Re: parabola - conic section

From: The Last Danish Pastry (clivet_at_gmail.com)
Date: 09/01/04 

Date: Wed, 1 Sep 2004 12:09:43 +0100

"NKProductionZ" <nkproductionz@aol.com> wrote in message
news:20040901010741.25959.00000104@mb-m28.aol.com...

> I was just reading a book on conic sections. How can one prove that if
cone is
> sliced by a plane parallel to a lateral side of the cone, we get a
parabola?

Let pi be some fixed plane.
Let D be some fixed line in pi.
Let F be some fixed point in pi, but not in D.

A parabola may be defined as the locus of a point, in pi, which moves so
that its distance from F is equal to its distance from D.

F is called the focus of the parabola.
D is called the directrix of the parabola.

We may use a "Dandelin sphere" to prove that if a cone, C, is intersected by
a plane, pi, which is parallel to one of the generators, G, of C, then the
curve of intersection is a parabola.

Let S be a sphere (a Dandelin sphere) inscribed in C and touching pi. [I
will not prove the existence (or uniqueness) of such a sphere.]

Let delta be the plane in which S touches C.

Let D be the line of intersection of delta and pi.

Let F be the point at which S touches pi.

Let the vertex of the cone be the point V.

Let A be the point at which G (the generator of C to which pi is parallel)
intersects delta.

Let L be the line in pi, and through F, which is parallel to G.

Let B be the point of intersection of L and D.

Clearly, angle VAB = angle ABF (since VA [the line G] is parallel to BF [the
line L]).

Let us call this angle t.

Clearly, all the generators of C meet delta at angle t.

Let P be any point on the intersection of C and pi.

Let the generator of C which passes through P intersect delta at Q.

Let the foot of the perpendicular from P to delta be the point R.

Let the foot of the perpendicular from P to D be the point C.

Clearly, the plane PCR is parallel to the plane FBA and so the angle PCR is
equal to t.

Hence the triangles PCR and PQR are congruent (they are both right-angled
with a common side PR and equal corresponding angles [since angle PQR = t =
PCR]).

So PQ=PC.

Also, PF=PQ (they are both tangents from P to S, and hence of equal length).

Hence, PF=PC.

So, the distance from P to F is the same as the distance from P to D.

Thus, P lies on a parabola with focus F and directrix D.

Q.E.D.

~~~~

For info on Dandelin, see:
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Dandelin.html
http://scienceworld.wolfram.com/biography/Dandelin.html

For a slightly whimsical take on all this, see:
http://www.clowder.net/hop/Dandelin/Dandelin.html
by Hop David 

-- 
Clive Tooth
http://www.clivetooth.dk

Relevant Pages

  • Re: Geometry with locus...
    ... Let P be the intersection point between L and xy-plane. ... The locus of point P is a parabola. ... For a given c, L describes a cone. ... Intersect of cone and plane is a parabola if plane is parallel to ...
    (sci.math)
  • Re: Geometry with locus...
    ... Let P be the intersection point between L and xy-plane. ... The locus of point P is a parabola. ... For a given c, L describes a cone. ... Intersect of cone and plane is a parabola if plane is parallel to ...
    (sci.math)
  • Re: Where is the flaw?
    ... Slice a plane through each ... Pair the divided oranges. ... The intersection is a circle, not a cone. ...
    (sci.physics.relativity)
  • Re: Geometry with locus...
    ... Line L is tangent to this sphere. ... Let P be the intersection point between L and xy-plane. ... The locus of point P is a parabola. ... A cone intersects a plane in a parabola when only one line in the ...
    (sci.math)
  • Re: parabola - conic section
    ... >I was just reading a book on conic sections. ... How can one prove that if cone is ... >sliced by a plane parallel to a lateral side of the cone, ... Equation is a parabola. ...
    (sci.math)