QL-Tf3 CSCe Transformation
The CSCe Transformation is a linear transformation mapping lines into points.
Consequently its inverse transformation, here called QL-Tf3R or CSCeR, is also a linear transformation mapping points into lines.
This transformation is related to QL-Tf1 (Clawson-Schmidt Conjugate), often abbreviated with CSC. CSC transforms lines into “circles-through-QL-P1” and transforms “circles-through-QL-P1” into lines. CSCe is transforming a line into the center of the “circle-through-QL-P1”.
CSCe was introduced in Ref-34, QFG-message #430.
Special about this transformation is this property:
- Let Li1 and Li2 be two random lines.
- Let Li12 be new line CSCe(n1).CSCe(n2).
- Now CSCe(Li12) will be Li1^Li2 !
A corresponding property is valid for CSCeR.
The transformation CSCe even becomes more beautiful when dealing with 3 random lines:
- Let Li1, Li2 and Li3 be three random lines.
- Let Tr1 = Triangle (Li1,Li2,Li3) and Tr2 = Triangle (CSCe(Li1), CSCe(Li2), CSCe(Li3)).
- Now each vertex or line from these triangles will be mutually related by CSCe or CSCeR. When transforming a triangle the vertices or the sidelines can be transformed (resp. by CSCeR and CSCe), the outcome will be the same transformed triangle!
- And Tr1 will be perspective with Tr2 with some perspector Pe and some perspectrix Px.
- Moreover CSCe(Px) = Pe and correspondingly CSCeR(Pe) = Px.
- Another consequence is that every QL-Triangle can be transformed into a point Pe and line Px (using CSCe or CSCeR). This can be considered as another QL-transformation mapping a QL-Triangle into a QL-point and a QL-line.
Corresponding properties are valid for CSCeR.
Consequently CSCe and CSCeR have a Dual-function, meaning:
Each line has a corresponding point and each point has a corresponding line.
Lines are transformed into points and points into lines.
The Trilinear Pole and the Trilinear Polar have corresponding functions in a triangle.
These special properties occur when the transformation is an involutary correlation or also called a polarity. See Ref-53, page 61.
This can be checked in the transformation matrix in general form:
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a11
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a12
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a13
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a21
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a22
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a23
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a31
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a32
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a33
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when the matrix equals its transpose matrix meaning that a12=a21, a23=a32, a13=a31.
See Ref-54, page 80, section 3.4.
For QL-Tf3 this is true as can be deducted from following transformed coordinates.
More information about QL-Tf3 can be found at Ref-34, QFG-messages #430, #435, #436, #442, #453, #454, #456, #465, #470, #471, #480, #489, #492, #1682, #1684, #1687, #1691, #1697.
CT-coordinates of the CSCe Transformation of some line (x:y:z):
(-a2 m n (a4 l2 - a2 b2 l2 - a2 c2 l2 - a4 l m + a2 b2 l m + 2 a2 c2 l m + b2 c2 l m - c4 l m - a4 l n + 2 a2 b2 l n - b4 l n + a2 c2 l n + b2 c2 l n + a4 m n - 2 a2 b2 m n + b4 m n - 2 a2 c2 m n - 2 b2 c2 m n + c4 m n) x
- a2 b2 l m n (-a2 l + b2 l - c2 l + a2 m - b2 m - c2 m + 2 c2 n) y
- a2 c2 l m n (-a2 l - b2 l + c2 l + 2 b2 m + a2 n - b2 n - c2 n) z :
-a2 b2 l m n (-a2 l + b2 l - c2 l + a2 m - b2 m - c2 m + 2 c2 n) x -
b2 l n (a2 b2 l m - b4 l m + a2 c2 l m + 2 b2 c2 l m - c4 l m - a2 b2 m2 + b4 m2 - b2 c2 m2 + a4 l n - 2 a2 b2 l n + b4 l n - 2 a2 c2 l n - 2 b2 c2 l n + c4 l n - a4 m n + 2 a2 b2 m n - b4 m n + a2 c2 m n + b2 c2 m n) y -
b2 c2 l m n (2 a2 l - a2 m - b2 m + c2 m - a2 n + b2 n - c2 n) z :
-a2 c2 l m n ( -a2 l - b2 l + c2 l + 2 b2 m + a2 n - b2 n - c2 n) x -
