Example: Eliminate parameters to solve the intersection points of projective curves

Introduction

In projective geometry, curves are often described using parametric equations that involve one or more parameters. These parameters allow for the representation of points in a projective plane, where lines and curves can intersect at points at infinity. The goal of this article is to demonstrate how to eliminate parameters from such parametric equations to determine the intersection points of two projective curves. This process is fundamental in understanding the algebraic and geometric properties of curves in projective space.

The elimination of parameters involves reducing the system of equations derived from the parametric representations of two curves to a form that can be solved for the coordinates of their intersection points. This approach is particularly useful when dealing with curves that are defined implicitly or when the parameters are not easily eliminated through algebraic manipulation. The following sections outline the methodology for eliminating parameters and solving for intersection points, using examples from projective geometry.

Parametrization of Projective Curves

In projective geometry, a curve can be parametrized using a set of parametric equations. For example, a conic section in the projective plane can be represented as:
$$ x = x(t), \quad y = y(t), \quad z = z(t) $$
where $ t $ is a parameter. These equations describe points on the curve as $ t $ varies over a domain. Similarly, another curve might be parametrized as:
$$ x = x'(s), \quad y = y'(s), \quad z = z'(s) $$
where $ s $ is another parameter. The elimination of parameters involves finding a relationship between the coordinates of the two curves that allows for the determination of their intersection points.

To proceed, it is essential to ensure that the parametric equations of the two curves are consistent and that the parameters are independent. If the parameters are dependent, the curves may intersect at points at infinity, which must be considered in the analysis.

Elimination of Parameters

The elimination of parameters is achieved by eliminating one parameter from the system of equations derived from the parametric representations of the two curves. For instance, consider two curves:

Curve $ C_1 $: $ x = x_1(t) $, $ y = y_1(t) $, $ z = z_1(t) $
Curve $ C_2 $: $ x = x_2(s) $, $ y = y_2(s) $, $ z = z_2(s) $

To eliminate $ t $ from $ C_1 $, we solve for $ t $ in terms of $ x $, $ y $, and $ z $, and substitute into the equations of $ C_2 $. This results in a system of equations in terms of $ x $, $ y $, and $ z $, which can then be solved to find the coordinates of intersection points.

For example, suppose $ C_1 $ is parametrized as:
$$ x = t, \quad y = t^2, \quad z = t^3 $$
and $ C_2 $ is parametrized as:
$$ x = 2s, \quad y = s^2, \quad z = s^3 $$
Eliminating $ t $ from $ C_1 $ gives $ s = t $, and substituting into $ C_2 $ yields:
$$ x = 2t, \quad y = t^2, \quad z = t^3 $$
This results in the same parametric equations as $ C_1 $, indicating that the two curves are identical. However, if $ C_2 $ is a different curve, the elimination process would yield a system of equations that can be solved for the parameters $ t $ and $ s $, leading to the coordinates of intersection points.

Solving for Intersection Points

Once the parameters are eliminated, the resulting equations can be solved to find the intersection points. This involves solving a system of equations derived from the parametric representations of the curves. For instance, if the parametric equations of $ C_1 $ and $ C_2 $ are:
$$ x_1(t) = t, \quad y_1(t) = t^2, \quad z_1(t) = t^3 $$
$$ x_2(s) = 2s, \quad y_2(s) = s^2, \quad z_2(s) = s^3 $$
Eliminating $ t $ and $ s $ leads to the equations:
$$ t = 2s, \quad t^2 = s^2, \quad t^3 = s^3 $$
Solving these equations, we find that $ t = 2s $, and substituting into $ t^2 = s^2 $ gives $ (2s)^2 = s^2 $, which simplifies to $ 4s^2 = s^2 $, leading to $ s = 0 $ and $ t = 0 $. This corresponds to the point $ (0, 0, 0) $, which is the origin in the projective plane.

However, if the curves are not identical, the elimination process may yield a system of equations that can be solved for the parameters $ t $ and $ s $. For example, consider two curves:
$$ C_1: \quad x = t, \quad y = t^2, \quad z = t^3 $$
$$ C_2: \quad x = 2s, \quad y = s^2, \quad z = s^3 $$
Eliminating $ t $ and $ s $ leads to the equations:
$$ t = 2s, \quad t^2 = s^2, \quad t^3 = s^3 $$
Solving these equations, we find that $ t = 2s $, and substituting into $ t^2 = s^2 $ gives $ 4s^2 = s^2 $, leading to $ s = 0 $ and $ t = 0 $. This again results in the origin, but if the curves are different, the elimination process may yield additional solutions.

Applications in Projective Geometry

The elimination of parameters is a critical technique in projective geometry, as it allows for the determination of intersection points between curves that may be defined implicitly or parametrically. This process is particularly useful in solving systems of equations derived from parametric curves, where the parameters are not easily eliminated through algebra