geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, two

diameter 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 (geometry), chord of the circle. Both definitions a ...

s of a

conic section 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, tho ...

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 A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

are conjugate

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

they are

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

Of ellipse

For an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), 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 ty ...

, two diameters are conjugate if and only if the

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

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 In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

, sometimes called a bounding parallelogram (skewed compared to a bounding rectangle). In his manuscript

De motu corporum in gyrum (from Latin: "On the motion of bodies in an orbit"; abbreviated ) is the presumed title of a manuscript by Isaac Newton sent to Edmond Halley in November 1684. The manuscript was prompted by a visit from Halley earlier that year when he had qu ...

, and in the ' Principia',

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

cites as a lemma proved by previous authors that all (bounding) parallelograms for a given ellipse have the same

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

. It is possible to

reconstruct Reconstruction may refer to: Politics, history, and sociology *Reconstruction (law), the transfer of a company's (or several companies') business to a new company *''Perestroika'' (Russian for "reconstruction"), a late 20th century Soviet Union ...

an ellipse from any pair of conjugate diameters, or from any bounding parallelogram. For example, in

proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...

14 of Book VIII of his ''Collection'',

Pappus of Alexandria Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...

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 In geometry, Thales's theorem states that if , , and are distinct points on a circle where the line is a diameter, the angle is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as pa ...

for finding the directions and lengths of the major and minor axes of an ellipse regardless of its

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

shearing Sheep shearing is the process by which the woollen fleece of a sheep is cut off. The person who removes the sheep's wool is called a '' shearer''. Typically each adult sheep is shorn once each year (depending upon dialect, a sheep may be sai ...

. In

analytic geometry 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. Analytic geometry is used in physics and engineering, and als ...

, if we let the vectors of the two conjugate half-diameters be

\vec a, \vec b

, then the ellipse is parameterized by

\cos\theta \vec a + \sin\theta \vec b

\theta

varies over

, 2\pi /math>.

Of hyperbola

Similar to the elliptic case, diameters of a

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, 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 component ( ...

are conjugate when each bisects all chords parallel to the other. In this case both the hyperbola and its conjugate are sources for the chords and diameters.

Apollonius of Perga 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 o ...

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." Thomas Heath (1896) ''Apollonius of Perga: Treatise on Conic Sections'', page 64 In

, if

\vec a,\ \vec b

are vectors of the two conjugate half-diameters, then the hyperbola is parameterized by

\vec a \ \cosh \theta + \vec b \ \sinh \theta

\theta

varies over

\R

. In the case of a rectangular hyperbola, its conjugate is the reflection across an

asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

. 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 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 hyp ...

is the relation of conjugate diameters of rectangular hyperbolas. The placement of tie rods reinforcing a square assembly of

girder A girder () is a Beam (structure), beam used in construction. It is the main horizontal support of a structure which supports smaller beams. Girders often have an I-beam cross section composed of two load-bearing ''flanges'' separated by a sta ...

s is guided by the relation of conjugate diameters in a book on

. Conjugate diameters of hyperbolas are also useful for stating the

principle of relativity In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the framework of special relativity, the Maxwell equations ...

in the modern physics of

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

. The concept of relativity is first introduced in a plane consisting of a single dimension in

space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...

, the second dimension being

time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

. 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.

In projective geometry

Every line in

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

contains a

point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...

, 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 In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into i ...

between points and lines. Using these concepts, "two diameters are conjugate when each is the polar of the figurative point of the other." G. B. Halsted (1906) ''Synthetic Projective Geometry'', #135, #141 Only one of the conjugate diameters of a hyperbola cuts the curve. The notion of

point-pair separation In a cyclic order, such as the real projective line, two pairs of points separate each other when they occur alternately in the order. Thus the ordering ''a b c d'' of four points has (''a,c'') and (''b,d'') as separating pairs. This point-pair ...

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.

References

External links

* * {{cite book , first=W. H. , last=Besant , authorlink=W. H. Besant , year=1895 , url=http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;q1=109;rgn=full%20text;idno=00630002;didno=00630002;view=image;seq=127 , title=Conic sections treated geometrically , page=109 , chapter=Properties of Conjugate Diameters , series=Historical Math Monographs , location=London; Ithaca, NY , publisher=G. Bell;

Cornell University Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...

Conic sections Affine geometry

Of ellipse

Of hyperbola

In projective geometry

References

Further reading

External links