Dandelin spheres
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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Dandelin spheres are one or two
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s that are
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
both to a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
and to a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
that intersects the plane. The intersection of the cone and the plane is a
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
, and the point at which either sphere touches the plane is a
focus Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ...
of the conic section, so the Dandelin spheres are also sometimes called focal spheres.Taylor, Charles. ''An Introduction to the Ancient and Modern Geometry of Conics''
page 196 ("focal spheres")pages 204–205 (history of discovery)
(Deighton, Bell and co., 1881).
The Dandelin spheres were discovered in 1822. They are named in honor of the French mathematician
Germinal Pierre Dandelin Germinal Pierre Dandelin (12 April 1794 – 15 February 1847) was a French mathematician, soldier, and professor of engineering. Life He was born near Paris to a French father and Belgian mother, studying first at Ghent then returning to Par ...
, though
Adolphe Quetelet Lambert Adolphe Jacques Quetelet FRSF or FRSE (; 22 February 1796 – 17 February 1874) was a Belgian astronomer, mathematician, statistician and sociologist who founded and directed the Brussels Observatory and was influential in intro ...
is sometimes given partial credit as well. The Dandelin spheres can be used to give elegant modern proofs of two classical theorems known to
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
. The first theorem is that a closed conic section (i.e. 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 ...
) is the
locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...
of points such that the sum of the distances to two fixed points (the foci) is constant. The second theorem is that for any conic section, the distance from a fixed point (the focus) is proportional to the distance from a fixed line (the directrix), the constant of proportionality being called the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
. A conic section has one Dandelin sphere for each focus. An ellipse has two Dandelin spheres touching the same
nappe In geology, a nappe or thrust sheet is a large sheetlike body of rock that has been moved more than or above a thrust fault from its original position. Nappes form in compressional tectonic settings like continental collision zones or on the o ...
of the cone, while
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, cal ...
has two Dandelin spheres touching opposite nappes. A
parabola 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. One descript ...
has just one Dandelin sphere.


Proof that the intersection curve has constant sum of distances to foci

Consider the illustration, depicting a cone with apex ''S'' at the top. A plane ''e'' intersects the cone in a curve ''C'' (with blue interior). The following proof shall show that the curve ''C'' is an ellipse. The two brown Dandelin spheres, ''G''1 and ''G''2, are placed tangent to both the plane and the cone: ''G''1 above the plane, ''G''2 below. Each sphere touches the cone along a circle (colored white), k_1 and k_2. Denote the point of tangency of the plane with ''G''1 by ''F''1, and similarly for ''G''2 and ''F''2 . Let ''P'' be a typical point on the curve ''C''. ''To Prove:'' The sum of distances d(P,F_1) + d(P,F_2) remains constant as the point ''P'' moves along the intersection curve ''C''. (This is one definition of ''C'' being an ellipse, with F_1 and F_2 being its foci.) *A line passing through ''P'' and the
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...
''S'' of the cone intersects the two circles, touching ''G''1 and ''G''2 respectively at points ''P''1 and ''P''2. *As ''P'' moves around the curve, ''P''1 and ''P''2 move along the two circles, and their distance ''d''(''P''1, ''P''2) remains constant. *The distance from ''P'' to ''F''1 is the same as the distance from ''P'' to ''P''1, because the line segments ''PF''1 and ''PP''1 are both
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the same sphere ''G''1. *By a symmetrical argument, the distance from ''P'' to ''F''2 is the same as the distance from ''P'' to ''P''2. *Consequently, we compute the sum of distances as d(P,F_1) + d(P,F_2) \ =\ d(P,P_1) + d(P,P_2) \ =\ d(P_1,P_2), which is constant as ''P'' moves along the curve. This gives a different proof of a theorem of
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
. If we define an ellipse to mean the locus of points ''P'' such that ''d''(''F''1, ''P'') + ''d''(''F''2, ''P'') = a constant, then the above argument proves that the intersection curve ''C'' is indeed an ellipse. That the intersection of the plane with the cone is symmetric about the perpendicular bisector of the line through ''F''1 and ''F''2 may be counterintuitive, but this argument makes it clear. Adaptations of this argument work for
hyperbolas In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, cal ...
and
parabolas 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. One descript ...
as intersections of a plane with a cone. Another adaptation works for an ellipse realized as the intersection of a plane with a right circular
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
.


Proof of the focus-directrix property

The directrix of a conic section can be found using Dandelin's construction. Each Dandelin sphere intersects the cone at a circle; let both of these circles define their own planes. The intersections of these two parallel planes with the conic section's plane will be two parallel lines; these lines are the directrices of the conic section. However, a parabola has only one Dandelin sphere, and thus has only one directrix. Using the Dandelin spheres, it can be proved that any conic section is the locus of points for which the distance from a point (focus) is proportional to the distance from the directrix. Ancient Greek mathematicians such as
Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
were aware of this property, but the Dandelin spheres facilitate the proof.Heath, Thomas. ''A History of Greek Mathematics'',
page 119 (focus-directrix property)page 542 (sum of distances to foci property)
(Clarendon Press, 1921).
Neither Dandelin nor Quetelet used the Dandelin spheres to prove the focus-directrix property. The first to do so may have been Pierce Morton in 1829, or perhaps Hugh Hamilton who remarked (in 1758) that a sphere touches the cone at a circle which defines a plane whose intersection with the plane of the conic section is a directrix. The focus-directrix property can be used to prove that astronomical objects move along conic sections around the Sun.Hyman, Andrew. "A Simple Cartesian Treatment of Planetary Motion", ''European Journal of Physics'', Vol. 14
page 145
(1993).


Notes


External links



*{{MathWorld , id=DandelinSpheres , title=Dandelin Spheres
Math Academy page on Dandelin's spheres
by Xavier Hubaut (in French).

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