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Strips along Space Curves

For a space curve X(t) we have already considered curves obtained by pushing the curve off a fixed distance r along a particular vector, for example, the unit tangent, the principal normal, or the binormal vector. (See, for example, the section on Parallel Curves in lab 3.) We now consider the surface swept out by the one-parameter family of curves as r varies. These strips along the different vector fields are called developables.

For example, the tangential surface of a curve is defined by the formula

    Y(t,u) =X(t) +uT(t)
where u ranges over all real numbers. By restricting to non-negative values of u, we obtain the forward tangential surface , while the part of the surface corresponding to negative values of u gives the backward tangential surface . The way the two parts of the tangential surface fit together along the curve is one of the most significant phenomena associated with this surface.

Demonstration 4: Surfaces generated from Curves
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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. To have no component along some vector, simply make the function zero.

This demo is quite extensive and allows you to create various types of surfaces associated with a curve.

    Try graphing all of the surfaces described above with various generating curves.

    Does the tangential surface have any particular properties independent of the generating curve?

    Type 0 in all the unit vector components but type -curve(t)+u*curve(t) in the Second Curve type-in. This will give you the cone through the curve. Try it for various curves.

Next: Tubes Around Space Curves