Intersection of a sphere and a cone. (projection onto the xy-plane)

In summary, to find the equation of the intersection between a sphere with radius 'a' centered at the origin and a cone with its point at the origin and phi equal to ∏/3, you can set the z values of the two shapes equal to each other. This will give you the equation for the projection onto the xy plane, which can also be used to find the volume of the cone in polar form.
  • #1
Beamsbox
61
0
Part of a chapter review problem.

Say you have a sphere centered at the origin and of radius 'a'.

And you have a (ice-cream) cone which has it's point at the origin and phi equal to ∏/3.

How do I find the equation of their intersection? Which is the projection onto the xy plane.

Basically, I have to find the volume of the cone given the radius of the sphere (curvature of ice-cream) and phi = ∏/3... My answer needs to be in polar form, so I need to find the region of the projection, which has the same equation as their intersection, where z=0...


Now I'm confusing myself... Any help would be nice.

Cheers.
 
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  • #2
Set the z value on the cone equal to the z value of the sphere. (Actually, the z2 values are easier). That will give you the xy equation of the projection onto the xy plane.
 

1. What is the equation for the intersection of a sphere and a cone projected onto the xy-plane?

The equation for the intersection of a sphere and a cone projected onto the xy-plane is x^2 + y^2 = r^2, where r is the radius of the sphere.

2. How does the angle of the cone affect the size and shape of the intersection on the xy-plane?

The angle of the cone determines the size and shape of the intersection on the xy-plane. A smaller angle will result in a smaller and narrower intersection, while a larger angle will result in a larger and wider intersection.

3. What is the significance of the intersection of a sphere and a cone in 3-dimensional space?

The intersection of a sphere and a cone represents the points where the sphere and cone intersect in 3-dimensional space. It is important in various fields of mathematics and physics, such as in the study of conic sections and in geometric optics.

4. How does the location of the sphere and cone in 3-dimensional space affect the intersection on the xy-plane?

The location of the sphere and cone in 3-dimensional space can affect the size, shape, and orientation of the intersection on the xy-plane. Moving the sphere or cone closer or further apart can change the size of the intersection, while rotating the sphere or cone can change its orientation.

5. Can the intersection of a sphere and a cone on the xy-plane be used to solve real-world problems?

Yes, the intersection of a sphere and a cone on the xy-plane can be used to solve real-world problems, such as in the design of optical systems or in predicting the path of a projectile. It can also be used in computer graphics to create three-dimensional objects and simulations.

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