2nd order curves - CONICS
Within our course we will mostly study the 2nd order curves, called conics.
Figure 5
The line at infinity of the plane of the conic also intersects the conic in
two points. We classify conics based on the type of this intersection: Figure 6
Figure 7 Plane algebraic curves can be sorted not just by their degree, but also by class of the curve. Class of a plane algebraic curve equals the number of tangent lines to the curve from any point of the plane of the curve.
Tangents to the curve can be real but can also be conjugate imaginary. Imaginary
lines are very hard to imagine, just like imaginary points. We state that all
conics are curves of class 2 - that through any point in the plane we have two
tangent lines of a conic. These tangent lines are real and different if the
point lies in the exterior of the conic, real but coincide if the point lies on
the conic and conjugate imaginary if the point lies in the interior of the
conic. If the degree and the class of the algebraic curve coincide, than we say
that curve has order. Conics are 2nd order curves.
Move the point X in Image 8 to observe in which areas we have two real tangents
or two imaginary tangents. In the boundary of these areas are the points that
give unique tangent; but it is counted twice in the determination of the class
of the curve.
Figure 8 PARABOLA - ENVELOPE of STRAIGHT LINES All nondegenerate conics (ellipses, hyperbolas, parabolas) have no double points. The expression for the bound of the number of double points (n−1)(n−2)/2 equals 0 when n=2. If a conic has a double point, than it is decomposed into two lines that can be real and different, real and coincide or a pair of conjugate imaginary lines that intersect in a single point. Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb, made with GeoGebra |