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Dandelin's theorem - Ellisse

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The conic curve along which a round cone is sectioned by a plane has its two focuses in the points where the plane is tangent to the spheres inscribed in the cone. The straight lines obtained intersecting the secant plane with the cone-sphere contact circle are the directirxes of the conic curve (Dandelin - Quetelet, 1822). In the case of the ellipse, two are the spheres with this property: respect to the plane to which the conic belong, one is on the side of the vertex and the other is on the opposite side. Based on the notorious properties of the tangents to a sphere running through a point, the definition of focus of a round cone section that arises from the Dandelin's theorem brings to the main properties of the focus points of a conic curve. While most of the studies on conics, as curves concieved in the 3D space, are based on projective goemetry, Dandelin and Quetelet go back to classical geometry finding new properties missed by previous researchers. A meaningfull difference has anyway to be pointed out: while the original wording describes the theorem as "static" (and the same is for the model that illustrates it), Quetelet and Dandelin research is based on geometry of movement and more precisely on the study of "focal curves", therefore those curves (strophoids) described by the ellips focuses when the secant plane rotates around one of the two straight lines tangent to the cone.

In the picture below, VAB is the axial triangle obtained by cutting the cone with a plane to which the axes of the cone itself belongs.

Circumferences Υ and φ are the intersections between those planes and the Dandelin spheres. Straight lines cc are the intersections between the plane containing the triangle and the plane containing the cone-sphere contact circle (two in this case). The dashed segments are the diameters of the contact circles. Straight line ss is the intersection between the plane containing the triangle and the plane containing the conic section. The dashed segment is the majior axes of the ellipse. F and G are the focal points of the ellipse. The directixes of the ellipse, relative to the focal points, go through D and E, perpendicularly respet to the plane containing the triangle.

 

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