A function assigns one output to every allowed input. Its codomain is the set where those outputs are declared to belong—much like an output type in programming. The domain and codomain together specify a function type. section 2.4.1 section 2.4.2
Do not confuse the codomain with the range. For sine, both inputs and the codomain are real numbers, ℝ, while the values sine actually reaches lie only in [-1, 1]. A codomain can therefore be larger than a function’s attained values. section 2.4.1
In geometric modeling, the codomain says what kind of result a rule produces:
- A geometric transformation maps points in ℝN to points in the same ℝN space. section 2.4.2 section 4.1
- A vector field maps points in ℝN to vectors in ℝM; the vector space may have different dimensionality. section 2.4.2
- A scalar field and a signed-distance function map points in space to real numbers. section 2.4.3
From a plane to a surface
A parametric surface uses a two-dimensional input space and a three-dimensional point codomain: f(u, v) → (x, y, z). Thinking this way separates where a surface is parameterized from the 3D space where it appears. section 8.2.1
Table 2.1 Function types used in geometry (view table figure)
Compare common geometry functions by the spaces they accept and the kinds of values they return. section 2.4.3
