euclidean.cppm File Reference

The Study of Distance, Metrics, and Curvature. More...

#include <concepts>
#include <cstddef>
#include <functional>
import dedekind.algebra;
import dedekind.geometry:inner_product;

Include dependency graph for euclidean.cppm:

Concepts
concept	dedekind::geometry::IsMetricSpace
	A set equipped with a distance function (metric) d(a, b).
concept	dedekind::geometry::IsEuclideanSpace
	A Vector Space that is also a Metric Space where distance is the Norm.

Functions
template<dedekind::algebra::IsFloatingScalar F, std::size_t N>
constexpr F	dedekind::geometry::distance (const Vector< F, N > &a, const Vector< F, N > &b)
	Canonical Euclidean distance induced by the norm.
template<typename Threshold, typename Compare = std::greater<>>
constexpr auto	dedekind::geometry::compare_euclidean_norm_squared (Threshold threshold, Compare compare={})
	Predicate factory over squared Euclidean norm.
template<typename RadiusSquared>
constexpr auto	dedekind::geometry::outside_closed_euclidean_ball_squared (RadiusSquared radius_squared)
	Predicate factory for points outside the closed Euclidean ball, using squared radius to avoid square roots.

Detailed Description

The Study of Distance, Metrics, and Curvature.

@partition :euclidean

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

euclidean__Geometry

This partition bridges Algebra and Topology by introducing the concept of a "Metric." It ensures that our numerical species can be measured not just as sets or fields, but as physical points on a 1D, 2D, or n-dimensional manifold. Wikipedia: Geometry, Metric space, Euclidean space

Copyright

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

Note

"With the aid of visual imagination we can illuminate the manifold facts and problems." – David Hilbert and S. Cohn-Vossen, Geometry and the Imagination (1932)

Function Documentation

compare_euclidean_norm_squared()

template<typename Threshold, typename Compare = std::greater<>>

auto dedekind::geometry::compare_euclidean_norm_squared ( Threshold threshold, Compare compare = {} )

constexprexport

Predicate factory over squared Euclidean norm.

Returns x |-> compare(euclidean_norm_squared(x), threshold). Works for any carrier providing euclidean_norm_squared(x) via ADL.