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Suppose I have a cylinder defined by the endpoints $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ in Cartesian coordinates. These endpoints represent the centres of the circles at each end of the cylinder. Each circle has a radius of $r$. What is the best way of finding the maximum and minimum coordinates of any point on the surface of the sphere, i.e. $x_{min}$,$y_{min}$,$z_{min}$,$x_{max}$,$y_{max}$ and $z_{max}$?

One method would be to find the equation of the circle at each end of the cylinder and find the minimum and maximum coordinates that lie on this, but I'm not sure how to go about finding the equation of this circle, and even if this is the best method?

With many thanks,

Froskoy.

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What does your given coordinates represent? Are those endpoints the centers of both circles of your cylinder? This is not clear to me... –  Smajl Nov 21 '12 at 7:49
    
Indeed, the coordinates represent the centres of both the circles. I have edited the question to make this clearer. –  Froskoy Nov 21 '12 at 8:26
    
and I expect you know the radius of your circle, right? If so, you can compute the angle between the line connecting your endpoints and the plane. That gives you the tilt of your cylinder. Once u have it, you can compute the max point that lies on your cylinder with Pythagorean theorem –  Smajl Nov 21 '12 at 8:36
    
Oh excellent, I can't believe I didn't think of that! Just to clarify though, when you're finding the maximum and minimum points, do you work in two dimensions separately? I've edited the question again to clarify the radius. Thanks. –  Froskoy Nov 21 '12 at 8:55
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