Landmark shape¶

The equivalence class of corresponding Euclidean landmarks after location, uniform size, and proper rotation are removed.

Version: v1 · 2026-10-31 · History
Domain-specific #: 14528
Domain group: Formal Sciences
Origin domain: Experimental Design & Statistics
Subdomain: Landmark Shape Space → Experimental Design & Statistics
Aliases: Kendall landmark shape, Labelled landmark shape

Core Idea¶

A landmark shape is what remains of a labelled arrangement of points after moving, uniformly resizing and properly rotating the whole arrangement are ignored.^[1] Two arrangements have the same shape only if one common translation, one positive scale factor and one proper rotation carry each point to its corresponding point.^[1] The shape is the entire class of arrangements related in this way, not a preferred picture or a fitted alignment.^[1]

The point labels matter. In Kendall's basic construction, reflecting a non-collinear labelled triangle in its own plane can produce a different shape; reflection and swapping vertex labels are extra identifications if an application chooses them.^[1]

How would you explain it like I'm…

Slide, Grow, Turn Match

Put stickers on the same spots of two faces, like the nose and both eyes. If you can slide one picture, make it bigger or smaller, and turn it so every sticker lands on its matching sticker, the two have the same landmark shape. Flipping it like a mirror doesn't count, and the stickers have to match up in order.

Shape Without Place or Size

Scientists sometimes mark matching points, called landmarks, on things like faces or bodies. To compare their shapes, they ignore where the points are, how big the whole thing is and which way it's turned. Two sets of points have the same landmark shape if you can slide, resize (evenly) and rotate one to sit exactly on the other, point for point. Flipping it in a mirror is not allowed, so a triangle and its mirror image can be different shapes. Swapping which point is which can also change the shape.

Shape Modulo Position, Scale, Rotation

A landmark shape is the geometry of a labelled set of corresponding points once you remove position, overall size, and rotation. Start with an ordered list of at least two points (not all the same) in ordinary space. Two lists have the same shape if one translation, one positive uniform scaling and one proper rotation map each point to its partner. The shape is the whole group of lists that match this way, an 'equivalence class', not any one drawing or alignment. Reflections are not included, so a mirror-image triangle with the same side lengths can be a different shape, and relabelling points can matter too. Researchers can later choose to ignore mirroring, but that makes a coarser grouping.

A landmark shape is the equivalence class of a labelled configuration of corresponding points under translation, positive uniform scaling and proper rotation. Formally, take an ordered list of at least two points in a Euclidean space with at least two distinct; two such lists have the same shape when a single common translation, scale factor and proper rotation carry each point to its correspondent. The shape is this orbit, not any drawing, coordinate array or alignment procedure; Kendall defines the carrier and quotient explicitly for labelled configurations. The choice of transformation group is essential: Kendall's base shape excludes reflections, so a reflected planar labelled triangle may be a distinct shape despite equal side lengths, and relabelling the points can change the class. An analysis may later identify mirror images or relabellings, but that yields a coarser class without altering equalities under proper similarity. The same object appears in formal settings, like Kendall's shape space of labelled planar triangles, and empirical ones, like corresponding points on standardized 3D human-body meshes analyzed with translation, scale and special-orthogonal rotations; in both, pose and size are removed while relative geometry is kept.

Scope of Application¶

Kendall builds a mathematical shape space for labelled triangles in a plane.^[1] Valero and colleagues use corresponding points sampled from standardized three-dimensional human-body meshes and define their shape classes using translation, positive scale and proper rotations.^[2] These examples differ in dimension, number of points and purpose, but the same rule compares corresponding landmark arrangements. A study must establish what point \(i\) in one specimen corresponds to in another before a comparison has an empirical meaning.

Clarity¶

Imagine a triangle with vertices labelled \(A\), \(B\) and \(C\). Slide the paper, turn it within the plane or zoom the whole triangle uniformly: its landmark shape stays the same. Flip the paper over while keeping the vertex labels: the triangle's handedness reverses, and a proper planar rotation cannot undo that flip.^[1]

Manages Complexity¶

The class gathers many coordinate pictures of the same relative geometry into one object. To compare two configurations, first fix the point correspondence and the allowed transformations; then ask whether a single allowed transformation maps all points together.^[1]^[2] A small fitting error may mean two shapes are close, but exact class equality is a stronger claim.^[1]^[2]

Abstract Reasoning¶

Take labelled planar points \(A=(0,0)\), \(B=(1,0)\) and \(C=(0,1)\). Replace only \(C\) by \((0,-1)\). The side lengths match, yet the new triangle is the original's mirror across \(AB\). Since a proper planar rotation that fixes both \(A\) and \(B\) cannot also move \(C\) across that line, the two labelled triangles lie in different shape classes under Kendall's proper-rotation rule.^[1]

Knowledge Transfer¶

The triangle and body-scan examples share a rule: corresponding Euclidean points, a common reversible transformation group, and the orbit of a whole configuration. Kendall's special triangle-sphere picture does not automatically describe the much larger body shape space, and empirical findings about bodies do not follow from the geometry of three points.^[1]^[2]

Example¶

In the human-body study, the researchers prepare meshes so that sampled surface points correspond across people, then compare the resulting three-dimensional shapes and summaries.^[2]

Mapped back: the indexed mesh points are the carrier; common translation, uniform positive scaling and proper 3D rotation are ignored; each person's configuration represents one orbit; differences in relative form remain.^[2] Mesh preparation and statistical fitting help apply or compare the classes, but do not define the mathematical equality rule.

Relationships to Other Abstractions¶

Current abstraction Landmark shape Domain-specific

Parents (2) — outgoing typed relations

Landmark shape presupposes Equivalence Relation Prime

A landmark shape is an orbit class only because proper-similarity sameness is an equivalence relation on labelled configurations.
Landmark shape includes Group Prime

The reversible proper-similarity group and its action are internal to every labelled landmark-shape orbit.

Neighborhood in Abstraction Space¶

Landmark shape sits in a sparse region of the domain-specific corpus (87^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Manifolds, Stacks & Classifying Spaces (19 abstractions)

Nearest neighbors

Fixed points of isometry groups in Euclidean space — 0.81
Reach (Mathematics) — 0.81
Brandt matrix — 0.80
Bundle of Principal Parts — 0.80
Deligne–Lusztig theory — 0.80

Computed from structural-signature embeddings · 2026-10-31

Not to Be Confused With¶

A silhouette: it may have no corresponding labelled landmarks.
Congruence: it usually preserves size, which this identity removes.
A mirror-folded display: Kendall treats reflection and relabelling as later, coarser quotients of the labelled base shape.^[1]
Procrustes fitting: an alignment operation or distance calculation, rather than the shape class itself.

References¶

[1] D. G. Kendall, “Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces,” Bulletin of the London Mathematical Society 16 (1984), 81–121, especially §§2, 5–6 and 8.3. https://doi.org/10.1112/blms/16.2.81. Full author article. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k

[2] Jorge Valero et al., “Statistical Shape Analysis of Human Bodies,” Statistical Analysis and Data Mining: The ASA Data Science Journal (2025), especially §§2, 3.1 and 4.2. https://doi.org/10.1002/sam.70024. Publisher full text. registry ↩a ↩b ↩c ↩d ↩e ↩f