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QA-7: QA-Conversion DT → CT – coordinates


Let S1.S2.S3 be the QA-Diagonal Triangle of the Reference Quadrangle P1.P2.P3.P4.
Let S1.S2.S3 be the Reference Triangle.
Let P4 be an arbitrary point of the Quadrangle with coordinates (p:q:r) wrt the DT.
The Component Triangle P1.P2.P3 is the Anticevian Triangle of P4 wrt S1.S2.S3.
Let Q be some point to be converted from DT- to CT-coordinates.


Let Qc (xc : yc : zc) be the presentation of Q in barycentric coordinates wrt the Component Triangle. Let Qd (xd : yd : zd) be the presentation of Q in barycentric coordinates wrt the Diagonal Triangle.
Now Qd = xd·cfd1·S1 + yd·cfd2·S2 + zd·cfd3·S3 wrt the Reference Diagonal Triangle
also Qc = xc·cfc1·P1 + yc·cfc2·P2 + zc·cfc3·P3 wrt the Diagonal Triangle,
where:

  • (xd : yd : zd) are the barycentric coordinates of Q wrt the Diagonal Triangle,
  • (xc : yc : zc) are the barycentric coordinates of Q wrt the Component Triangle,
  • cfc1, cfc2, cfc3 are the Compliance Factors of the Component Triangle,
  • cfd1, cfd2, cfd3 are the Compliance Factors of the Diagonal Triangle.

Explanation of Compliance Factors can be found at [26b page 40].
Since the Diagonal Triangle is the Reference Triangle, the Compliance Factors of the Component Triangle are all equal 1.
The Compliance Factors of the Component Triangle are:

  • cfc1 = Det [Gc, P2, P3] / Det [P1, P2, P3]
  • cfc2 = Det [P1, Gc, P3] / Det [P1, P2, P3]
  • cfc3 = Det [P1, P2, Gc] / Det [P1, P2, P3]

where Gc = the Centroid of the Component Triangle and “Det” is abbreviation for “Determinant”.
Calculation gives 2 presentations of the coordinates of Q wrt the Diagonal Triangle:

  • Qd = (xd : yd : zd),
  • Qc = (−p (p2 xc − q2 xc + 2 q r xc − r2 xc + p2 yc − q2 yc − 2 p r yc + r2 yc + p2 zc − 2 p q zc + q2 zc − r2 zc) :
    −q (−p2 xc + q2 xc − 2 q r xc + r2 xc − p2 yc + q2 yc + 2 p r yc − r2 yc + p2 zc − 2 p q zc + q2 zc − r2 zc) :
    −r (−p2 xc + q2 xc − 2 q r xc + r2 xc + p2 yc − q2 yc − 2 p r yc + r2 yc − p2 zc + 2 p q zc − q2 zc + r2 zc))

Since Qc and Qd present the same point we can now calculate the coordinates of Q wrt the Component Triangle:

  • (xc : yc : zc) =
(p (p − q − r)(r·yd + q·zd) : q (−p + q − r)(r·xd + p·zd) : r (−p − q + r)(q·xd + p·yd))

However we have to bear in mind that the variables in these coordinates are expressions in (a, b, c) and (p, q, r), which are variables wrt the Diagonal Triangle.
Therefore the DT → CT-conversion of P(x : y : z) consists of 3 consecutive steps:

  1. Transform Point (x : y : z) →
    (p (p − q − r)(r·y + q·z) :
    q (−p + q − r)(r·x + p·z) :
    r (−p − q + r)(q·x + p·y))
  1. Replace:
    p → (q + r)
    q → (p + r)
    r → (p + q)
  1. Replace:
    a2 → (p2(q − r)2 SA + q2(p + r)2 SB + (p + q)2 r2 SC) / ((p + q)2 (p + r)2)
    b2 → (p2(q + r)2 SA + q2(p − r)2 SB + (p + q)2 r2 SC) / ((p + q)2 (q + r)2)
    c2 → (p2(q + r)2 SA + q2(p + r)2 SB + (p − q)2 r2 SC) / ((p + r)2 (q + r)2)


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