Flattening

A dimensionless axial-compression measure for an ellipse or spheroid, ordinarily the semiaxis difference divided by the semimajor axis, with explicitly convertible alternative normalizations.

Version

v3 · 2026-09-19 · History

Domain-specific #

1851

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Geometry → Mathematics

Aliases

First flattening, Ellipticity, Oblateness parameter

Core Idea

Flattening quantifies how far an ellipse or spheroid departs from circular or spherical shape through axial compression. For semimajor axis \(a\) and semiminor axis \(b\), with \(a\ge b>0\), the ordinary or first flattening is

\[ f=\frac{a-b}{a}=1-\frac ba. \]

It is dimensionless, equals zero for a circle or sphere, and increases as the minor axis shrinks relative to the major axis. Geodetic reference ellipsoids are often specified by \(a\) and reciprocal flattening \(1/f\).

Alternative conventions include second flattening \(f'=(a-b)/b\) and third flattening \(n=(a-b)/(a+b)\). Names vary, so the formula or symbol must accompany any value.

How would you explain it like I'm…

How Squished Is It?

Think of a round ball, and then imagine squishing it a little from the top so it gets shorter. Flattening is a number that says how squished it is. A perfect ball gets zero, and the more you squish it, the bigger the number gets.

How Squashed Is It?

An oval or a squished ball has a long half-width, called a, and a short half-width, called b. Flattening measures how far the shape is from perfectly round using the formula f = (a − b) / a. A circle or a perfect sphere has flattening 0, because a and b are equal. The shorter b gets compared to a, the bigger the flattening. It has no units, so it works the same whether you measure in centimeters or kilometers.

Ellipse Flattening Ratio

Flattening measures how far an ellipse or spheroid departs from a circle or sphere because it's compressed along one axis. With semimajor axis a and semiminor axis b (a ≥ b > 0), the usual (first) flattening is f = (a − b)/a = 1 − b/a. It's dimensionless, equals 0 for a circle or sphere, and increases as b shrinks relative to a. Map makers often describe reference ellipsoids for the Earth by a and the reciprocal flattening 1/f. There are other versions with different denominators—second flattening (a − b)/b and third flattening (a − b)/(a + b)—so you should always state which formula you're using.

 

Flattening is a dimensionless measure of how much an ellipse or spheroid is compressed along one axis relative to a circle or sphere. For semiaxes a ≥ b > 0, the first (ordinary) flattening is f = (a − b)/a = 1 − b/a, which is zero exactly when a = b and increases as the minor axis shrinks relative to the major one. In geodesy, reference ellipsoids are commonly specified by the semimajor axis a together with the reciprocal flattening 1/f. Alternative conventions normalize the same axial difference by other quantities: second flattening f′ = (a − b)/b and third flattening n = (a − b)/(a + b). Since naming varies across sources, any reported value should come with its formula or symbol. The essential ingredients are ordered semiaxes, a normalized axial difference, a declared denominator, and a dimensionless measure of departure from equality.

Scope of Application

Flattening parameterizes terrestrial and planetary reference ellipsoids, map-projection formulae, geodesics, orbit and figure calculations, and comparisons of rotating bodies. Because Earth's flattening is small, series in \(f\) or third flattening \(n\) are efficient in high-accuracy geodesy.

The simple two-axis definition applies directly to spheroids. A fully triaxial ellipsoid needs more than one axial ratio or a clearly selected pair of axes.

Clarity

Report the formula, axes, units for axes, and whether the value is \(f\), \(1/f\), \(f'\), or \(n\). State whether the body is oblate or prolate and whether dimensions are fitted geometric parameters or physical equilibrium predictions. Do not call two values inconsistent until their conventions have been converted.

Manages Complexity

One scalar captures the leading departure from spherical symmetry and allows a large family of geodetic formulae to be organized as perturbation series. Conversion identities translate among standards without reconstructing the entire ellipse. Reciprocal flattening also makes small terrestrial departures easier to tabulate.

Abstract Reasoning

1. Identify the relevant figure and its principal semiaxes.
2. Order or label the axes consistently.
3. Choose the flattening convention and write it explicitly.
4. Check dimensional cancellation and the spherical limit.
5. Convert other reported parameters before comparison.
6. Propagate uncertainty from axis estimates, including covariance if material.
7. Test whether a spheroidal reduction is adequate for a triaxial body.
8. Use expansion formulae only within their stated error range.

Knowledge Transfer

The portable pattern is normalize a directional deficit by a reference extent to obtain a scale-free departure from isotropy. It transfers to aspect ratios, anisotropy indices, strain measures, and normalized geometric residuals. The proposed immediate parent is Measurement.

Relationships to Other Abstractions

Current abstraction Flattening Domain-specific

Parents (1) — outgoing typed relations

• Flattening is a kind of Measurement Prime

  Measurement is the broader abstraction this entry instantiates.

Hierarchy path (1) — routes to 1 parentless root

• Flattening → is a kind of Measurement

Neighborhood in Abstraction Space

Flattening sits in a sparse region of the domain-specific corpus (92^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

• Spheroid — 0.82
• Superegg — 0.80
• Ellipsoid packing — 0.78
• Dihedral Angle — 0.78
• Ellipse — 0.77

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