Parameterization of hyperbola intersecting cone

In summary: However, you will need to use the inverse cosh(t) and sinh(t) to get back down to the real world. In summary, you are trying to solve a problem that is parametrized by z=x*cosh(t) and y=x*sinh(t).
  • #1
jake2009
3
0
Hello. I am having some trouble with the following problem and would be thankful if any of you could help me out.

Homework Statement



Let C be the hyperbola formed by intersecting the cone

[tex]x^2+y^2=z^2[/tex], [tex]z>0[/tex]

with the plane [tex]x+y+z=1[/tex], and let

[tex]\textbf{f}(x,y,z)=<0,0,1/z^2>[/tex].

I am trying to calculate [tex]\int_C \textbf{f} \wedge d\textbf{r}[/tex].

Homework Equations



Here [tex]\wedge[/tex] is the cross product in 3-space.

The Attempt at a Solution



I am able to solve the same sort of problem for the cylinder
[tex]x^2+y^2=1[/tex]
and the plane
[tex]z=2y+1[/tex].
In which case we let [tex]x = \cos t, y = \sin t[/tex] and [tex]z = 2\sin t +1[/tex] yeilding

[tex]r(t) = <\cos t,\sin t,2\sin t +1>[/tex]

from which the result follows from integration.

I attempted to do something simular for the problem I am trying to figure out. Using [tex]r(t,z) = <z \cos t, z\sin t, z>[/tex] to represent points on the cone, I tried to move forward, but did not find a good way to combine this with the equation for the plane as I did in the above example for the cylinder. Perhaps it is because I have z in r(t,z)? Following this line of attack I can comebine this with the equation for the plane z = 1-x-y = 1-z cos t - sin t, solving for z and combining with r(t,z) yeilds r(t) as

[tex]r(t) = (1+\cos t+\sin t)^{-1}<\cos t,\sin t,1>[/tex]

but then I end up with horrible singularities!

Would be most greatful if you would let me know how to solve this one!
 
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  • #2
You meant to post x^2+y^2=z^2. x^2+y^2+z^2=z^2 is a line. The problem is that the hyperbola is unbounded and sin(t) and cos(t) are bounded functions. Try using their unbounded hyperbolic cousins, sinh(t) and cosh(t), in a similar way.
 
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  • #3
Thanks, I fixed the equation you mention. I am not sure how I would use hyperbolic functions to represent the cone in the manor done with r(t,z) = <z cos t, z sin t, z>. I guess a hyperbola can be expressed as (+/- a cosh(u),b sinh(u)) but I am not sure how that helps me. Arg, this is confusing.
 
  • #4
Perhaps what I did is actually correct, privided I do not integrate past the singularities? This should not be so hard, but it is.
 
  • #5
Not entirely sure if this helps you but, the equation for a circle [itex]x^2+y^2=r^2[/itex] is parametrized by [itex]x=r \cos \theta, y =r \sin \theta[/itex], which you of course already know. Similarly the equation for a hyperbola x^2-y^2=r^2 is parametrized by [itex] x=r \cosh \theta, y=r \sinh \theta[/itex], note that [itex]\cosh^2 \theta-\sinh^2 \theta=1[/itex].
 
  • #6
jake2009 said:
Thanks, I fixed the equation you mention. I am not sure how I would use hyperbolic functions to represent the cone in the manor done with r(t,z) = <z cos t, z sin t, z>. I guess a hyperbola can be expressed as (+/- a cosh(u),b sinh(u)) but I am not sure how that helps me. Arg, this is confusing.

Your posted quadratic is now a sphere. I still think you want x^2+y^2=z^2. If so try z=x*cosh(t) and y=x*sinh(t). That satisfies x^2+y^2=z^2 for any choice of x. Now substitute into the plane equation to find x in terms of t. It's really almost exactly the same thing you did with the cylinder example.
 

FAQ: Parameterization of hyperbola intersecting cone

1. What is the purpose of parameterization in the intersection of a hyperbola and a cone?

Parameterization is used to express the coordinates of points on the hyperbola and cone in terms of one or more variables, known as parameters. This allows for a more general and flexible representation of the intersection, making it easier to analyze and manipulate mathematically.

2. How do you determine the parameters for a hyperbola and cone intersection?

The parameters can be determined by setting up a system of equations using the equations of the hyperbola and cone. Solving for the common variables will give the parameters that can be used in the parameterization.

3. Can a hyperbola intersect a cone at more than two points?

Yes, it is possible for a hyperbola to intersect a cone at more than two points. The number of intersections will depend on the specific values of the parameters and the orientation of the hyperbola and cone.

4. How does the orientation of the hyperbola and cone affect the parameterization of their intersection?

The orientation of the hyperbola and cone will determine the equations used to parameterize their intersection. Depending on the orientation, the parameters may represent different values or have different ranges of values.

5. What are some applications of parameterization in the intersection of a hyperbola and cone?

Parameterization of the intersection can be useful in fields such as physics, engineering, and computer graphics. It allows for a more efficient and accurate representation of the intersection, which can aid in solving problems or creating visualizations.

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