In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a plane intersects a cone to form a curve that is called a
conic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
. There exist either one or two spheres, called
Dandelin spheres,
tangentIn geometry, the tangent line to a curve at a given point is the straight line that "just touches" the curve at that point...
to both the plane and the cone. The point at which either sphere touches the plane is a
focusIn geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, ellipse, parabola, and hyperbola....
of the conic section, so such spheres are also sometimes called
focal spheres.
The Dandelin spheres were discovered in 1822. They are named in honor of the
BelgianThe Kingdom of Belgium is a country in northwest Europe. It is a founding member of the European Union and hosts its headquarters, as well as those of other major international organizations, including NATO...
mathematician
Germinal Pierre DandelinGerminal Pierre Dandelin was a mathematician, soldier, and professor of engineering. He was born near Paris to a French father and Belgian mother, studying first at Ghent then returning to Paris to study at the École Polytechnique. He was wounded fighting under Napoleon. He worked for the...
, though
Adolphe QueteletLambert Adolphe Jacques Quételet was a Belgian astronomer, mathematician, statistician and sociologist. He founded and directed the Brussels Observatory and was influential in introducing statistical methods to the social sciences...
is sometimes given partial credit as well.
In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a plane intersects a cone to form a curve that is called a
conic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
. There exist either one or two spheres, called
Dandelin spheres,
tangentIn geometry, the tangent line to a curve at a given point is the straight line that "just touches" the curve at that point...
to both the plane and the cone. The point at which either sphere touches the plane is a
focusIn geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, ellipse, parabola, and hyperbola....
of the conic section, so such spheres are also sometimes called
focal spheres.
The Dandelin spheres were discovered in 1822. They are named in honor of the
BelgianThe Kingdom of Belgium is a country in northwest Europe. It is a founding member of the European Union and hosts its headquarters, as well as those of other major international organizations, including NATO...
mathematician
Germinal Pierre DandelinGerminal Pierre Dandelin was a mathematician, soldier, and professor of engineering. He was born near Paris to a French father and Belgian mother, studying first at Ghent then returning to Paris to study at the École Polytechnique. He was wounded fighting under Napoleon. He worked for the...
, though
Adolphe QueteletLambert Adolphe Jacques Quételet was a Belgian astronomer, mathematician, statistician and sociologist. He founded and directed the Brussels Observatory and was influential in introducing statistical methods to the social sciences...
is sometimes given partial credit as well. The Dandelin spheres can be used to prove at least two important theorems. Both of those theorems were known for centuries before Dandelin, but he made it easier to prove them.
The first theorem is that a closed conic section (i.e. an
ellipseIn mathematics, an ellipse is the bounded case of a conic section, the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane...
) is the
locusIn mathematics, a locus is a collection of points which share a property. The term locus is typically used of a condition which defines a continuous figure or figures, that is, a curve...
of points such that the sum of the distances to two fixed points (the foci) is constant. This was known to
Ancient GreekAncient Greece is the civilisation belonging to the period of Greek history lasting from the Greek Dark Ages ca. 1100 BC and the Dorian invasion, to 146 BC and the Roman conquest of Greece after the Battle of Corinth. It is generally considered to be the seminal culture which provided the...
mathematicians such as
Apollonius of PergaApollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
, but the Dandelin spheres facilitate the proof.
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
eccentricityIn mathematics, the eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,* The eccentricity of a circle is zero....
. Again, this theorem was known to the Ancient Greeks, such as
Pappus of AlexandriaPappus of Alexandria was one of the last great Greek mathematicians of antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
, but the Dandelin spheres facilitate the proof.
A conic section has one Dandelin sphere for each focus. In particular, an ellipse has two Dandelin spheres, both touching the same
nappeIn geology a nappe is a complex recumbent fold system.In geometry, a nappe is half of a double cone.In engineering a nappe can also refer to a sheet of water flowing over a dam or similar structure....
of the cone. A
hyperbolaIn mathematics a hyperbola is a smooth planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bows aimed at each other as seen in the figure below...
has two Dandelin spheres, touching opposite nappes of the cone. A
parabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
has just one Dandelin sphere.
Proof that the curve has constant sum of distances to foci
Consider the illustration, depicting a plane intersecting a cone to form an ellipse (the interior of the ellipse is colored light blue). The two Dandelin spheres are shown, one (G1) above the ellipse, and one (G2) below. The intersection of each sphere with the cone is a circle (colored white).
- Each sphere touches the plane at a point, and let us call those two points F1 and F2.
- Let P be a typical point on the ellipse.
- Prove: The sum of distances d(F1, P) + d(F2, P) remain constant as the point P moves along the curve.
- A line passing through P and the vertex
In geometry, an apex is a descriptive label for a visual singular highest or most distant point or vertex in an isosceles triangle, pyramid or cone, usually contrasting with the opposite side called the base....
S of the cone intersects the two circles at points P1 and P2.
- As P moves along the ellipse, P1 and P2 move along the two circles.
- The distance from F1 to P is the same as the distance from P1 to P, because both are tangent
In geometry, the tangent line to a curve at a given point is the straight line that "just touches" the curve at that point...
to the same sphere (G1).
- Likewise, the distance from F2 to P is the same as the distance from P2 to P, because both are tangent
In geometry, the tangent line to a curve at a given point is the straight line that "just touches" the curve at that point...
to the same sphere (G2).
- Consequently, the sum of distances d(F1, P) + d(F2, P) must be constant as P moves along the curve because the sum of distances d(P1, P) + d(P2, P) also remains constant.
- This follows from the fact that P lies on the straight line from P1 to P2, and the distance from P1 to P2 remains constant.
This proves a result that had been proved in a different manner by
Apollonius of PergaApollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
.
If (as is often done) one takes the definition of the ellipse to be the locus of points
P such that
d(
F1,
P) +
d(
F2,
P) = a constant, then the argument above proves that the intersection of a plane with a cone is indeed an ellipse. That the intersection of the plane with the cone is symmetric about the perpendicular bisector of the line through
F1 and
F2 may be counterintuitive, but this argument makes it clear.
Adaptations of this argument work for hyperbolas and parabolas 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
cylinderA cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
.
Proof of the focus-directrix property
The directrix of a conic section can also 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 planes with the conic section's plane will be two parallel lines; these lines are the directrices of the conic section. 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 AlexandriaPappus of Alexandria was one of the last great Greek mathematicians of antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
were aware of this property, but the Dandelin spheres facilitate the proof.
Neither Dandelin nor Quetelet used the Dandelin spheres to prove the focus-directrix property. The first to do so was apparently Pierce Morton in 1829. The focus-directrix property is essential to proving that astronomical objects move along conic sections around the Sun.
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