Hilbert metric¶

A projectively invariant metric on the interior of a bounded convex domain, defined by a logarithmic cross ratio of the two boundary intersections on the line through a point pair.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4884
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Geometry, Projective Metrics → Mathematics

Core Idea¶

Hilbert distance between two interior points is one-half or another conventional multiple of the logarithm of their boundary cross ratio. The line through the points reduces the geometry to an interval, and projective invariance of cross ratio makes the resulting distance independent of affine coordinates. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of geometry. It is convex-domain intrinsic geometry induced solely by projective boundary position. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that domain is convex and boundary-point ordering and normalization follow one convention fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

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Faraway Walls Distance

Picture a room with no dents in its walls, like a circle or a triangle. To measure how far apart two spots are, you look at where the walls are along the straight line through them. The closer you get to a wall, the bigger each little step counts. So you can never reach the wall: it is infinitely far away.

Edge-Based Distance Inside Shapes

Take a shape with no dents, called a convex shape, and two points inside it. Draw the straight line through the two points until it hits the edge of the shape on both sides. The Hilbert metric decides how far apart the two points are by comparing where they sit relative to those two edge points. Two points near the edge can be very far apart even if they look close. The edge itself is endlessly far away. Only the shape's boundary decides the distances.

Cross-Ratio Distance in Convex Sets

In a convex region, take two interior points x and y. Extend the line through them until it meets the boundary at points a and b. The Hilbert distance is a fixed multiple, usually one-half, of the logarithm of the cross ratio of the four points a, x, y, b. Because the cross ratio is preserved by projective transformations, the distance does not depend on which coordinates you use. As a point approaches the boundary, its distance to any fixed interior point grows without limit. Unlike ordinary Euclidean distance, this measure depends only on where the boundary is. You also have to fix one convention for the order of the boundary points and for the constant in front.

Let Ω be a convex domain and let p, q be interior points. The line through them meets ∂Ω at a and b, so the problem reduces to a one-dimensional interval. The Hilbert metric is d(p,q) = c·log [a,p,q,b], where [a,p,q,b] is the cross ratio of the four collinear points. The constant c is a convention, often 1/2. Cross ratio is a projective invariant, so d is independent of affine or projective coordinates. It is an intrinsic geometry induced solely by where the boundary lies. For it to be well defined, the domain must be convex, and the ordering of boundary points and the normalization must follow one fixed convention. Distances go to infinity near the boundary.

Scope of Application¶

Hilbert metric belongs to geometry and is useful where the analyst can specify a bounded open convex domain, two interior points, line through them, ordered boundary intersections, Euclidean segment ratios, cross ratio, logarithm and projective transformations, then evaluate domain is convex and boundary-point ordering and normalization follow one convention. The scope is broad within that domain but bounded by the need for domain is convex and boundary-point ordering and normalization follow one convention. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making domain is convex and boundary-point ordering and normalization follow one convention the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Hilbert metric can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Hilbert metric. Hilbert metric compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: a bounded open convex domain, two interior points, line through them, ordered boundary intersections, Euclidean segment ratios, cross ratio, logarithm and projective transformations. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express domain is convex and boundary-point ordering and normalization follow one convention independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of geometry because they reuse a bounded open convex domain, two interior points, line through them, ordered boundary intersections, Euclidean segment ratios, cross ratio, logarithm and projective transformations, The line through the points reduces the geometry to an interval, and projective invariance of cross ratio makes the resulting distance independent of affine coordinates., and type the carrier, state every parameter and convention in the definition, test that domain is convex and boundary-point ordering and normalization follow one convention, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Hilbert metric Domain-specific

Parents (1) — outgoing typed relations

Hilbert metric is a kind of Measurement Prime

The metric measures intrinsic separation by boundary cross ratio.

Hierarchy path (1) — routes to 1 parentless root

Hilbert metric → is a kind of Measurement

Neighborhood in Abstraction Space¶

Hilbert metric sits in a crowded region of the domain-specific corpus (15^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Metric Geometry & Curvature (14 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25