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Cone-Sphere Intersection

ConeSphereIntersection

ConeSphereIntersectionCurv

Let a cone of opening parameter c and vertex at (0,0,0) intersect a sphere of radius r centered at (x_0,y_0,z_0), with the cone oriented such that its axis does not pass through the center of the sphere. Then the equations of the curve of intersection are

Combining (1) and (2) gives

(x-x_0)^2+(y-y_0)^2+(x^2+y^2)/(c^2)-(2z_0)/csqrt(x^2+y^2)+z_0^2=r^2

(3)

x^2(1+1/(c^2))-2x_0x+y^2(1+1/(c^2))-2y_0y+(x_0^2+y_0^2+z_0^2-r^2)-(2z_0)/csqrt(x^2+y^2)=0.

(4)

Therefore, x and y are connected by a complicated quartic equation, and x, y, and z by a quadratic equation.

If the cone-sphere intersection is on-axis so that a cone of opening parameter c and vertex at (0,0,z_0) is oriented with its axis along a radial of the sphere of radius r centered at (0,0,0), then the equations of the curve of intersection are

Combining (5) and (6) gives

c^2(z-z_0)^2+z^2=r^2

(7)

c^2(z^2-2z_0z+z_0^2)+z^2=r^2

(8)

z^2(c^2+1)-2c^2z_0z+(z_0^2c^2-r^2)=0.

(9)

Using the quadratic equation gives

So the curve of intersection is planar. Plugging (11) into (◇) shows that the curve is actually a circle, with radius given by

a=sqrt(r^2-z^2).

(12)

See also

Cone, Sphere

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References

Kenison, E. and Bradley, H. C. Descriptive Geometry. New York: Macmillan, pp. 282-283, 1935.

Referenced on Wolfram|Alpha

Cone-Sphere Intersection

Cite this as:

Weisstein, Eric W. "Cone-Sphere Intersection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cone-SphereIntersection.html

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