Sunday, October 6, 2013

Cubic sections

I noticed that you can plot the surface of the cone that they use in describing conics as:
I don't know why maple didn't show points close to the origin but they would just follow the same pattern of smaller circles. 
So an easy proof that the degree 2 equations are conic sections is suppose you were solving the following two equations, the first for this cone and the second for the plane. 
You could in solving these equations add equation 2 to equation 1 and solve equation 2 for z to get these 2:
Then you could plug the expression for z in equation 2 in for z in equation 1

Since a,b,c,d were arbitrary we can group like terms and rename the constant part multiplying each term to f,g,h,j,k,l:
Which is just the general quadratic equation. So the plane intersecting x^2+y^2-z^2=0 are general quadratics, or they could be called conic sections as I wanted to prove. 

Now maybe the more interesting thing is that the same reasoning applies equally well to a surface x^3+y^3-z^3=0 intersecting with a plane. It looks like:
Sections of this surface are apparently the cubic sections. Also I believe the reasoning could apply to more variable, but hyperplanes intersecting higher dimensional surfaces. 

No comments:

Post a Comment