How To Find Distance Between Skew Lines

How to Find the Distance Between Skew Lines

Skew lines are lines in three-dimensional space that are neither parallel nor intersecting. Finding the distance between them is a crucial concept in various fields, including computer graphics, physics, and engineering. This comprehensive guide will walk you through different methods to calculate this distance, explaining the underlying principles and providing step-by-step instructions. We'll cover both vector and parametric approaches, ensuring a thorough understanding for all levels of mathematical proficiency.

Understanding Skew Lines

Before diving into the methods, let's solidify our understanding of skew lines. Two lines are considered skew if they fulfill two conditions:

1. They are not parallel: Their direction vectors are not scalar multiples of each other.
2. They do not intersect: There's no single point that lies on both lines.

Imagine two train tracks that are not parallel and never cross; that's a perfect representation of skew lines.

Method 1: Using Vectors

This method leverages the power of vector algebra to elegantly calculate the distance. We'll break down the process into manageable steps.

Step 1: Defining the Lines

Let's represent our skew lines using vector notation. We'll use two points on each line and their respective direction vectors.

• Line 1: Point A(x₁, y₁, z₁) and direction vector v = <a, b, c>
• Line 2: Point B(x₂, y₂, z₂) and direction vector w = <d, e, f>

The parametric equations for these lines are:

• Line 1: r₁ = A + tv = <x₁ + ta, y₁ + tb, z₁ + tc>
• Line 2: r₂ = B + sw = <x₂ + sd, y₂ + se, z₂ + sf>

Where 't' and 's' are scalar parameters.

Step 2: Finding the Vector Connecting the Lines

We need a vector connecting a point on Line 1 to a point on Line 2. Let's choose points A and B for simplicity. The vector connecting these points is:

AB = B - A = <x₂ - x₁, y₂ - y₁, z₂ - z₁>

Step 3: Finding the Normal Vector

The shortest distance between the two lines will be perpendicular to both direction vectors, v and w. This means we need to find the cross product of these vectors:

n = v x w = <(bf - ce), (cd - af), (ae - bd)>

n is the normal vector representing the direction perpendicular to both lines.

Step 4: Calculating the Distance

The distance 'd' between the two skew lines is the scalar projection of the vector AB onto the normal vector n:

d = |AB • n| / ||n||

Where:

• AB • n is the dot product of AB and n.
• ||n|| is the magnitude (length) of n.

Let's break down the calculation:

• AB • n = (x₂ - x₁)(bf - ce) + (y₂ - y₁)(cd - af) + (z₂ - z₁)(ae - bd)
• ||n|| = √[(bf - ce)² + (cd - af)² + (ae - bd)²]

Finally, substitute these values into the distance formula:

d = |(x₂ - x₁)(bf - ce) + (y₂ - y₁)(cd - af) + (z₂ - z₁)(ae - bd)| / √[(bf - ce)² + (cd - af)² + (ae - bd)²]

This formula provides the shortest distance between the two skew lines.

Method 2: Using Parametric Equations and Minimization

This method uses the parametric equations of the lines and calculus to find the minimum distance.

Step 1: Defining the Lines (Parametric Form)

As before, we define the lines parametrically:

• Line 1: x = x₁ + at, y = y₁ + bt, z = z₁ + ct
• Line 2: x = x₂ + ds, y = y₂ + es, z = z₂ + fs

Where 't' and 's' are parameters.

Step 2: Distance Formula

The distance between any two points on the lines is given by:

D²(s,t) = (x₂ + ds - (x₁ + at))² + (y₂ + es - (y₁ + bt))² + (z₂ + fs - (z₁ + ct))²

Step 3: Minimization

To find the minimum distance, we need to find the values of 's' and 't' that minimize D²(s,t). This involves taking partial derivatives with respect to 's' and 't', setting them to zero, and solving the resulting system of equations. This leads to a system of two linear equations in 's' and 't', which can be solved using standard techniques like substitution or elimination.

Step 4: Calculating the Distance

Once 's' and 't' are determined, substitute them back into the distance formula D²(s,t) to find the minimum distance squared. Taking the square root gives the shortest distance between the skew lines. This method requires a stronger background in calculus but provides a powerful alternative approach.

Illustrative Example

Let's work through an example using Method 1 (Vector Method).

Line 1: Point A(1, 2, 3), direction vector v = <1, 1, 0> Line 2: Point B(4, 5, 6), direction vector w = <2, 0, 1>

1. AB = <4 - 1, 5 - 2, 6 - 3> = <3, 3, 3>
2. n = v x w = <(11 - 00), (02 - 11), (10 - 12)> = <1, -1, -2>
3. AB • n = (3)(1) + (3)(-1) + (3)(-2) = -6
4. ||n|| = √(1² + (-1)² + (-2)²) = √6
5. d = |-6| / √6 = 6 / √6 = √6

Therefore, the shortest distance between these skew lines is √6.

Conclusion

Finding the distance between skew lines involves understanding vector algebra or calculus. Both methods described above provide a robust way to calculate this crucial distance. The vector method offers a more direct and computationally efficient approach, while the parametric method provides a deeper understanding of the underlying principles. Choosing the method that best suits your mathematical background and computational resources will ensure accurate and efficient solutions. Remember to always double-check your calculations to avoid errors. This detailed guide provides a complete framework for tackling this important geometrical problem.