Function type

As explained in Geometry for Programmers

A function type specifies the kinds of inputs and outputs a function accepts, including its domain and codomain.

A function type describes the inputs a function accepts and the outputs it can return. Its domain names the input set, while its codomain names the output set; in programming, these correspond closely to input and output types. section 2.4.2 section 2.8

Geometric modeling makes the idea concrete. A transformation maps points from ℝᴺ to ℝᴺ, such as translation, which shifts every point by the same distance and direction. A vector field instead maps each point in ℝᴺ to an M-dimensional vector in ℝᴹ. section 2.4.2 section 2.4.2

Common function types in geometric modeling include:

• Geometric transformation: turns a point into another point, normally in the same space. section 2.4.2
• Vector field: assigns a vector to every point, indicating movement, color, or another value. section 2.4.2
• Scalar field and distance function: appear among the principal function types used for geometric work. section 2.8

Translation

Applying a translation to the vertices of a square illustrates the rule, but the function is defined for every point in ℝ², not just that square. section 2.4.2

Figure 2.23 The simplest geometric transformation: translation

Visualizes translation as the simplest geometric transformation. section 2.4.2

How Geometry for Programmers covers it

How deep it goes

Defines domains and codomains, then distinguishes transformations and vector fields, without developing their later chapters’ mechanics.

Examples

Uses translation of a square in ℝ² and vector-field illustrations to ground abstract function types.

What it assumes

Assumes familiarity with real numbers and introduces ℝ², ℝ³, and ℝᴺ as one-, two-, three-, and N-dimensional spaces.

What you can do after

Evaluate/understand geometric functions by identifying their input spaces, output spaces, and per-input values.

How it connects

Connects domain and codomain with geometric transformations, vector fields, scalar fields, distance functions, points, and vectors.

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Also explained in

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  Kotlin in Action, 2EA function type describes the parameter types and return type of callable values.
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  Learn Haskell by ExampleA function type specifies the input and output types of a function value.
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  100 Go Mistakes and How to Avoid ThemA function type is a Go type whose values are functions with a specified parameter and result signature.

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Related concepts

• Kotlinused by · Kotlin is a statically typed general-purpose programming language that targets the JVM and other platforms and interoperates with Java.
• data typekind of · A data type classifies values and specifies their representation or interpretation and the operations applicable to them.
• atomic typekind of · An atomic type is a type whose supported accesses and operations are indivisible with respect to concurrent execution and provide defined synchronization semantics.
• factorialequivalent to · The factorial of a nonnegative integer n, written n!, is the product of all positive integers from 1 through n, with 0! equal to 1.
• FunctionN interfaceequivalent to · A FunctionN interface is a compiler-generated Kotlin interface representing a function value with a specific number of parameters and an invoke operation.

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.