Vector field

As explained in Geometry for Programmers

A vector field is a function that assigns a vector to each point in a domain.

A vector field assigns an M-dimensional vector to every point in N-dimensional space. Its domain is ℝN and its codomain is ℝM, so the two dimensions may differ. Imagine a field as a collection of geometric pointers: each pointer describes a direction and magnitude attached to a location. section 2.4.2

A vector field differs from a geometric transformation. A transformation moves points directly; a vector field describes where points could move and can also carry information such as colors in an unrelated color space. A continuous field contains vectors between the regularly spaced arrows shown in an illustration, even when those vectors are not drawn. section 2.4.2

From field to deformation

When a vector field specifies how each location should translate, it becomes a deformation field. Applying it to a square can move, bend, and stretch different points by different amounts, making the idea useful for 3D modeling. section 2.4.2

Figure 2.24 An illustration of a vector field. You have to imagine that it’s continuous.

Use the continuous-field illustration to read the arrows as sampled views of vectors that exist throughout the plane. section 2.4.2

Figure 2.25 The vector field from figure 2.24 used as a deformation field. It moves, bends, and stretches the square.

Use the deformation-field illustration to connect local vector directions and magnitudes with a model’s resulting movement, bending, and stretching. section 2.4.2

How Geometry for Programmers covers it

How deep it goes

Explains domain, codomain, continuity, visualization, and deformation use, but stops short of computational algorithms.

Examples

Illustrates regularly spaced arrows, a continuous field, and a square moved, bent, and stretched by a deformation field.

What it assumes

Assumes familiarity with points, vectors, functions, domains, and codomains while distinguishing their geometric roles.

What you can do after

Evaluate vector fields as location-dependent geometric pointers and recognize their use for describing deformations.

How it connects

Connects vector fields with geometric transformations, deformation fields, continuous functions, vectors, scalar fields, and distance functions.

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Related concepts

• continuous deformation fieldincludes · A continuous deformation field is a spatially continuous assignment of transformations or displacement vectors that deforms geometric objects.
• geometric transformationcontrasts with · A geometric transformation is a mapping or operation that changes or expresses the position, orientation, scale, shape, or coordinate representation of geometric objects.
• meanaddresses · A mean is a measure of central tendency, calculated as the arithmetic average of values or, for a random variable, as the probability-weighted average of its possible values.
• scalar fieldcontrasts with · A scalar field is a function that assigns a scalar value to each point in a domain.

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.