Squeeze Mapping¶
A planar linear map that stretches one fixed axis by a positive factor and contracts the other by its reciprocal, preserving area and coordinate product.
Core Idea¶
A squeeze mapping is \(S_a(x,y)=(ax,y/a)\) for a positive factor \(a\) and fixed planar axes. Its determinant is one, so it preserves region area; it also preserves \(xy\) and hence each hyperbola \(xy=c\). Nonidentity squeezes change lengths, slopes and many angles. They compose by multiplying factors: \(S_aS_b=S_{ab}\).[^ref-eb9dc406d7e9]
Scope of Application¶
In visualization, Talbot, Gerth and Hanrahan use the equivalent matrix \(\operatorname{diag}(1/\sqrt q,\sqrt q)\) to vary a chart's aspect ratio while holding display area fixed, then select \(q\) by minimizing total curve arc length. In special relativity, a one-dimensional Lorentz boost becomes reciprocal scaling \(u\mapsto e^\eta u\), \(v\mapsto e^{-\eta}v\) after changing to null coordinates \(u=ct+x\), \(v=ct-x\). The chart objective and physical interval interpretation are additional, distinct structures.[ref-eb9dc406d7e9][ref-53d4a56c73e4]
Clarity¶
Area preservation alone does not identify a squeeze: a shear and a rotation can have determinant one without the reciprocal diagonal rule. A generic scale \(\operatorname{diag}(a,b)\) qualifies only when \(b=1/a\) under the positive convention. The chart paper's \(q\) is an aspect-ratio parameter, not the same parameter as the direct stretch \(a=1/\sqrt q\).[^ref-eb9dc406d7e9]
Manages Complexity¶
One positive parameter fixes both axis scales and guarantees determinant one. This compresses area-preserving aspect-ratio comparison and makes inverse and composition calculations immediate. It does not settle which chart looks best, nor does it preserve Euclidean shape; those questions need separate criteria.[^ref-eb9dc406d7e9]
Abstract Reasoning¶
Write the map in a declared basis, check for \(\operatorname{diag}(a,a^{-1})\), compute determinant and \(xy\), and distinguish the result from other area-preserving matrices. For repeated maps multiply the factors; for a null-coordinate Lorentz application, first derive the basis change and retain the interval and boost-direction assumptions.[ref-eb9dc406d7e9][ref-53d4a56c73e4]
Knowledge Transfer¶
The same reciprocal matrix appears in area-fixed plot adjustment and null-coordinate boost representation, allowing its algebraic invariants to transfer. Perceptual chart quality does not transfer to spacetime, and physical boost conclusions do not transfer to an arbitrary plot. The live Equiareal Map is the proposed strict DAG parent: every squeeze preserves area, while many equiareal maps are not squeezes.[ref-eb9dc406d7e9][ref-53d4a56c73e4]
[^ref-eb9dc406d7e9]: Justin Talbot, John Gerth and Pat Hanrahan, “Arc Length-Based Aspect Ratio Selection”, original author-hosted 2011 paper, §3.2, squeeze matrix and equation (1). [^ref-53d4a56c73e4]: Tevian Dray, Geometry of Special Relativity, “Lorentz transformation”, boost equations and hyperbolic matrix; the null-coordinate squeeze form follows by change of basis.
Relationships to Other Abstractions¶
Current abstraction Squeeze Mapping Domain-specific
Parents (1) — more general patterns this builds on
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Squeeze Mapping is a kind of Equiareal map Domain-specific
A reciprocal-axis squeeze has determinant one and is an equiareal planar map.
Hierarchy path (1) — routes to 1 parentless root
- Squeeze Mapping → Equiareal map
Neighborhood in Abstraction Space¶
Squeeze Mapping sits in a sparse region of the domain-specific corpus (65th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Dynamical Systems & Differential Structures (37 abstractions)
Nearest neighbors
- Linear Canonical Transformation — 0.85
- Schreinemakers Analysis — 0.84
- Stiefel Manifold — 0.84
- Planar ternary ring — 0.84
- Complex representation — 0.84
Computed from structural-signature embeddings · 2026-10-08