Radius of A Circle inside a Sphere

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SUMMARY

The shortest distance between the end of a vector v rotated about the y-axis by angle theta and the z-axis in a sphere of radius r is definitively calculated as r*cos(theta). This conclusion is based on the geometric relationship formed by the right triangle involving the y-axis and the vector v'. The discussion confirms this calculation as correct for the context of a programming assignment related to spherical geometry.

PREREQUISITES
  • Understanding of spherical geometry
  • Knowledge of vector rotation in three-dimensional space
  • Familiarity with trigonometric functions, specifically cosine
  • Basic principles of right triangles
NEXT STEPS
  • Study vector rotation techniques in three-dimensional geometry
  • Learn about the applications of trigonometric functions in geometry
  • Explore the properties of spheres and their geometric relationships
  • Investigate programming techniques for geometric calculations in 3D space
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PAR
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Homework Statement



Say you have a sphere of radius r centered at the origin, and a vector v <r,0,0>.

Let v' be the vector v rotated about the y-axis by angle theta.

What is the shortest distance between the end of the vector and the z-axis?

Homework Equations


The Attempt at a Solution



I drew a picture:

[PLAIN]http://img253.imageshack.us/img253/9459/circleinsphere.png

Obviously the shortest distance would be the line normal to the z-axis that would complete a right triangle with the y-axis and the vector v'. The distance is also equal to the radius of a circle, which I drew on the picture.

Because of this, I believe the answer is r*cos(theta), however I am not sure, and I need to know this for a programming assignment. Thank You!
 
Last edited by a moderator:
Physics news on Phys.org
Hi PAR! :wink:

Yes, rcosθ. :smile:
 

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