Helix Lying on a CylinderDate: 11/26/97 at 14:38:11 From: Celina Alvarez Subject: Equation of a helix lying on a cylinder I am trying to find a parametric equation for a helix (in 3 space) lying on the cylinder, x^2+z^2 = 4, and winding three times around the y-axis. The helix starts at the point (-2,3,0) and ends at the point (-2,9,0). I have the general equation: x(T) = {2Cos[T],2Sin[T],T} and T goes from 0 to 6Pi (so it goes 3 times around the y-axis) However, I can not figure out how to get the helix to start and end at the given points or how to get the helix to lie on the given cylinder. Date: 11/26/97 at 21:57:44 From: Doctor Pete Subject: Re: Equation of a helix lying on a cylinder Hi, Since (Cos[t])^2+(Sin[t])^2 = 1, we see that (2 Cos[t])^2 + (2 Sin[t])^2 = 4, so if we let x[t] = 2 Cos[t], z[t] = 2 Sin[t], then any point of the form (x[t],y,z[t]) will lie on the given cylinder. So the helix will have some parameterization h[t] = (x[t],y[t],z[t]) = (2 Cos[t], y[t], 2 Sin[t]), for t in some interval [a,b]. Now, we require at the start point t = a, h[a] = (2 Cos[a], y[a], 2 Sin[a]) = (-2, 3, 0), and one possible value of a which satisfies this is a = Pi. But since we also require the helix to wind around three times, b = 6 Pi + a = 7 Pi. Indeed, h[b] = (2 Cos[7 Pi], y[7 Pi], 2 Sin[7 Pi]) = (-2, y[7 Pi], 0). So we conclude that y[Pi] = 3, and y[7 Pi] = 9. But we know that y[t] must be a linear function in order to get a helix; that is, y[t] = mt + n for some values m, n. So y[Pi] = m(Pi) + n = 3 y[7 Pi] = 7m(Pi) + n = 9, whereupon subtracting the first from the second, we find 6m(Pi) = 6, or m = 1/Pi, and substituting this into the first equation, 1 + n = 3, so n = 2. Therefore, y[t] = t/Pi + 2, for t between Pi and 7 Pi. Hence h[t] = (2 Cos[t], t/Pi + 2, 2 Sin[t]), t = [Pi, 7 Pi]. This gives one parameterization of the helix. There is another, namely the one which goes counterclockwise with respect to the positive y-axis, g[t] = (2 Cos[t], t/Pi + 2, -2 Sin[t]), t = [Pi, 7 Pi]. Note we can substitute u = t - Pi in either of these to give a different parameterization H[u] = (2 Cos[u+Pi], (u+Pi)/Pi + 2, 2 Sin[u+Pi]) = (-2 Cos[u], u/Pi + 3, -2 Sin[u]), u = [0, 6 Pi], G[u] = (-2 Cos[u], u/Pi + 3, 2 Sin[u]), u = [0, 6 Pi]. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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