Point

As explained in Geometry for Programmers

A point is a zero-dimensional geometric location that can serve as an element of a curve or surface.

A point is a geometric location: a building block from which curves and surfaces are made. On a coordinate plane, it can be written as a tuple of real-number coordinates, such as A = (0.5, 1.5). The tuple records where the point is; it does not turn the point into a number or give it vector arithmetic. section 2.1.3 section 2.8

Figure 2.1 A point in Cartesian coordinates: a = (0.5, 1.5)

Read the ordered pair as the point’s position in Cartesian coordinates: 0.5 in one coordinate direction and 1.5 in the other. section 2.1.1

Points and vectors often look alike in code because both may appear as coordinate tuples. Their jobs differ. A point is part of a geometric object; a vector has direction and length and can show where a point lies relative to a chosen origin. Treating the notation as interchangeable can obscure that distinction. section 2.1 section 2.1.2

Keep these practical distinctions clear:

• Do not apply vector algebra directly to points. Vectors can be added and scaled; points have no algebra of their own. section 2.1.2 section 2.8
• A point vector is a vector conventionally drawn from (0, 0) to a point. It represents the point’s location, but the vector itself has only direction and length. section 2.1.2
• Choose coordinate and numeric representations deliberately in programs: programming values are models rather than true real numbers, and each model has trade-offs. section 2.1.3 section 2.8

Points, vertices, and triangle membership

For a triangle, its three vertices define it, but they are not its only points. Interior locations and locations along its edges are also triangle points. A vertex is always a point, while a point need not be a vertex. Also distinguish a point on a triangle from a point in its interior: an edge point is on the triangle but not in its interior. section 2.2.1

Figure 2.8 On a triangle and in a triangle have different meanings. Here, point D is on the triangle but not in it.

The edge case matters in geometry code and documentation: a location on an edge belongs to the triangle but is not in its interior. section 2.2.1

Precise point vocabulary prevents small ambiguities from spreading through geometric algorithms. Name vertices when you mean the defining corners of a triangle, and say whether a test accepts points on a boundary or only points in an interior. section 2.2.1

How Geometry for Programmers covers it

How deep it goes

Defines coordinate notation, separates points from vectors, and explains triangle membership; defers deeper understanding to later practice.

Examples

Uses A = (0.5, 1.5), point vectors from the origin, and triangle vertices, edges, interiors, and boundary points.

What it assumes

Teaches the basic terminology without requiring deep prior understanding, while assuming readiness to use coordinate tuples.

What you can do after

Evaluate/understand whether geometry terminology and coordinate representations correctly distinguish points, vectors, vertices, and interior membership.

How it connects

Connects points with coordinate systems, vectors, curves, surfaces, triangles, vertices, and real-number models.

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Also explained in

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  Just Use Postgres!Point is a PostGIS geometric subtype representing a single position with coordinate values.
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  Advanced Algorithms and Data StructuresA point is a record representing a position in a multidimensional space through an ordered collection of coordinate values.
• 

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  Math for Web DesignA point is a location in space represented without length, width, or height, commonly by coordinates.

Compare how 4 books explain point →

Related concepts

• vectorcontrasts with · A vector is an ordered collection of components representing an element of a vector space.
• ST_MakePointproduced by · ST_MakePoint is a PostGIS function that constructs a point geometry from X and Y coordinates.
• nodeincludes · A node is a discrete element in a structured system, such as a graph, tree, network, or distributed system, representing an entity, state, component, or computational unit and potentially connected to other elements.
• control pointincludes · A control point is a point whose position influences the shape, direction, or local behavior of a curve or other parametric geometric representation.

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.