 + m4 Sb Sc2) ( m2 y3 + 3 l2 x2 y + 3 n2 y z2)
+ (4 m2 n2 Sa2 Sb + 4 n4 Sa2 Sb + 4 l2 n2 Sa Sb2 + 4 n4 Sa Sb2 + m4 Sa2 Sc + 6 m2 n2 Sa2 Sc + n4 Sa2 Sc + 2 l2 m2 Sa Sb Sc + 6 l2 n2 Sa Sb Sc + 6 m2 n2 Sa Sb Sc + 2 n4 Sa Sb Sc + l4 Sb2 Sc + 6 l2 n2 Sb2 Sc + n4 Sb2 Sc + 2 l2 m2 Sa Sc2 + 2 m4 Sa Sc2 + 2 l2 n2 Sa Sc2 + 2 m2 n2 Sa Sc2 + 2 l4 Sb Sc2 + 2 l2 m2 Sb Sc2 + 2 l2 n2 Sb Sc2 + 2 m2 n2 Sb Sc2 + l4 Sc3 - 2 l2 m2 Sc3 + m4 Sc3) (n2 z3 + 3 l2 x2 z + 3 m2 y2 z)
+ 3 (m6 Sa3 - m4 n2 Sa3 - m2 n4 Sa3 + n6 Sa3 + 3 l2 m4 Sa2 Sb + 2 l2 m2 n2 Sa2 Sb + 3 m4 n2 Sa2 Sb + 3 l2 n4 Sa2 Sb + 2 m2 n4 Sa2 Sb + 3 n6 Sa2 Sb + 3 l4 m2 Sa Sb2 + 3 l4 n2 Sa Sb2 + 2 l2 m2 n2 Sa Sb2 + 2 l2 n4 Sa Sb2 + 3 m2 n4 Sa Sb2 + 3 n6 Sa Sb2 + l6 Sb3 - l4 n2 Sb3 - l2 n4 Sb3 + n6 Sb3 + 3 l2 m4 Sa2 Sc + 3 m6 Sa2 Sc + 2 l2 m2 n2 Sa2 Sc + 2 m4 n2 Sa2 Sc + 3 l2 n4 Sa2 Sc + 3 m2 n4 Sa2 Sc + 6 l4 m2 Sa Sb Sc + 6 l2 m4 Sa Sb Sc + 6 l4 n2 Sa Sb Sc - 4 l2 m2 n2 Sa Sb Sc + 6 m4 n2 Sa Sb Sc + 6 l2 n4 Sa Sb Sc + 6 m2 n4 Sa Sb Sc + 3 l6 Sb2 Sc + 3 l4 m2 Sb2 Sc + 2 l4 n2 Sb2 Sc + 2 l2 m2 n2 Sb2 Sc + 3 l2 n4 Sb2 Sc + 3 m2 n4 Sb2 Sc + 3 l4 m2 Sa Sc2 + 2 l2 m4 Sa Sc2 + 3 m6 Sa Sc2 + 3 l4 n2 Sa Sc2 + 2 l2 m2 n2 Sa Sc2 + 3 m4 n2 Sa Sc2 + 3 l6 Sb Sc2 + 2 l4 m2 Sb Sc2 + 3 l2 m4 Sb Sc2 + 3 l4 n2 Sb Sc2 + 2 l2 m2 n2 Sb Sc2 + 3 m4 n2 Sb Sc2 + l6 Sc3 - l4 m2 Sc3 - l2 m4 Sc3 + m6 Sc3) x y z

Properties:
-        the 27 centers of Morley’s Multiple Cardioids QL-27Qu1.
  • It’s three asymptotes meet in QL-P1 (Miquel Center) at angles of 60°.
  • The asymptotes are parallel to the three axes of the Kantor-Hervey Deltoid QL-Qu2 (Eckart Schmidt, May 9, 2013)
  • The intersection points of QL-Cu2 with its three asymptotes lie collinear on a line parallel to QL-L1 (Newton Line) in 2/3 distance to QL-P1 (Miquel Point)See Ref-34, QFG message # 94 (Eckart Schmidt, June 29, 2013).