conics general question

[mr.mechanics]

could one please let me know how we write the equation of a conic passing thru two conics?

for egs., the equation of a circle 'S' passing thru the points of intersetion of two circles S1 & S2 is

'S= S1+$S2'

can we use this even for parabola and other conics?

if no, then what is the general form of the above equation?

please let me know the meaning of all the unfamiliar signs you use.

and also the concept behind it.

please help me..

thanks in advance.

[Bignose]

The general conic equation can be written in Cartesian coordinates as:

[math]Ax^2 + Bxy + Cy^2 + Dx +Ey + F =0[/math].

 

The problem is that, to identify any specific conic, you need to know the values of all six constants there, A through F. So, in general, to identify any individual conic, you will need to know 6 points that it goes through. Now, there are special cases, depending on the specific conic you want. The circle only needs three points to identify a unique circle, for example. But, in general, having only 2 points is not enough. It won't restrict the type or specific shape of the conic section going through two points. That is, you can find more than one parabola to go through any given two points, as well as find multiple hyperbolas and ellipses that will go through those two points, so two is not enough to uniquely identify any unique curve.

[EdEarl]

I am trying to understand how to determine the formula of a parabola from constructing a cone and an intersecting plane. I have found this reference, which labels the conic section in figures 8, 9, and 10. It uses the letters A-F, but they are points not constants. Thus, these figures apparently do not help to explain the following formula:

 

I have looked elsewhere on the internet, and can find nothing to explain how to obtain the values of A-F based on the cone & plane dimensions & rotation. Does anyone know of a reference that derives this formula from the geometry? TYVM.

[studiot]

 

So, in general, to identify any individual conic, you will need to know 6 points that it goes through

 

If you divide the general equation through by one of the constants you can reduce the number of points to 5, that is you can fit a conic to any 5 points.

 

I am not sure about three intersecting conics but two overlaid conics will intersect at four points. These points may be real or imaginary depending upon conic type and orientation.

 

Ed Earl

 

A parabola results if the intersecting plane is parallel to any line passing through both the base and the vertex of the cone, ie parallel to the side of the cone.

 

Does this help or do you need more, What exactly are you looking for?

Really this thread is about the two dimensional co-ordinate geometry of conics, not their generation by intersecting three D objects. So if you want more you should really start your own thread.

You will note that the general formula above only refers to two axes (x and y).

Edited August 30, 2013 by studiot