Page 84 - Geometric Modeling and Algebraic Geometry

82 R. Krasauskas and S. Zube

Fig. 5.4. Conics of type σ =(+−) in canonical position.

• an affine type T = E, P, H: ellipse, parabola, or hyperbola;
• a signature σ = (++), (+0), (+−) of the spanned affine 2-plane;
• positions of infinite points of C with respect to Ω, e.g. a number n =#(C ∩Ω).
We denote the class T σ , where we skip [n] if n =0, or change [n] to the list of
[n]
signatures α =(α 1 ,α 2 ) of asymptotic directions when it is necessary. In hyperbolic
case H sometimes it is necessary to distinguish conjugated hyperbolas having the
α
α
same asymptotes. We mark with a tilde H a case that is conjugated to H .
σ σ
Theorem 1. All PE equivalence classes of irreducible conics (i.e. not a pair of lines)
with non-empty set of real points in R are listed in the following table, except three
4
1
[2] (0−) (−−)

conjugated cases with totally negative tangents H ,H , H :

+− +− +−
σ =++ E [2]
++ E ++ P ++ H ++
σ =+0 P [2]
E +0
+0 P +0
[1]

H
H +0
H +0
+0
(−) (+) (0)
σ =+− E +− P P P
+− +− +−
(++) (++) (+0) (+0)
H H H H
+− +− +− +−
(0−) (−−) (+−) [2]
H H H H
+− +− +− +−
Conics with different signatures σ from this table are illustrated in Fig. 5.2–
5.4. Arcs of curves with negative tangents are shown in grey. Points with isotropic