Compute the intersection points

Oct 13, 2025 by thesweetdesigns.com 97 views

Introduction

In the realm of geometry and linear algebra, finding common tangents between two ellipses is a complex problem that has garnered significant attention in recent years. The goal of this article is to explore the concept of degenerate conics and their application in determining the common tangents of two ellipses. We will delve into the mathematical framework, provide a step-by-step approach, and offer a Python implementation to facilitate the computation.

Background

An ellipse is a closed curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant. The general equation of an ellipse is given by:

${ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }$

where ${ (h,k) }$ is the center of the ellipse, and ${ a }$ and ${ b }$ are the semi-major and semi-minor axes, respectively.

A tangent to an ellipse is a line that touches the ellipse at exactly one point. The problem of finding common tangents between two ellipses involves determining the lines that are tangent to both ellipses simultaneously.

Degenerate Conics

Degenerate conics are a special class of conic sections that have a specific form. They can be represented by the equation:

${ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 }$

where ${ A, B, C, D, E, }$ and ${ F }$ are constants. Degenerate conics can be classified into three types:

Point conic: A degenerate conic that represents a single point.
Line conic: A degenerate conic that represents a line.
Parabola conic: A degenerate conic that represents a parabola.

In the context of finding common tangents between two ellipses, we are interested in line conics, which can be represented by the equation:

${ Ax + By + C = 0 }$

Mathematical Framework

To determine the common tangents between two ellipses, we can follow these steps:

Compute the intersection points: Find the intersection points of the two ellipses by solving the system of equations formed by the two ellipse equations.
Compute the tangent lines: Compute the tangent lines to each ellipse at the intersection points.
Find the common tangent lines: Find the common tangent lines between the two ellipses by solving the system of equations formed by the tangent lines.

Step 1: Compute the Intersection Points

Let's consider two ellipses with equations:

${ E_1: \frac{(x-h_1)^2}{a_1^2} + \frac{(y-k_1)^2}{b_1^2} = 1 }$

${ E_2: \frac{(x-h_2)^2}{a_2^2} + \frac{(y-k_2)^2}{b_2^2} = 1 }$

To find the intersection points, we need to solve the system of equations formed by the two ellipse equations.

Step 2: Compute the Tangent Lines

Once we have the intersection points, we can compute the tangent lines to each ellipse at the intersection points.

Step 3: Find the Common Tangent Lines

Finally, we can find the common tangent lines between the two ellipses by solving the system of equations formed by the tangent lines.

Python Implementation

Here is a Python implementation to facilitate the computation:

import numpy as np
from scipy.optimize import fsolve

def ellipse_equation(x, h, k, a, b):
    return (x-h)**2/a**2 + (x-k)**2/b**2 - 1

def tangent_line(x, h, k, a, b):
    return (x-h)/a**2 + (x-k)/b**2

def common_tangents(tm):
    # Compute the intersection points
    intersection_points = []
    for i in range(len(tm)):
        for j in range(i+1, len(tm)):
            x = fsolve(lambda x: ellipse_equation(x, tm[i][0], tm[i][1], tm[i][2], tm[i][3]) - ellipse_equation(x, tm[j][0], tm[j][1], tm[j][2], tm[j][3]), 0)[0]
            intersection_points.append((x, tm[i][1]))

    # Compute the tangent lines
    tangent_lines = []
    for point in intersection_points:
        for i in range(len(tm)):
            tangent_lines.append((tangent_line(point[0], tm[i][0], tm[i][1], tm[i][2], tm[i][3]), tangent_line(point[1], tm[i][0], tm[i][1], tm[i][2], tm[i][3])))

    # Find the common tangent lines
    common_tangent_lines = []
    for line in tangent_lines:
        count = 0
        for i in range(len(tm)):
            if line[0] == tangent_line(line[1], tm[i][0], tm[i][1], tm[i][2], tm[i][3]):
                count += 1
        if count == 2:
            common_tangent_lines.append(line)

    return common_tangent_lines

# Define the toy models
tm = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]]

# Compute the common tangent lines
common_tangent_lines = common_tangents(tm)

print(common_tangent_lines)

Conclusion

In this article, we explored the concept of degenerate conics and their application in determining the common tangents of two ellipses. We provided a step-by-step approach and a Python implementation to facilitate the computation. The code can be used to find the common tangent lines between two ellipses, given their equations.

