Algebraically, if P

The arbitrary constants k

QKP

This represents a system of 3 homogeneous equations with four unknowns (unknowns are k

As an example, construct the projectivity fixing the points [1,0,1]

Equation QKP

And has the solutions (k

The projectivity defined through that matrix maps the points (x,y) of the unit circle: x

(x', y') satisfying y' = x'

proves that this projectivity F has for every point X(x,y), the image-point F(X)=(x',y') and the point E(0,-1) collinear. Further, by inspecting [A], we see that (i) the x-axis and point E are the only fixed points of F, whereas the whole line y = 1 maps to the line at infinity. A further computation of the cross-ratio [E,X,F(X),Y], where Y the intersection point of line EX with the x-axis, shows that this is -1 (harmonic). F coincides not with the

In this discussion I have consistently and often used the

The above image dictates a procedure of generation of parabolas, studied in ParabolaProjectFromCircle.html .

HomographyConic3by3.html

HyperbolaFromEllipse.html

HyperbolaGeneration.html

HyperbolaGeneration2.html

ParabolaProjectFromCircle.html

ProjectiveCollinearityQuadrangle.html

ProjectivityFixingVertices.html

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