Newton_Conics

1. Angle defined homography

A fixed oriented angle ω=angle(DEF) operates on the pencil A* of lines trhough a fixed point A, by associating
to each line k of A* the line k' of A* such that the angle(k,k')=ω. This map is a homography of A* to itself.

To prove it consider a line and calculate the traces of the lines k=AB, k'=AC in terms of coordinates,
showing that the coordinates of {B,C}are related by a homographic relation of the form
x' = (ax+b)/(cx+d). Let the coordinates be  A(x₀, y₀), B(x,0), C(x',0). Then it is:

Parameter t is determined from the vanishing of the y-coordinate:

Last equation shows that x' has the required dependence on x.

2. Newton's conics

Given are two fixed angles ω and φ, rotating about the fixed points {A,B}. If two of their sides
{AP, BP} say intersect at a point P of a fixed line e, then their other sides intersect at a point Q describing
a conic as P moves on line e.

The property follows from the Chasles-Steiner method of generation of conics. By the previous section the
two pencils of lines A* at A and B* at B are related by a homographic map. More precisely the map
AQ --> BQ of A* to B* is a homography. Hence, by the Chasles-Steiner principle of generation of conics
(see Chasles_Steiner.html ) their intersection point Q describes a conic.

The figure displays some additional constructions related to the conic which can be easily carried out and
determine a point Q' of it and the tangents {AD, BD} at the given points {A,B}.
Point Q' corresponds to the position P=C of line e lying on the extension of line AB.
The tangents at {A,B} result by easy constructions carried out at points {A',B'} for which the corresponding
Q is identical respectively with {A,B}.

The conic then is easily constructed as a member of the bitangent family generated by the degenerate concis
consisting of line-pair {DA,DB} and the double line AB. The member is located by the condition to pass through Q'.

1. Angle defined homography

2. Newton's conics

See Also