QG-Co1: Inscribed Harmonic Conic


There is a projective transformation from 4 lines in a square to any other set of 4 lines.
See chapter QG-Tf1: QG-Projective Square Transformation.
The Inscribed Harmonic Conic is the projective transformation of the inscribed circle in a square to the Reference Quadrigon. This conic touches the sidelines of the Quadrigon in their perspective midpoints.
See picture below.
 
QG-Co1-InscribedHarmonicConic-01mut


Construction:
QG-Co1-Inscribed-Harmonic-Conic-20-Constr
Equations: 
Equation QG-Co1 in 3 QA-Quadrigons in CT-notation:
  • q2 r2 x2 + p2 r2 y2 + p2 q2 z2 - 6 p q r2 x y + 2 p2 q r y z + 2 p q2 r x z = 0
  • q2 r2 x2 + p2 r2 y2 + p2 q2 z2 + 2 p q r2 x y - 6 p2 q r y z + 2 p q2 r x z = 0
  • q2 r2 x2 + p2 r2 y2 + p2 q2 z2 + 2 p q r2 x y + 2 p2 q r y z - 6 p q2 r x z = 0
 
Equation QG-Co1 in 3 QL-Quadrigons in CT-notation:
  • l2 x2 + m2 y2 + 4 n2 z2 - 2 l m x y + 4 m n y z + 4 l n x z = 0
  • 4 l2 x2 + m2 y2 + n2 z2 + 4 l m x y - 2 m n y z + 4 l n x z = 0
  • l2 x2 + 4 m2 y2 + n2 z2 + 4 l m x y + 4 m n y z - 2 l n x z = 0
 
Equation QG-Co1 in 3 QA-Quadrigons in DT-notation:
  •   q2 r2 x- p2 r2 y2 + p2 q2 z2 = 0
  • -r2 q2 x2 + r2 p2 y2 + p2 q2 z2 = 0
  •   q2 r2 x2 + r2 p2 y- q2 p2 z2 = 0
 
Equation QG-Co1 in 3 QL-Quadrigons in DT-notation:
  • 2 x2 l-     y2 m2 + 2 z2 n2 = 0
  • - x2 l + 2 y2 m2 + 2 z2 n2 = 0
  • 2 x2 l2 + 2 y2 m-     z2 n2 = 0

Properties: