Representation of Conics in the Oriented Projective Plane

Representing Conics Using the Oriented Projective Plane

Conics on the Projective Plane

In this paper, we discuss a special property of conics on the projective plane and answer questions in enumerative algebraic geometry such as ”How many points determine a conic?” and ”How many conics do we expect to pass through m points and tangent to n lines?”

Plane Conics in Algebraic Geometry

We first examine points and lines within projective spaces. Then we classify affine conics based on the classification of projective conics. Based on the parametrization of conics, we also prove two easy cases of Bézout’s Theorem. In the end we turn to the discussion of the space of conics and the notion of pencils of conics.

Tutorial I: The Algebraic Approach to Invariance

I Transformations and Projections 5 2 Camera Models 5 3 Plane to plane transformations 9 3.1 Plane projectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Homogeneous line representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Projective tra...

Special Conics in a Hyperbolic Plane

In Euclidean geometry we find three types of special conics, which are distinguished with respect to the Euclidean similarity group: circles, parabolas, and equilateral hyperbolas. They have on one hand special elementary geometric properties (c.f. [7]) and on the other they are strongly connected to the “absolute elliptic involution” in the ideal line of the projectively enclosed Euclidean pla...