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bossk171

Member

Registered: 2007-07-16

Posts: 305

Equivalence of classical and modern definitions of Parabola

Classically, a parabola is the intersection of a cone and a plane parallel to the surface of the cone. Modernly, a parabola is the curve y = Ax² +Bx + C. Can anyone provide a sketch of a proof of the equivalence of these definitions?

Thanks

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: Equivalence of classical and modern definitions of Parabola

hi bossk171

I think I can do this; but I've got to derive the right equations from scratch so it'll take more than one post.

step one.  The cartesian equation in 3D for a cone. 

If you are not familiar with this, a simpler example would be a sphere:

where R is a constant

See diagram below.

I've simplified by choosing to show only the top half of a double (right-)cone, apex at the origin, axis along the z axis.

At a point (x,y,z) on the surface of this cone

where r is the radius.  Note. r is also a variable here.

Now a cone has the property that it gets wider at a steady rate so

where k is a constant.

So substituting to eliminate r

Step two. The equation of any plane in 3D is of the form

Step 3. Where does the plane intersect the cone?

Making z the subject and substituting to eliminate z will give an equation of the form

where a, b, c, d, e, and f are constants.

All the conic sections, circle, ellipse, parabola, hyperbola and pair of straight lines, can be fitted to this equation.

Step 4.  the circle

If the plane is parallel to the xy plane then it's equation simplifies to

and substituting in (1)

gives

which is the equation of a circle.

That's as far as I've got for the moment.

My next step will be to get the equation of a plane parallel to the slope of the surface of the cone and substitute that in (1). 
That should give a parabola.

More will hopefully follow. 

Bob

Last edited by Bob (2011-05-21 17:21:52)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Equivalence of classical and modern definitions of Parabola

This is great, thank you. I look forward to part 2.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: Equivalence of classical and modern definitions of Parabola

hi bossk171,

Yes, I'm looking forward to it too!  But so far it has eluded me.  Something strange about getting the right plane to cut the conic.

But I'm still 'on the case'.

And if all else fails, I have an old maths text book that has a proof that all conic sections lead to curves governed by the equation

where SP is the distance from a point on the curve to a fixed point (=the focus), PM is the distance from a point on the curve to a fixed line (=the directrix)  and 'e' is the eccentricity.

I don't know whether any of that means anything but

when e = 0, the curve is a circle
when e < 1, the curve is an ellipse
when e = 1, the curve is a parabola
when e > 1, the curve (now in two parts) is a hyperbola.

I'll keep trying my approach, but in a few days, if it still won't yield, I'll post the above proof.

Bob

Last edited by Bob (2011-05-23 19:52:36)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: Equivalence of classical and modern definitions of Parabola

hi again,

I think I have it.  I found it much easier to start with a fresh diagram (see below).  This time, I am looking square-on to the y-z plane with the (positive) x axis coming forward out of the plane of the picture.  I have moved the apex of the cone from the origin to a point 'p' across in the y direction and 'q' down in the (negative) z direction. (q > 0)

The section plane that cuts the cone is shown dotted and going through the origin.  The x axis lies fully in this plane so that, if (0,Y,Z) lies in the plane, then so does (x,Y,Z) for all x.

The red line also lies in the y-z plane and is normal (perpendicular) to the section plane.

The angle a is the semi vertical angle of the cone.  It is also the angle that the section plane makes with the z axis (because the section must be parallel to the cone for a parabola) and the angle that the normal makes with the y axis.

Moving the cone like this makes it possible to get the parabola going through the origin which greatly simplfies the algebra!

The new equation for the conic is

and the new equation of slope is

To get the new equation of the section plane I am using a property of 3D vector geometry that the equation of a plane is r dot n = 0 where n is the normal .

see

http://www.mathisfunforum.com/viewtopic.php?id=14856

if you need the background to this.

If (0,y,z) is a point on that normal, then

so the equation of the plane is

so

Note: the y and z in (3) are for points on the line, whereas, in (4) they are points in the section plane.

Substituating in (2)

and this in (1)

but, from the translational shift of the apex

so

Which is what you wanted.  (I hope!)

Bob

Last edited by Bob (2011-05-24 01:32:38)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Equivalence of classical and modern definitions of Parabola

Thank you very much.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.