The passage explains why three seemingly different geometric definitions of an ellipse—stretching a circle, the thumb‑tack‑and‑string (constant sum of distances to two foci) construction, and slicing a cone with a plane—are actually the same family of curves. It focuses on proving the equivalence of the cone‑slice and thumb‑tack definitions by introducing two spheres (the Dandelin spheres) that are tangent both to the cone and to the cutting plane. The points where the spheres touch the plane become the ellipse’s foci. Using the fact that tangents from a point to a sphere have equal length, one shows that for any point q on the intersection curve, the sum of its distances to the two foci equals the constant distance between the spheres’ circles of tangency on the cone. This constant‑sum property proves that the conic section is an ellipse. The proof is highlighted as an elegant example of mathematical beauty: it requires little background, showcases creative construction (adding the spheres), and illustrates how mathematics often proceeds by revealing equivalences among different viewpoints rather than seeking a single “most fundamental” definition.
1. There are at least three main geometric ways to define an ellipse.
2. One definition is to stretch a circle in one dimension by multiplying the x‑coordinate of all points by a constant factor.
3. Another definition is the two‑thumbtack‑and‑string construction: the set of points where the sum of distances to two fixed points (foci) is constant.
4. A third definition is to intersect a cone with a plane whose angle is smaller than the cone’s slope; the intersection curve is an ellipse (a conic section).
5. Ellipses form a family of shapes ranging from a perfect circle to an infinitely stretched curve.
6. The shape of an ellipse is quantified by its eccentricity.
7. A circle has eccentricity 0; as an ellipse becomes more squashed, its eccentricity approaches 1.
8. Earth’s orbital eccentricity is approximately 0.0167.
9. Halley’s comet’s orbital eccentricity is approximately 0.9671.
10. In the thumbtack definition, eccentricity equals the distance between the foci divided by the length of the ellipse’s major axis.
11. When an ellipse is obtained by slicing a cone, its eccentricity is determined by the slope of the slicing plane.
12. The Dandelin‑sphere proof shows that slicing a cone yields a curve with the constant‑focal‑sum property.
13. Two spheres are placed, one above and one below the cutting plane, each tangent to the cone along a circle and tangent to the plane at a single point.
14. The points where the spheres touch the plane are the foci of the ellipse.
15. For any point q on the ellipse, the distance from q to a focus equals the distance from q to the corresponding circle of tangency on the cone.
16. Consequently, the sum of the distances from q to the two foci equals the constant distance between the two circles of tangency measured along the cone.
17. This sum does not depend on which point q is chosen on the ellipse.
18. Therefore the curve produced by slicing a cone satisfies the thumbtack definition of an ellipse.
19. The proof using two tangent spheres was published by Dandelin in 1822.
20. The same sphere method shows that slicing a cylinder at an angle also produces an ellipse.
21. Projecting a shape from one plane onto another tilted plane simply stretches the shape, demonstrating that the stretched‑circle definition of an ellipse is equivalent to the other two definitions.