Conjugate diameters

Last updated

In geometry, two diameters of a conic section are said to be conjugate if each chord parallel to one diameter is bisected by the other diameter. For example, two diameters of a circle are conjugate if and only if they are perpendicular.

Contents

Of ellipse

Two conjugate diameters of an ellipse. Each edge of the bounding parallelogram is parallel to one of the diameters. Conjugate Diameters.svg
Two conjugate diameters of an ellipse. Each edge of the bounding parallelogram is parallel to one of the diameters.

For an ellipse, two diameters are conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram , sometimes called a bounding parallelogram (skewed compared to a bounding rectangle). In his manuscript De motu corporum in gyrum, and in the 'Principia', Isaac Newton cites as a lemma proved by previous authors that all (bounding) parallelograms for a given ellipse have the same area.

It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any bounding parallelogram. For example, in proposition 14 of Book VIII of his Collection, Pappus of Alexandria gives a method for constructing the axes of an ellipse from a given pair of conjugate diameters. Another method is using Rytz's construction, which takes advantage of the Thales' theorem for finding the directions and lengths of the major and minor axes of an ellipse regardless of its rotation or shearing.

In analytic geometry, if we let the vectors of the two conjugate half-diameters be , then the ellipse is parameterized by as varies over .

Of hyperbola

For any ph, the indicated diameters of the circles and hyperbolas are conjugate. Orthogonality and rotation.svg
For any φ, the indicated diameters of the circles and hyperbolas are conjugate.

Similar to the elliptic case, diameters of a hyperbola are conjugate when each bisects all chords parallel to the other. [1] In this case both the hyperbola and its conjugate are sources for the chords and diameters.

Apollonius of Perga gave the following construction of conjugate diameters, given the conjugate hyperbola: "If Q be any point on a hyperbola and CE be drawn from the centre parallel to the tangent at Q to meet the conjugate hyperbola in E, then (1) the tangent at E will be parallel to CQ and (2) CQ and CE will be conjugate diameters." [2]

In analytic geometry, if we let the vectors of the two conjugate half-diameters be , then the hyperbola is parameterized by as varies over .

In the case of a rectangular hyperbola, its conjugate is the reflection across an asymptote. A diameter of one hyperbola is conjugate to its reflection in the asymptote, which is a diameter of the other hyperbola. As perpendicularity is the relation of conjugate diameters of a circle, so hyperbolic orthogonality is the relation of conjugate diameters of rectangular hyperbolas.

The placement of tie rods reinforcing a square assembly of girders is guided by the relation of conjugate diameters in a book on analytic geometry. [3]

Conjugate diameters of hyperbolas are also useful for stating the principle of relativity in the modern physics of spacetime. The concept of relativity is first introduced in a plane consisting of a single dimension in space, the second dimension being time. In such a plane, one hyperbola corresponds to events a constant space-like interval from the origin event, the other hyperbola corresponds to events a constant time-like interval from it. The principle of relativity can be formulated "Any pair of conjugate diameters of conjugate hyperbolas can be taken for the axes of space and time". This interpretation of relativity was enunciated by E. T. Whittaker in 1910. [4]

In projective geometry

Every line in projective geometry contains a point at infinity, also called a figurative point. The ellipse, parabola, and hyperbola are viewed as conics in projective geometry, and each conic determines a relation of pole and polar between points and lines. Using these concepts, "two diameters are conjugate when each is the polar of the figurative point of the other." [5]

Only one of the conjugate diameters of a hyperbola cuts the curve.

The notion of point-pair separation distinguishes an ellipse from a hyperbola: In the ellipse every pair of conjugate diameters separates every other pair. In a hyperbola, one pair of conjugate diameters never separates another such pair.

Related Research Articles

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

<span class="mw-page-title-main">Diameter</span> Straight line segment that passes through the centre of a circle

In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere.

<span class="mw-page-title-main">Ellipse</span> Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

<span class="mw-page-title-main">Hyperbola</span> Plane curve: conic section

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

<span class="mw-page-title-main">Parabola</span> Plane curve: conic section

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

<span class="mw-page-title-main">Perpendicular</span> Relationship between two lines that meet at a right angle (90 degrees)

In geometry, two geometric objects are perpendicular if their intersection forms right angles at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines, between a line and a plane, and between two planes.

<span class="mw-page-title-main">Parallelogram</span> Quadrilateral with two pairs of parallel sides

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

<span class="mw-page-title-main">Hyperboloid</span> Unbounded quadric surface

In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system.

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.

<span class="mw-page-title-main">Apollonius of Perga</span> Ancient Greek geometer and astronomer (c. 240–190 BC)

Apollonius of Perga was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. With his predecessors Euclid and Archimedes, Apollonius is generally considered among the greatest mathematicians of antiquity.

<span class="mw-page-title-main">Degenerate conic</span> 2nd-degree plane curve which is reducible

In geometry, a degenerate conic is a conic that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers as the product of two linear polynomials.

<span class="mw-page-title-main">Hyperbolic orthogonality</span> Relation of space and time in relativity theory

In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular timeline. This dependence on a certain timeline is determined by velocity, and is the basis for the relativity of simultaneity.

<span class="mw-page-title-main">Squeeze mapping</span> Linear mapping permuting rectangles of the same area

In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.

<span class="mw-page-title-main">Centre (geometry)</span> Middle of the object in geometry

In geometry, a centre or center of an object is a point in some sense in the middle of the object. According to the specific definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of isometry groups, then a centre is a fixed point of all the isometries that move the object onto itself.

<span class="mw-page-title-main">Conic section</span> Curve from a cone intersecting a plane

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

<span class="mw-page-title-main">Unit hyperbola</span> Geometric figure

In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length

<span class="mw-page-title-main">Conjugate hyperbola</span> A symmetric figure defined by a hyperbola

In geometry, a conjugate hyperbola to a given hyperbola shares the same asymptotes but lies in the opposite two sectors of the plane compared to the original hyperbola.

<span class="mw-page-title-main">Dupin's theorem</span>

In differential geometry Dupin's theorem, named after the French mathematician Charles Dupin, is the statement:

References

  1. Spain, Barry (1957). Analytical Conics. International series of monographs in pure and applied mathematics.v.3. New York: Pergamon Press. p. 49.
  2. Thomas Heath (1896) Apollonius of Perga: Treatise on Conic Sections, page 64
  3. Osgood, William F.; Graustein, William C. (1921). Plane and solid analytic geometry. New York: The Macmillan Company. p.  307.
  4. Whittaker, E.T. (1910). A History of the Theories of Aether and Electricity (1 ed.). Dublin: Longman, Green and Co. p.  441.
  5. G. B. Halsted (1906) Synthetic Projective Geometry, #135, #141

Further reading