affine.cppm File Reference

Affine spaces — points, vectors, dimension witnesses. More...

#include <algorithm>
#include <array>
#include <concepts>
#include <cstddef>
#include <initializer_list>
#include <limits>
#include <type_traits>
#include <utility>
import dedekind.algebra;
import dedekind.category;

Include dependency graph for affine.cppm:

Classes
struct	dedekind::geometry::FiniteDim
struct	dedekind::geometry::CountablyInfiniteDim
class	dedekind::geometry::Vector< F, N >
	An element of a Coordinate Space over a Field F. More...
class	dedekind::geometry::FunctionalVector< F, Fn >
	A vector-like object defined by coordinate evaluation x[i]. More...

Concepts
concept	dedekind::geometry::HasFiniteDimension
	A vector type whose coordinate index set is finite.
concept	dedekind::geometry::HasCountableDimension
	A vector type whose coordinate index set is at most countably infinite.
concept	dedekind::geometry::HasDimension
	A vector type with a specific compile-time dimension N.
concept	dedekind::geometry::IsAffine
	Affine-space witness over scalar carrier S.

Typedefs
template<IsFloatingScalar F, typename Fn>
using	dedekind::geometry::InfiniteVector = FunctionalVector<F, Fn>
	Alias highlighting intent for potentially infinite-dimensional vectors.
template<IsFloatingScalar F, typename Fn>
using	dedekind::geometry::SequenceVector = FunctionalVector<F, Fn>
	Alias for well-behaved infinite sequences viewed as vectors.

Functions
template<IsFloatingScalar F, typename Fn> requires requires(const Fn& fn, std::size_t i) { { fn(i) } -> std::convertible_to<F>; }
constexpr auto	dedekind::geometry::functional_vector (Fn &&fn)
	Factory helper for function-based vectors.
template<IsFloatingScalar F, typename Fn> requires requires(const Fn& fn, std::size_t i) { { fn(i) } -> std::convertible_to<F>; }
constexpr auto	dedekind::geometry::sequence_vector (Fn &&fn)
	Factory helper emphasizing sequence semantics.
template<IsFloatingScalar F>
constexpr Vector< F, 1 >	dedekind::geometry::as_vector (const F &x)
	Embed a scalar as a 1-dimensional vector.

Detailed Description

Affine spaces — points, vectors, dimension witnesses.

@partition :affine

Copyright

2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.

Note

"Space and time are commonly regarded as the forms of existence of the real world, matter as its substance." – Hermann Weyl, Space-Time-Matter (1918)

Typedef Documentation

SequenceVector

template<IsFloatingScalar F, typename Fn>

using dedekind::geometry::SequenceVector = FunctionalVector<F, Fn>

export

Alias for well-behaved infinite sequences viewed as vectors.

A sequence (x_n) with coordinate access x[n] is treated as an infinite-dimensional vector in the same carrier.

Function Documentation

as_vector()

template<IsFloatingScalar F>

Vector< F, 1 > dedekind::geometry::as_vector ( const F & x )

constexprexport

Embed a scalar as a 1-dimensional vector.

Formalises the view that ℝ (or any floating scalar) is a 1-dimensional vector space over itself. Combined with the tower embeddings (bool → ℕ → ℤ → ℚ → ℝ), any number in the tower can be lifted to a Vector<F,1>.