or

In particular lines, defined by homogeneous equations of degree 1:

ax

and conics, defined by homogeneous equations of degree two:

(0) ax

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) dx

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 x

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) x

The first represents the degenerate conic which is the product of two lines:

(5) BC (x

The second represents similarly the conic which is the product of:

(6) AB(x

Notice that (d,e) are defined modulo a non zero multiplicative constant, so that they determine a one parameter

passing through all five points {A,B,C,D,E}.

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

Thus the conic reduces to the form:

(7) ax

Thus it belongs to the so-called

by linear combinations of two degenerate conics. The first conic consists of the product of lines:

(8) x

The second conic consists of the

(9) x

In particular, if D belongs to the conic, then a + f = 0, and the conic is represented by:

(10) x

This shows that

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).

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) ax

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

The polar of a point (u

Thus, if BC (x

0. Analogous arguments have been applied to the case of bitangent conic, to reduce it to the

form x

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.

ProjectiveBasis.html

Conic_Equation.html

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