right|thumb|330px|Two Dandelin spheres touch the pale yellow plane that intersects the cone. The points of tangency F<sub>1</sub>, F<sub>2</sub> are the foci of the blue ellipse. The spheres are also tangent to the cone at circles k<sub>1</sub>, k<sub>2</sub>. <small>For a point P on the ellipse, the tangent segments PF<sub>1</sub> and PF<sub>2</sub> can each be reflected to other tangents of equal length, PF<sub>1</sub> = PP<sub>1</sub> and PF<sub>2</sub> = PP<sub>2</sub>, with PP<sub>1</sub> and PP<sub>2</sub> colinear along the ray SP. Their combined length P<sub>1</sub>P + PP<sub>2</sub> = P<sub>1</sub>P<sub>2</sub> = L is the distance between circles k<sub>1</sub> and k<sub>2</sub>, and is independent of the choice of P; thus any point on the ellipse has PF<sub>1</sub> + PF<sub>2</sub> = L.</small>
thumb|330px|This construction shows how the focal points of an ellipse can be found using the Dandelin spheres. <small>The [[angle bisector between the line representing the plane and a line representing the cone surface leads to the center of the respective sphere.</small>]]
In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.
The Dandelin spheres were discovered in 1822. They are named in honor of the French mathematician Germinal Pierre Dandelin, though Adolphe Quetelet 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. The first theorem is that a closed conic section (i.e. an ellipse) is the locus 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. Ancient Greek mathematicians such as Pappus of Alexandria 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 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.