b2 c2 l m n (2 a2 l - a2 m - b2 m + c2 m - a2 n + b2 n - c2 n) y -
c2 l m (a4 l m - 2 a2 b2 l m + b4 l m - 2 a2 c2 l m - 2 b2 c2 l m + c4 l m + a2 b2 l n - b4 l n + a2 c2 l n + 2 b2 c2 l n - c4 l n - a4 m n + a2 b2 m n + 2 a2 c2 m n + b2 c2 m n - c4 m n - a2 c2 n2 - b2 c2 n2 + c4 n2) z )
CT-coordinates of the Inverse CSCe Transformation CSCeR of some point (x:y:z):
(b2 c2 l (-a2 b2 l2 m2 + b4 l2 m2 - a2 c2 l2 m2 - 2 b2 c2 l2 m2 + c4 l2 m2 + a2 b2 l m3 - b4 l m3 + b2 c2 l m3 - a4 l2 m n + 2 a2 b2 l2 m n - 2 b4 l2 m n + 2 a2 c2 l2 m n + 4 b2 c2 l2 m n - 2 c4 l2 m n + a4 l m2 n - a2 b2 l m2 n + 2 b4 l m2 n - b2 c2 l m2 n - c4 l m2 n - a2 b2 m3 n - a2 b2 l2 n2 + b4 l2 n2 - a2 c2 l2 n2 - 2 b2 c2 l2 n2 + c4 l2 n2 + a4 l m n2 - b4 l m n2 - a2 c2 l m n2 - b2 c2 l m n2 + 2 c4 l m n2 - a4 m2 n2 + a2 b2 m2 n2 + a2 c2 m2 n2 + a2 c2 l n3 + b2 c2 l n3 - c4 l n3 - a2 c2 m n3) x
+ a2 b2 c2 l m (-a2 l2 m + b2 l2 m - c2 l2 m + a2 l m2 - b2 l m2 - c2 l m2 - b2 l2 n + c2 l2 n + a2 l m n + b2 l m n + 3 c2 l m n - a2 m2 n + c2 m2 n - 2 c2 l n2 - 2 c2 m n2 + c2 n3) y
+ a2 b2 c2 l n (b2 l2 m - c2 l2 m - 2 b2 l m2 + b2 m3 - a2 l2 n - b2 l2 n + c2 l2 n + a2 l m n + 3 b2 l m n + c2 l m n - 2 b2 m2 n + a2 l n2 - b2 l n2 - c2 l n2 - a2 m n2 + b2 m n2) z :
a2 b2 c2 l m (-a2 l2 m + b2 l2 m - c2 l2 m + a2 l m2 - b2 l m2 - c2 l m2 - b2 l2 n + c2 l2 n + a2 l m n + b2 l m n + 3 c2 l m n - a2 m2 n + c2 m2 n - 2 c2 l n2 - 2 c2 m n2 + c2 n3) x
+ a2 c2 m (-a4 l3 m + a2 b2 l3 m + a2 c2 l3 m + a4 l2 m2 - a2 b2 l2 m2 - 2 a2 c2 l2 m2 - b2 c2 l2 m2 + c4 l2 m2 - a2 b2 l3 n + 2 a4 l2 m n - a2 b2 l2 m n + b4 l2 m n - a2 c2 l2 m n - c4 l2 m n - 2 a4 l m2 n + 2 a2 b2 l m2 n - b4 l m2 n + 4 a2 c2 l m2 n + 2 b2 c2 l m2 n - 2 c4 l m2 n + a2 b2 l2 n2 - b4 l2 n2 + b2 c2 l2 n2 - a4 l m n2 + b4 l m n2 - a2 c2 l m n2 - b2 c2 l m n2 + 2 c4 l m n2 + a4 m2 n2 - a2 b2 m2 n2 - 2 a2 c2 m2 n2 - b2 c2 m2 n2 + c4 m2 n2 - b2 c2 l n3 + a2 c2 m n3 + b2 c2 m n3 - c4 m n3) y
+ a2 b2 c2 m n (a2 l3 - 2 a2 l2 m + a2 l m2 - c2 l m2 - 2 a2 l2 n + 3 a2 l m n + b2 l m n + c2 l m n - a2 m2 n - b2 m2 n + c2 m2 n + a2 l n2 - b2 l n2 - a2 m n2 + b2 m n2 - c2 m n2) z :
a2 b2 c2 l n (b2 l2 m - c2 l2 m - 2 b2 l m2 + b2 m3 - a2 l2 n - b2 l2 n + c2 l2 n + a2 l m n + 3 b2 l m n + c2 l m n - 2 b2 m2 n + a2 l n2 - b2 l n2 - c2 l n2 - a2 m n2 + b2 m n2) x
+ a2 b2 c2 m n (a2 l3 - 2 a2 l2 m + a2 l m2 - c2 l m2 - 2 a2 l2 n + 3 a2 l m n + b2 l m n + c2 l m n - a2 m2 n - b2 m2 n + c2 m2 n + a2 l n2 - b2 l n2 - a2 m n2 + b2 m n2 - c2 m n2) y
+ a2 b2 n (-a2 c2 l3 m + a2 c2 l2 m2 + b2 c2 l2 m2 - c4 l2 m2 - b2 c2 l m3 - a4 l3 n + a2 b2 l3 n + a2 c2 l3 n + 2 a4 l2 m n - a2 b2 l2 m n - b4 l2 m n - a2 c2 l2 m n + c4 l2 m n - a4 l m2 n - a2 b2 l m2 n + 2 b4 l m2 n - b2 c2 l m2 n + c4 l m2 n + a2 b2 m3 n - b4 m3 n + b2 c2 m3 n + a4 l2 n2 - 2 a2 b2 l2 n2 + b4 l2 n2 - a2 c2 l2 n2 - b2 c2 l2 n2 - 2 a4 l m n2 + 4 a2 b2 l m n2 - 2 b4 l m n2 + 2 a2 c2 l m n2 + 2 b2 c2 l m n2 - c4 l m n2 + a4 m2 n2 - 2 a2 b2 m2 n2 + b4 m2 n2 - a2 c2 m2 n2 - b2 c2 m2 n2) z )
Examples of CSCe Transformations:
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Line
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Point
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QL-L2 Steiner Line
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QL-L3 Pedal Line
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QL-L5 NSM Line
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Point on QL-L2
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Properties:
- CSCe(QL-P3.QL-P4) is the intersection of QL-L2 and CSC(QL-Ci6). See Ref-34, Eckart Schmidt, QFG-message #1687.
- CSC-images of two points and the CSCe-image of lines through the circumcenter of the two points and QL-P1 are collinear:
- If two lines are parallel, their CSCe-images and QL-P1 are collinear:
- If lines have a common point, their CSCe-images are collinear:
- If two lines intersect in P, their CSCe-images and the CSC-images of points on a circle round P through QL-P1 are collinear: