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Semi-major and Semi-minor Axes of an Ellipse

Definitions, properties, and uses of an ellipse's semi-major and semi-minor axes, with key formulas (area, eccentricity) and examples including the circle and orbital applications.

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Overview

The semi-major and semi-minor axes are the two principal half-diameters of an ellipse. The semi-major axis is the longest radius from the ellipse's center to its perimeter; the semi-minor axis is the shorter radius, perpendicular to the semi-major axis at the center. These concepts appear in elementary geometry and in the study of the ellipse more specifically.

Definitions and basic properties

By convention the semi-major axis is denoted a and the semi-minor axis b, with a ≥ b. The full lengths across the ellipse are the major axis (2a) and the minor axis (2b). The two semi-axes are orthogonal: the semi-minor axis meets the semi-major axis at a right angle at the center of the shape, a perpendicular relation commonly noted in Euclidean geometry (perpendicular).

Key relations and formulas

  • Standard Cartesian equation (axes aligned with coordinate axes): x²/a² + y²/b² = 1.
  • Area of an ellipse: πab.
  • When a ≥ b, the focal distance c from center to each focus satisfies c² = a² − b²; the eccentricity is e = c/a = sqrt(1 − b²/a²).
  • Circle as a special case: a = b = r, where r is the radius of the circle (radius), and the two semi-axes coincide.

History and mathematical role

The ellipse and its axes have been studied since antiquity but became central to analytic geometry and celestial mechanics after the development of coordinate methods. In algebraic and differential treatments the semi-axes serve to normalize and classify second-degree curves and to express invariant measures such as eccentricity and area.

Applications and examples

Semi-major and semi-minor axes are used across mathematics, physics, and engineering. In orbital mechanics the semi-major axis of an object's elliptical orbit determines its characteristic size and period under Kepler's laws; in optics and mechanical design, ellipses with prescribed semi-axes control focusing and clearance properties. In statistics, the axes of an error ellipse reflect variances and covariances of two-dimensional estimates.

Notable distinctions

Although notation varies, the convention a ≥ b avoids ambiguity: the semi-major axis is always the larger of the two. For rotated or translated ellipses the semi-axes still indicate the principal half-diameters but are no longer aligned with the coordinate axes; they remain orthogonal and pass through the ellipse's center. Practical calculations often use the semi-axes to compute bounding boxes, areas, and other geometric quantities.

Parameters of an ellipse:

S1,S2

Main title

S3,S4

Parting

\overline {S_{1}S_{2}}

Main axis

\overline {S_{3}S_{4}}

Minor axis

a

Large semi-axis

b

Small half-axis

F1,F2

Focal point (ellipse)

e

lin. Eccentricity

M

Center

p

Parameter (semi-latus rectum)

Astronomy

In astronomy, the major semimajor axis of a Keplerian orbit is one of the six so-called orbital elements and is often also inaccurately specified as "mean distance" and usually abbreviated as a. It characterizes - together with the eccentricity - the shape of elliptical orbits of various celestial bodies.

Such bodies are first of all the planets and their moons, artificial earth satellites, the asteroids and thousands of double stars.

According to Kepler's third law, the orbital period U of an elliptical orbit is coupled with a ( {\displaystyle U^{2}/a^{3}=\mathrm {const} }). The constant is related to the mass of the central body - so in a planetary system, it is related to the mass of the central star.

The two main vertices are called apsides, the main axis is the apside line: When a body lies at the focal point F1 and a smaller body orbits it on an ellipse, the shortest distance ( {\displaystyle {\overline {S_{1}F_{1}}}}= a-e) is called the periapsis, and the longest distance ( \overline {S_{2}F_{1}}= a+e) is called the apoapsis (perihelion, aphelion for the Sun).

In the periapsis (pericenter, main vertex close to the gravicenter) the orbital velocity is maximal, in the apocenter minimal.

In addition to the major semimajor axis, the actual mean distance depends on the numerical eccentricity ε \varepsilon=e/aand is

{\displaystyle a\cdot \left(1+{\frac {\varepsilon ^{2}}{2}}\right)}

Geodesy

In geodesy, the axes of the so-called error ellipses are an important means of representing the mean or maximum/minimum point errors. In the adjustment of geodetic networks, the accuracy with which the individual survey points of the network are determined can be represented as an error ellipse.

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AlegsaOnline.com Semi-major and Semi-minor Axes of an Ellipse

URL: https://en.alegsaonline.com/art/88740

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