Conics and Dandelin spheres
By intersecting a cone with a plane - not passing
through the vertex of the cone -, curves of different types can be
obtained, depending on the angle enclosed between the plane and the
axis of the cone. If the angle is equal to the half of the apex angle
of the cone, the intersection curve is a parabola; if it is
smaller than the half-angle, then the intersection is a hyperbola; if
the angle is greater than the half-angle, then the curve is an
ellipse; and finally, if the plane is perpendicular to the
axis, then the intersection is a circle.
The curves listed above can also be considered as the
loci of points of certain properties.
The parabola is the locus of those points in the
plane that are equidistant from a fixed point, called the
focus, and a fixed straight line, called the
The ellipse is the locus of those points in the plane
for which the sum of the distances from two given points - the
so-called foci - is constant.
The hyperbola is the locus of those points in the
plane for which the difference between the distances from two given
points - the so-called foci - is constant.
The most attractive proof for the equivalence of the
two types of definitions is the proof by G. P. Dandelin. His proof
will be demonstrated below.
Germinal Pierre Dandelin (1794-1847) was
a French engineer, who lived in Belgium. In 1822, he discovered the
relation between the intersection curve of the cone with a plane, the
foci of the intersecion conics, and the two inscribed spheres touching
the cone and the intersecting plane.
Choose a conic to see a demonstration of the Dandelin