Line–line intersection

Line-Line Intersection

skew lines shortest distance

Euclidian and non euclidian geometry

Line–line intersection is a fundamental concept in geometry and mathematics that deals with determining the point at which two lines intersect. This concept is widely applied in various fields such as computer graphics, geometric design, and engineering, where it is essential to calculate the exact point of intersection, if it exists, between two lines.

Overview

In Euclidean geometry, a line is defined as a straight one-dimensional figure having no thickness and extending infinitely in both directions. When two lines are placed on a plane, they can either intersect at a single point, be parallel and never intersect, or be coincident, where they lie on top of each other, effectively forming a single line. The line–line intersection algorithms are designed to find the intersection point of two lines defined by their equations.

Equations and Solutions

The most common representation of a line in a plane is the slope-intercept form, which is given by the equation \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. Given two lines:

• Line 1: \(y = m_1x + b_1\)
• Line 2: \(y = m_2x + b_2\)

The intersection point (\(x, y\)) of these two lines can be found by setting their equations equal to each other and solving for \(x\) and \(y\).

Determinant Method

Another method to find the intersection of two lines given by their general linear equations \(Ax + By = C\) is through the use of determinants. For two lines:

• Line 1: \(A_1x + B_1y = C_1\)
• Line 2: \(A_2x + B_2y = C_2\)

The point of intersection can be found using the formula: \[x = \frac{C_1B_2 - C_2B_1}{A_1B_2 - A_2B_1}\] \[y = \frac{A_1C_2 - A_2C_1}{A_1B_2 - A_2B_1}\]

This method is particularly useful when the equations of the lines are not in slope-intercept form.

Special Cases

• Parallel Lines: If the slopes (\(m_1\) and \(m_2\)) of the two lines are equal, the lines are parallel and do not intersect. The determinant method will result in a division by zero in this case, indicating no intersection.
• Coincident Lines: If the lines have the same slope and the same y-intercept (\(b_1 = b_2\)), they are coincident and intersect at an infinite number of points.

Applications

Line–line intersection calculations are crucial in many applications. In computer graphics, they are used for rendering and detecting collisions between objects. In surveying and civil engineering, they help in mapping and construction planning. Additionally, in robotics and automation, these calculations enable path planning and obstacle avoidance.

See Also

• Point (geometry)
• Slope
• Euclidean geometry
• Determinant