The Tight Clasp

John M Sullivan, University of Illinois
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This applet shows the shape of the tight clasp. A clasp is a pair of arcs, simply linked to each other: one arc (whose ends are pulled upwards) passing below the other arc (whose ends are pulled downwards).

The tight clasp is the length-minimizing clasp given the constraint that the two curves stay distance one from each other, and that each one has radius of curvature bounded below by a parameter lambda (in the range from 0 to 1). When lambda=1/2, the constraint can be rephrased to say that the normal disks of radius 1/2 around the curves are disjoint from each other.

A second parameter is the angle delta (from 0 to pi/2) at which the ends are pulled up and down. When delta=0, we just have two horizontal straight lines. When delta=pi/2, the ends go off vertically, and we can think of one rope attached to the ceiling (with ends free to move within that plane) and the other attached to the floor (a plane parallel to the ceiling).

Naively, one would expect that each curve in the clasp would be a circular arc around the tip point of the other, having radius 1 since it stays at distance 1 from the that tip, and that the shape would therefore be independent of lambda. But this turns out not to be true. Intuitively, if the entire force of one rope is pushing only against the tip of the other, the tips should move apart. Counterintuitively, this means that in the true tight clasp, the tips are further apart than distance 1! For instance, it is straightforward to check that perturbing the ends into slightly elongated elliptical shapes can allow the curves to get shorter. The tips move apart, but the sides of the arches can pull in and save length.

Recent work of Jason Cantarella, Joe Fu, Rob Kusner, John Sullivan and Nancy Wrinkle, gives a general balance criterion for criticality under thickness constraints. We were able to use this balance criterion to solve explicitly for the surprisingly intricate exact shape of the tight clasp.

The two curves do lie in perpendicular vertical planes, and each is convex as expected and symmetric about the plane of the other. But in general, each one is only piecewise smooth, consisting of 11 different pieces, joined in a C1 but not C2 fashion. By symmetry, this is 6 pieces to either side of the symmetry plane.

We start with the ends (shown in black), which are straight until the first contact with the other curve. Next come shoulders (shown in green), which are circular arcs of radius one around the tip of the other component. Then come straight segments (shown in red, and usually very short). These are followed by pieces of an analytic curve (shown in purple). Then we see straight segments (again in red, and even shorter). The final piece of the curve is a circular arc (shown in blue) near the tip; it fits smoothly with its symmetric image. This blue arc has radius lambda, meaning that it hist up against the curvature part of the thickness constraint. We call such an arc a kink.

The two sliders at the left control the two parameters of the problem: the opening angle delta (from 0 to pi/2) and the radius of curvature bound lambda (from 0 to 1).

Drag the mouse within the window to rotate the picture, and use the "Reset View" button to reset to the original view. Use the two sliders at the right to change the camera: the first will zoom in, and the second will change from orthogonal projection to increasing perspective (as if we used a wider-angle lens).

The boxes at the top let you choose which parts of the picture to display: the Struts connecting points on opposite components at distance 1; the component attached to the Floor; the component attached to the ceiling; the Excluded circles made by one component in the plane of the other, and the Normal circles (tube) around one component [which is not yet implemented].

The choice at the bottom lets you select which type of clasp to display: either the true Tight Clasp [not yet implemented], or the Gehring clasp (ignoring the value of lambda), or the fake clasp (with a shoulder balancing the tip, but with straight segments in between getting too close to each other).

The other choice at the bottom lets you switch from viewing the clasp curves themselves to viewing a plot of curvature against arclength. (In the latter mode, many of the other controls are ignored.) The thin gray horizontal lines in the background of the curvature plot are at curvatures k=0, k=1 (as in the shoulders), and k=2 (as in a kink, when lambda=1/2).

Printed near the bottom of the screen are the values of lambda and delta, and the vertical distance h>=1 between the two tip points.

The java source code for the applet is available online.