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Tubes around Space Curves

Another interesting set of surfaces that can be associated with curves is comprised of tubes, such as the normal tube of radius r about a curve X(t) . This may be defined as the boundary of the union of all balls of radius r centered at points X(t) on the curve. For sufficently small r , this tube can be described as the collection of points

    Y(t,u) =X(t) +rcos(u)P(t)+rsin( u)B( t)
giving a circle of radius r in the plane through each X(t) perpendicular to the unit tangent vector T(t) at that point. Just as the geometry of plane curves is connected to the collection of singularities of parallel curves, so too the geometry of space curves is connected to the collection of singularities of parallel tubes.

One of the most important surfaces we will study is a normal tube of constant radius around a circle, called a torus of revolution. More generally, the tube surface around a closed curve is called a torus, or in some books, the canal surface of the curve.

We can vary the radius r(t) of a tube as a function of t to obtain a more general form:

    X(t) +r(t)cos( u)P( t)+r(t)sin(u)B(t)
For example, we may set
    r(t)=1/{kappa}(t)
to obtain an analogue of the evolute curve. We may also display the surface with r(t)={tau}(t ) , the torsion of the curve at X(t) , as a good visual way of identifying the points where the torsion is zero and where it is large.

We can also consider the collection of osculating circles to the curve. This surface is called the osculating tube about a space curve. It is given by

    Y( t,u)=X(t)+1/{kappa}(t)P(t)+1/{kappa}(t)(t)cos( u)P( t)-1/{kappa}(t)sin(u)T(t)
    =X( t)+1/{kappa}(t)(1+cos( u))P(t)-1/{kappa}(t)sin( u)T( t)
For a curve in the plane with 0<({kappa}{_g}(t)) and 0<({kappa}{_g}(t))' we obtain a nested collection of osculating circles with no overlapping.

Demonstration 5: Tube-Maker
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In this demo, you specify the coefficients Ct, Cp, and Cb of T, P, and B, respectively, as functions of t and u. You may make use of k(t), the curvature of the curve at t, and tau(t), the torsion of the curve at t.

This demo owes much to the previous one. We bring it back so this time it can be used for creating tubes.