Distance

As explained in Geometry for Programmers

Distance is a nonnegative measure of separation between points or between a point and a geometric set.

A distance function assigns each point a nonnegative separation from another point or geometric set. In geometry and computation, it turns shapes into scalar fields that algorithms can evaluate and reuse. section 2.8 section 10.1.4

For a plane defined by three noncollinear points, distance can be understood as the height of a parallelepiped: signed volume divided by the base parallelogram’s area. The same result comes from projecting onto the plane’s normalized normal. section 9.4.3

A signed distance function (SDF) extends ordinary distance with a sign: it is negative inside a body and positive outside. This convention describes both a shape’s boundary and its surrounding field. section 10.1

Common geometric distance constructions include:

• Point-to-point distance uses the Euclidean square-root formula, or an equivalent self-dot-product implementation. section 10.1.4
• Point-to-line distance divides a parallelogram’s signed area by the line segment’s length; its absolute value gives unsigned distance. section 10.1.4
• Point-to-segment distance checks the normalized projection: use the line distance for an interior projection and an endpoint distance otherwise. section 10.1.4
• A rectangle SDF combines distances to its four edges and assigns the sign from the nearest edge’s supporting line. section 10.1.4

Figure 9.20 A parallelepiped formed by p₂-p₁, p₃-p₂, and p-p₁, and the parallelogram formed by p₂-p₁, p₃-p₁, too

Visualizes point-to-plane distance as the height of a parallelepiped over a parallelogram base. section 9.4.3

Figure 10.2 A disk and its SDF: negative inside, positive outside

Shows an SDF for a disk, with negative values inside and positive values outside. section 10.1

How Geometry for Programmers covers it

How deep it goes

Explains geometric derivations and Python implementations for points, lines, segments, planes, and rectangles, stopping short of mesh-wide construction.

Examples

Uses point-to-plane geometry, a disk SDF, point-to-point and point-to-segment Python functions, and a rectangular SDF.

What it assumes

Assumes familiarity with points, vectors, dot products, cross products, and signed geometric quantities.

What you can do after

Build it from scratch using vector products, projections, endpoint tests, and edge signs to compute basic distance fields.

How it connects

Connects distance functions with scalar fields, vector algebra, dot products, cross products, signed distance functions, and triangle meshes.

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Also explained in

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  Outlier Detection in PythonA function that quantifies dissimilarity or separation between observations in a data space.
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  Natural Language Processing in Action, Second EditionA distance metric is a function that quantifies dissimilarity between objects while satisfying the mathematical properties of a metric.
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  Math for Web DesignDistance is a numerical measure of the separation between two points or geometric objects.
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  Architecture Modernization: Socio-technical alignment of software, strategy, and structureDistance is the amount of technical and organizational coordination required for related parts to implement a change together.
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  Machine Learning Platform EngineeringA distance metric is a function that quantifies separation between vectors in a mathematical space.
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  Classic Computer Science Problems in Swift: Essential techniques for practicing programmersA measure of separation between locations or points in a feature space.
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  Machine Learning System DesignA function that assigns a numerical measure of separation or dissimilarity between mathematical objects or representations.

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Related concepts

• mahalanobis distanceincludes · Mahalanobis distance is a covariance-adjusted measure of separation between observations or between an observation and a reference distribution.
• Euclidean distanceincludes · Euclidean distance is the straight-line distance between two points in Euclidean space, calculated as the square root of the sum of squared differences between corresponding coordinates.
• similarity metricmeasures · A similarity measure is a rule that quantifies or characterizes how alike two objects or representations are according to selected properties.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.