All points on a coiled pipe may be represented by a two-parameter vector whose components are the Cartesian coordinates of the points. The derivation adopts the following strategy:
1) Compute a vector R(t) to a point on a helix (in the e3
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
e3 in this plane.
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 e1, e2 and e3 be an orthonormal set
of vectors spanning 3-space. The centre of the pipe describes a helix
about the e3 axis. The helical curve is given by the vector R(t)
as a function of the parameter t:
R(t) = a cos(t) e1 + a sin(t) e2 + ct e3
2) The vector tangent to the helix at t is
dR/dt = -a sin(t) e1 + a cos(t) e2 + c e3.
The unit vector tangent to R(t) is
V(t) = (dR/dt)(a2+c2)-1/2
The plane perpendicular to the pipe is given by the unit bivector
P(t) = IV(t), where I = e1e2e3. i.e.
P(t) =(-a sin(t) e2e3 + a cos(t) e3e1
+ c e1e2)(a2+c2)-1/2.
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 e3 onto plane P(t) and w = ||W||:
W(t) = e3.P P-1 = -e3.P P since
P2 = -1. Substituting the expression for P(t) from above, we get
W(t) = (a2e3 + ac(sin(t) e1
- cos(t) e2))/(a2 + c2).
w = a (a2 + c2)-1/2.
The desired reference vector, U(t), is then W(t)/w:
U(t) = (e3 +(c/a)(sin(t) e1 - cos(t) e2))(1 + (c/a)2)-1/2.
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) = e3.P(t)/w. The latter vector should lie in the
e1e2 plane. Substituting the expression for P(t)
verifies this:
U(t)P(t) = cos(t) e1 + sin(t) e2.
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!