In Mathematical, the theorem of Dandelin , or Belgian theorem on the conic section , shows equivalence between the two metric definitions of the Conique S (starting from the hearth S and of the eccentricity) and their geometrical definition (the section of a cone by a plan).

History

Apollonius, already in IIIe front century J. - C., defines the conical ones as being the forms obtained by slipping a plan through a cone into all the possible angles. One can then obtain a Cercle, a ellipse, a Hyperbole or a Parabole. He also discovers focal properties to them and the eccentricity defines some, but one unfortunately lost the traces of his works on this subject.

Equivalence between these three properties always appeared obvious, but one tested evil to show them clearly. It is the Belgian mathematician Germinal Pierre Dandelin which manages to do it, at the XIXe century, in an extremely elegant way. It is an big event in the history of the geometry, because the studies of Apollonius were sufficiently thorough so that little progress is done meanwhile.

Ellipse

The ellipse is:

1. the place of the points of the plan whose sum of the distances to two fixed points is a constant strictly higher than the distance between the two points;
2. the place of the points of the plan whose report/ratio of the distances at a fixed point and a line not passing by the point is constant and lower than 1;
3. one of the results obtained by the intersection of a cone by a plan.

Demonstrations

That is to say P an unspecified point of the conic section. Are also G₁ and G₂ the two spheres which are tangent with the cone and the secant plan. Their intersections with the cone form two named circles k₁ and k₂, and with the plan two named points F₁ and F₂, the whole respectively. Projections of P on k₁ and k₂ are named P₁ and P₂. As PP₁=PF₁ and PP₂=PF₂ (because two tangents with the same circle of a sphere are cut in a point located remotely equalizes of the two feet of the tangents), PF₁  +  PF₂  =  P₁P₂. However, the distance between P₁ and P₂ are constant, whatever the P. In other words, for any point P, PF₁+PF₂ is constant; and thus, by definition the conic section including/understanding P is a bifocal ellipse of hearths F₁ and F₂.

Dandelin proved the equality between bifocal ellipse and ellipse as a conic section. We can also prove the equality with an ellipse drawn starting from a hearth and of a director.

For this, let us take the intersection between the plan of the section and that of the small circle k₁. We will show that it is about the director. Now let us project P on the plan of the small circle and name this new point P'. Also let us project P on the presumedly direct one and name this point D. One notes whereas all the triangles PP' P₁ are similar, whatever the point P. Ainsi, the relationship between PP' and PP₁ (=PF₁) is always constant. It is also noted that the triangles PP' D are similar, which wants to say that the relationship between PP' and PD is another constant. However we want to know if DP/PF₁ is a constant, which is now shown. The three definitions of the ellipse thus meet.

Ring

The circle is an ellipse where the two hearths F1 and F2 are confused.

Parabola

August 1st

Hyperbole

August 1st

See too

Random links:

Hermann de Hauteville | Bocas del Toro | Hannes Bauer | County of Cloncurry | Adam Nightingale