4.7: Global Theory of Space Curves

Total Curvature

One global property that will prove interesting to study is the total curvature of a curve. There are theorems about it which become very important when generalized to higher-dimensional objects. We have T'(t) = k(t)s'(t), so the integral of the length of T with respect to t will be the integral of the curvature with respect to arclength.

The Total Curvature of a Knotted Curve

Some of the most interesting and complicated curves in mathematics are knotted curves. A curve is said to be a knot if it is impossible to deform the curve into a circle without having the curve pass through itself at some point during the deformation. A fundamental global theorem states that a knot must be at least twice as twisted as a circle.

One special kind of knot is a torus knot, so called because it lies on a torus of revolution. A torus knot can be parametrized in this way:

    X(t) =(cos(n1t)(n3+cos(n2t)),sin(n1t)(n3+cos(n2t)),sin(n2t))
This knot lies on a torus with large radius equal to n3 and small radius equal to 1 . The parameters n1 and n2 modify the number of times the curve winds around the torus.

Need to put an APPLET here! Curvature and Torsion of a Torus Knot Need to put an IMG here!

The parameter z , specified with the Stretch along z slider, stretches the curve in the z -direction for easier viewing. The curve will not be knotted for all values of n1 and n2 , but is guaranteed to be knotted if they are relatively prime. The parameter n3 , which represents the large radius of the torus on which the curve lies, is entered with a type-in in order to be able to choose accurately important values like 2 . This particular value has the property that the asymptotic lines in the innermost rim are perpendicular. These are defined in Lab 8.

The knot is more easily examined if a tube is drawn around it. The Do Tube checkbox toggles a tube around the curve, colored yellow where the torsion of the curve is negative, orange where it is zero, and red where it is positive. (The tube may occasionally be pinched in. This occurs because of the way the curve is twisting.) In the Torsion Function window the torsion of the curve is graphed as a function of t . The total curvature of the knot is displayed in the control panel.

This demo allows you to investigate the properties of a torus knot.

    For which values of n1 , n2 , n3 will the knotted curve have torsion everywhere non-zero? (Hint: n3 does play a role.)

    What is the total curvature for various knots? What happens to the total curvature of the knot as n3 gets very large? Why? Can you prove this analytically?

Curvature and Spherical Images

In the plane, a

    b
|T'(t)|dt=a
    b
|k(s)|s'(t)dt =0
    L
|k(s)|ds, where L is the length of the curve from a to b. This is also equal to a
    b
|U'(t)|dt, where U(t) is the unit normal obtained by 90 degree rotation. But in space it is a bit different. We still have a
    b
|T'(t)|dt=0
    L
|k(s)|ds, but a
    b
|P'(t)|dt is something different. In fact, it is a
    b
k(t)2+t(t)2s'(t)dt. The length of the binormal image, on the other hand, is equal to the integral of the absolute value of the torsion, since B'(t)=-t(t)P(t) by definition.

Inversion with Respect to a Sphere

This section introduces a technique similar to Inversion with Respect to a Circle , discussed in Lab 2. For a space curve we may explore the effect of inversion with respect to a sphere centered at a point C with radius r. The image of X(t) under this transformation is given by:

    YC(t )=r2( X-C)(X-C )2 +C

Need to put an APPLET here! Inversion with Respect to a Sphere Need to put an IMG here!

Choose the center C of the sphere by clicking the middle mouse button in the Original Curve window. Closed space curves are particularly interesting to study.

This demo shows the effect on a space curve of inversion with respect to a sphere.

    Examine the circular sine curves

      X(t) =(cos(t) ,sin(t),sin(ct))
    (the default family of curves).

    What can be said about the inverted images of a space cardioid?

    What can be said about the inverted images of a ellipse in the plane? Under what circumstances will the image of a plane curve remain planar?

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