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Any help would be appreciated.

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- Thread starter jaksweeney
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Any help would be appreciated.

Thanks

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cristo

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HallsofIvy

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1) Compute a vector R(t) to a point on a helix (in the e

direction) in terms of a parameter t.

2) Find the tangent to the helix at t and thereby the plane perpendicular

to the pipe at t, P(t).

3) Find the unit vector U(t) in plane P that lies along the projection of

e

4) Find the unit vector in plane P that results from U by rotating through

an angle phi, S(t,phi)

5) The desired vector to any point on the surface of the pipe is then

R(t) + qS(t,phi), where q is the radius of the pipe.

1) Let e

of vectors spanning 3-space. The centre of the pipe describes a helix

about the e

as a function of the parameter t:

R(t) = a cos(t) e

2) The vector tangent to the helix at t is

dR/dt = -a sin(t) e

The unit vector tangent to R(t) is

V(t) = (dR/dt)(a

The plane perpendicular to the pipe is given by the unit bivector

P(t) = IV(t), where I = e

P(t) =(-a sin(t) e

+ c e

3) We now wish to define a unit reference vector lying in the plane P(t). It

is convenient to take this as U(t) = W(t)/w, where W(t) is the projection

of vector e

W(t) = e

P

W(t) = (a

- cos(t) e

w = a (a

The desired reference vector, U(t), is then W(t)/w:

U(t) = (e

4) This reference vector can now be rotated to produce all points on a unit

circle in the plane P(t), which is perpendicular to the pipe at t:

S(t,phi) = U(t) exp(P(t)phi) = U(t) (cos(phi) + P(t) sin(phi)); i.e. S(t,phi)

is a vector in the plane P(t) written in terms of its componets in the

reference direction U(t) and in a direction perpendicular to U,

U(t)P(t) = e

e

verifies this:

U(t)P(t) = cos(t) e

5)Let the radius of the pipe be q. The points on the surface

of the coiled pipe are then given by

R(t) + q S(t,phi).

Please check my algebra before using this equation!

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