How can computational geometry libraries calculate the distance between two points?

1 Euclidean distance

The most intuitive way to measure the distance between two points is to use the Euclidean distance formula. This is based on the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In two dimensions, the Euclidean distance between points (x1, y1) and (x2, y2) is given by: d = sqrt((x2 - x1)^2 + (y2 - y1)^2) In three dimensions, the formula is similar, but with an additional term for the z-coordinates. Many computational geometry libraries implement this formula as a built-in function or a method of a point class. For example, in Python, you can use the scipy.spatial.distance.euclidean function or the numpy.linalg.norm function to compute the Euclidean distance.

2 Other distance metrics

The Euclidean distance is not the only way to define the distance between two points; depending on the context and application, different distance metrics may be more appropriate. For instance, the Manhattan distance measures the sum of absolute differences of coordinates, also known as taxicab distance, because it reflects how far a car would need to drive in a grid-like city. The Chebyshev distance, or chessboard distance, measures the maximum of absolute differences of coordinates and reflects how far a king piece can move in a chess game. Additionally, the Minkowski distance generalizes Euclidean, Manhattan, and Chebyshev distances by using a parameter p that controls how much weight is given to larger differences; Euclidean corresponds to p = 2, Manhattan to p = 1, and Chebyshev to p = infinity. Many computational geometry libraries provide functions or methods for calculating these and other distance metrics; for example, in Python, one can use scipy.spatial.distance.minkowski or numpy.linalg.norm with different values of p to compute the Minkowski distance.

3 Numerical issues

Calculating the distance between two points may seem like a straightforward task, however, it can involve numerical issues that affect the accuracy and stability of the results. Such issues include floating-point errors, which can cause discrepancies in the distance calculations; overflow and underflow, which can lead to incorrect or meaningless values; and degenerate cases, which can cause numerical instability or division by zero errors. To prevent or lessen these numerical issues, computational geometry libraries typically implement techniques such as normalizing the input points, using alternative formulas, and handling special cases. By doing so, they can ensure that the distance between two points is calculated with adequate accuracy and reliability.

4 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?