Conic Sections
    Conics, a Family of Similarly Shaped Curves - Properties of Conics
      Conics, a Family of Similarly Shaped Curves - Properties of Conics
      Dandelin's Spheres - proof of conic sections focal properties
         Proof that conic section curve is the ellipse
Conics, a family of similarly shaped curves - properties of conics
  By intersecting either of the two right circular conical surfaces (nappes) with the plane perpendicular to the axis of the cone the resulting intersection is a circle c, as is shown in the figure.
  When the cutting plane is inclined to the axis of the cone at a greater angle than that made by the generating segment or generator (the slanting edge of the cone), i.e., when the plane cuts all generators of   a single cone, the resulting curve is the ellipse e.
Thus, the circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone.
   If the cutting plane is parallel to any generator of one of the cones, then the intersection curve is the parabola p.
  When the cutting plane is inclined to the axis at a smaller angle than the generator of the cone, i.e., if the intersecting plane cuts both cones the hyperbola h is generated.
Dandelin's Spheres - proof of conic sections focal properties
Proof that conic section curve is the ellipse
In the case when the plane E intersects all generators of the cone, as in down figure, it is possible to inscribe two spheres which will touch the conical surface and the plane.
Upper sphere touches the cone surface in a circle k1 and the plane at a point F1. Lower sphere touches the cone surface in a circle k2 and the plane at a point F2.
  Arbitrary chosen generating line g intersects the     
circle
k1 at a point M, the circle k2 at a point N and  
the intersection curve
e at a point P.                        
  We see that points, M and F1 are the tangency     
points of the upper sphere and points,
N and F2 the  
tangency points of the lower sphere of the tangents  
drawn from the point
P exterior to the spheres.         
  Since the segments of tangents from a point           exterior to sphere to the points of contact, are equal  
PM = PF and  PN = PF2.
And since planes of circles k1and k2, are parallel,     then are all corresponding generating segments        equal                                                                    
MN = PM + PN  is constant.
  Thus, the intersection curve is the locus of points    in the plane for which sum of distances from the two  fixed points F1and F2, is constant, i.e., the curve is   the ellipse.                                                            
The proof due to the French/Belgian mathematician Germinal Dandelin (1794 1847).
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