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26 27 slIcIng cones Dandelins sphere trick What kind of curve do you get when you slice a circular cone with a plane into an upper and a lower part It may seem counterintuitive, but this shape will always be an ellipse, that is, the kind of curve you get when you pin the two ends of a piece of thread to a desk, pull the thread taut with a pen and draw a closed curve below. In other words, an ellipse is the set of all those points in the plane the sum of whose distances from two fixed points the focal points is a constant. To prove the slicingcone theorem, Germinal Dandelin 1794 1847 inscribed two spheres into the cone that touch the slicing plane at one point each opposite, top. He then observed that the cut is indeed an ellipse with these two points as focal points and associated constant the distance between the circles in which the two spheres touch the cone. Similar tricks show that a plane cuts the cone in an ellipse, a parabola, a hyperbola lower, opposite or, if it contains the vertex, a point, a line, or a pair of lines. Newton proved that two celestial bodies will always orbit each other on one of these conic sections, e.g., every planet orbits the Sun on an ellipse with the Sun at one of the focal points.
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