[alogo] 1. Conics in homogeneous coordinates

This is a sequel to ProjectiveCoordinates.html handling the description of conics in a homogeneous
or projective coordinate system. Projective coordinate systems are used to represent algebraic curves.
In particular lines, defined by homogeneous equations of degree 1:
                                                        ax1 + bx2 + cx3 = 0,
and conics, defined by  homogeneous equations of degree two:
(0)                                ax12  + bx22 + cx32 + 2dx1x2 + 2ex2x3 + 2fx3x1 = 0.

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If the conic passes through the basis points of the system, then {(1,0,0),(0,1,0),(0,0,1)} must satisfy
the equation. This implies a = b = c = 0, hence the equation obtains the form:
(1)                                                dx1x2 + ex2x3 + fx3x1 = 0.
If in addition the conic passes through the fourth (unit) point D calibrating the coordinate system,
then the coefficients must satisfy:
(2)                                                                d+e+f = 0.
Replacing f = -(d+e) into the equation, we find that the  conic is represented by an equation of the form:
(3)                                               d x1(x2 - x3) + e x3(x2 - x1) = 0.
This means that every conic passing through the four points is given as a linear combination of the two reducible
conics (i.e. conics that can be split into a product of lines):
(4)                                   x1(x2 - x3)  = 0,    and          x3(x2 - x1) = 0.
The first represents the degenerate conic which is the product of two lines:
(5)                                                                BC (x1 = 0)   and    AD (x2 - x3 = 0).
The second represents similarly the conic which is the product of:
(6)                                                                AB(x3 = 0)   and     CD(x2 - x1 = 0).
Notice that (d,e) are defined modulo a non zero multiplicative constant, so that they determine a one parameter
pencil or family of conics. This implies that a fifth point E completely determines the conic
passing through all five points {A,B,C,D,E}.

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[alogo] 2. Bitangent conics, projective equivalence

Another interesting case is that of the conic passing through two points B, C  and having there
tangents intersecting at a point A. Selecting a coordinate system as shown we have:
(i)    B, C lying on the conic ==> b = c = 0.
(ii)   AC resp. AB tangent at C resp. B ==> d = 0  resp.  e = 0,
see the remark in last section.
Thus the conic reduces to the form:
(7)                                             ax12 + fx2x3 = 0.
Thus it belongs to the so-called bitangent pencil or family of conics, represented
by linear combinations of two degenerate conics. The first conic consists of the product of lines:
(8)                                                 x2x3 = 0    (representing lines AB and AC).
The second conic consists of the double line:
(9)                                                 x12 = 0      (representing BC in product with itself).
In particular, if D belongs to the conic, then a + f = 0, and the conic is represented by:
(10)                                                 x12 = x2x3.
This shows that all non-degenerate conics of the projective plane are equivalent.
This because for every such conic we can select a system of coordinates representing the conic
through an equation like the one in (10).
The transformation of projective coordinates which matches two such coordinate systems,
matches also the  corresponding conics expressed in the form of (10).

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[alogo] 3. Self polar triangle of reference

Another interesting case is that of the conic referred to a self-polar triangle ABC.
This means that each vertex is the pole of the opposite side with respect to that conic.
Taking a projective basis {A,B,C,D} as shown (see ProjectiveBasis.html ), the polar of A being BC,
implies d = f = 0. Similarly the polar of B being AC implies d = e = 0. Thus the conic has the form:
(11)                                                  ax12 + bx22 + cx32 = 0.

The argument applied here is that the conic being given by an equation of the (matrix) form:


The polar of a point (u1,u2,u3) is given by the equation:


Thus, if BC (x1 = 0) is the polar of A (1,0,0) which is ax1+dx2+fx3=0, we must have d = f =
0. Analogous arguments have been applied to the case of bitangent conic, to reduce it to the
form x12 = x2x3.
The representation of the polar in this way is proved for the case of cartesian coordinates in the file
Conic_Equation.html . The proof for the case of homogeneous coordinates is essentially the same.

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