Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Intersection between two cones
Replies: 6   Last Post: Feb 7, 2013 4:36 AM

 Messages: [ Previous | Next ]
 Bill Whiten Posts: 8 Registered: 6/18/10
Re: Intersection between two cones
Posted: Feb 7, 2013 4:36 AM

"Doctor61" wrote in message <kevmfe\$gku\$1@newscl01ah.mathworks.com>...
> "Bill Whiten" <W.Whiten@uq.edu.au> wrote in message <keteb4\$gmc\$1@newscl01ah.mathworks.com>...
> > "Doctor61" wrote in message <ket0to\$214\$1@newscl01ah.mathworks.com>...
> > > I have two 3d circles (centre coordiantes and radii) with their normals passing through origin. If you consider the origin to be the vertex of a right cone having the circle as the base, how can I determine if there is an intersection between these cones?
> >
> > if centre is x1,y1,z1 and radius r set d^2=x1^2+y1^2+z1^2
> > Cone is intersection of spheres a^2 (d^2+r^2)=x^2+y^2+z^2 and
> > a^2 r^2 = (a x1-x)^2+(a y1-y)^2+(a z1-z)^2
> >
> > Eliminate a to get equation of cone:
> > x^2+y^2+z^2 = (r^2+d^2) (x1 x+y1 y+z1 z)^2/d^4
> > (check this)
> >
> > Set x^2+y^2+z^2=1 and solve for points on two cones and this sphere.
> > E.g. Solve linear equations (1= (r^2+d^2) (x1 x+y1 y+z1 z)^2/d^4 etc)
> > for x and y then substitute into x^2+y^2+z^2=1 to get quadratic for z.
> >
> > Regards

>
> Thanks Bill, but I do not get your method. You said "Cone is intersection of spheres a^2 (d^2+r^2)=x^2+y^2+z^2 and

> > a^2 r^2 = (a x1-x)^2+(a y1-y)^2+(a z1-z)^2". But intersection of two spheres is a circle. Could you please elaborate a little?

The free parameter a moves the circle thus forming the cone. Eliminating a then gives the equation of the cone.
You can generate a simpler case with circle origin [0,0,a] and diameter a*r, giving z=a and (a r)^2=x^2+y^2 as the circle, eliminate a to get formula of cone.

Regards

Date Subject Author
2/6/13 Roja
2/6/13 Bruno Luong
2/6/13 Bill Whiten
2/7/13 Roja
2/7/13 Bruno Luong
2/7/13 Roja
2/7/13 Bill Whiten