Previous: Surfaces Associated With Plane Curves
Strips along Space CurvesFor 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
Demonstration 4: Surfaces generated from Curves 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.
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