Cylinder of Arbitrary AxisDate: 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/ |
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