Cylinder of Arbitrary Axis

Date: 9/2/96 at 18:55:47
From: Darrin Vallis
Subject: Cylinder of Arbitrary Axis

Hi -

I have seen the equation for a cylinder given as

(x-x0)^2 + (y-y0)^2 = R^2

which will give a right circular cylinder parallel with the Z axis 
having center X0,Y0 and radius R (if we're dealing with XYZ 
coordinates).

What is the equation of a cylinder about an ARBITRARY axis, not 
necessarily parallel with the X,Y or Z axis?

Thanks,
Darrin Vallis

Date: 9/5/96 at 10:0:12
From: Doctor Jerry
Subject: Re: Cylinder of Arbitrary Axis

I can think of two ways of describing a cylinder about an arbitrary 
axis.  As a starting point, suppose we want to describe a cylinder of 
radius R about the line through the point (X0,Y0,0) and parallel to 
the Z-axis. A point (X,Y,Z) is on the cylinder when and only when the 
distance from (X,Y,Z) to the axis is equal to R.  The square of the 
distance from (X,Y,Z)  to the axis is (X-X0)^2+(Y-Y0)^2; this must be 
equal to R^2.

If the axis lies along an arbitrary line, the same idea can be used. 
Given a point (X,Y,Z), calculate the distance from (X,Y,Z) to the line 
and set the resulting expression equal to R.

If the line goes through the point (X0,Y0,Z0) and in the direction of 
the unit vector u = (u1,u2,u2), it is not difficult to show that for 
an arbitrary point (X,Y,Z), the point on the line closest to (X,Y,Z) 
is

   (X0,Y0,Z0) + ((X-X0)u1+(Y-Y0)u2+(Z-Z0)u3) (u1,u2,u3)

So, calculate the square of the distance from (X,Y,Z) to this point 
and set the result equal to R^2.

Here's a second way, based on vectors and parametric equations.  
Suppose the axis goes through the point (X0,Y0,Z0) and lies in the 
direction of the unit vector w; further suppose that u and v are unit 
vectors that are (1) mutually perpendicular and (2) are perpendicular  
to the axis.  Parametric equations for the cylindrical surface are

   x = X0 + cos(theta)u1 + sin(theta)v1 + t*w1
   y = Y0 + cos(theta)u2 + sin(theta)v2 + t*w2
   z = Z0 + cos(theta)u3 + sin(theta)v3 + t*w3

where theta varies over the interval 0 to 2pi and t ranges over the 
the set of real numbers.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/