Future Work

The problem of finding common tangents between two ellipses is a complex one, and there are many areas for future research. Some possible directions include:

Improving the computational efficiency: The current implementation has a time complexity of O(n^2), where n is the number of intersection points. Improving the computational efficiency would make the code more practical for large-scale problems.
Handling degenerate cases: The current implementation assumes that the ellipses are non-degenerate. Handling degenerate cases, such as when the ellipses intersect at a single point or are tangent to each other, would require additional modifications to the code.
Extending to higher-dimensional spaces: The current implementation is limited to two-dimensional spaces. Extending the code to higher-dimensional spaces would require significant modifications to the mathematical framework and the implementation.

Q: What is the main goal of this article?

A: The main goal of this article is to explore the concept of degenerate conics and their application in determining the common tangents of two ellipses.

Q: What is a degenerate conic?

A: A degenerate conic is a special class of conic sections that have a specific form. They can be represented by the equation:

${ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 }$

where ${ A, B, C, D, E, }$ and ${ F }$ are constants.

Q: What are the three types of degenerate conics?

A: The three types of degenerate conics are:

Point conic: A degenerate conic that represents a single point.
Line conic: A degenerate conic that represents a line.
Parabola conic: A degenerate conic that represents a parabola.

Q: How do you compute the intersection points of two ellipses?

A: To compute the intersection points of two ellipses, you need to solve the system of equations formed by the two ellipse equations.

Q: How do you compute the tangent lines to each ellipse at the intersection points?

A: To compute the tangent lines to each ellipse at the intersection points, you need to use the equation of the ellipse and the point of tangency.

Q: How do you find the common tangent lines between the two ellipses?

A: To find the common tangent lines between the two ellipses, you need to solve the system of equations formed by the tangent lines.

Q: What is the time complexity of the current implementation?

A: The time complexity of the current implementation is O(n^2), where n is the number of intersection points.

Q: How can you improve the computational efficiency of the code?

A: You can improve the computational efficiency of the code by using more efficient algorithms or data structures, such as using a more efficient method for solving the system of equations or using a data structure that allows for faster lookup and insertion of intersection points.

Q: What are some potential applications of this code?

A: Some potential applications of this code include:

Computer-aided design (CAD): The code can be used to determine the common tangent lines between two ellipses in CAD software, which can be useful for designing and analyzing complex shapes.
Computer vision: The code can be used to determine the common tangent lines between two ellipses in computer vision applications, such as object recognition and tracking.
Robotics: The code can be used to determine the common tangent lines between two ellipses in robotics applications, such as motion planning and control.

Q: What are some potential limitations of this code?

A: Some potential limitations of this code include:

Computational efficiency: The code may not be computationally efficient for large-scale problems, which can make it impractical for use in certain applications.
Degenerate cases: The code assumes that the ellipses are non-degenerate, which means that it may not work correctly for degenerate cases, such as when the ellipses intersect at a single point or are tangent to each other.
Higher-dimensional spaces: The code is limited to two-dimensional spaces, which means that it may not be applicable to higher-dimensional spaces.

Q: How can you extend this code to higher-dimensional spaces?

A: To extend this code to higher-dimensional spaces, you would need to modify the mathematical framework and the implementation to accommodate the additional dimensions. This would likely involve using more advanced mathematical techniques and data structures, such as tensors and tensor algebra.

Q: How can you improve the robustness of this code?

A: To improve the robustness of this code, you can add error checking and handling to ensure that the code can handle a wide range of inputs and edge cases. You can also use more robust algorithms and data structures to reduce the likelihood of errors and improve the overall performance of the code.