CClosed Constant Curvature Space Curves
Hermann Karcher, Bonn
Abstract: We use ODEs and symmetry arguments to con-struct closed constant curvature space curves, first on cylin-ders, next on tori, at last with the Frenet-Serret equations.While I was still teaching I only knew closed constant curvature space curveswhich were pieced together from circle segments and helix segments. Butsmooth examples are easily accessible.
1. Examples on Cylinders
First roll the plane isometrically onto a cylinder of radius R : F : µ xy ∂ xR cos( y/R ) R sin( y/R ) . In the plane we describe a curve by its rotation angle against the x-axis, α ( s ) = R s ∑ g ( σ ) dσ , where ∑ g is the curvature of the plane curve, or its geodesiccurvature when rolled onto the cylinder: c ( s ) := µ cos( α ( s ))sin( α ( s )) ∂ , c ( s ) := Z s c ( σ ) dσ. The cylinder has normal curvature 0 in the x-direction and 1 /R in the y-direction. The space curvature ∑ of F ◦ c is therefore given by ∑ = sin ( α ( s )) /R + ∑ g ( s ) = sin ( α ( s )) /R + ( α ( s )) . This is a first order ODE for α ( s ), if we want ∑ = const .This first order ODE is Lipschitz, if we look for curves with ∑ > /R : α ( s ) = + q ∑ − sin ( α ( s )) /R > . The solution curves are, in the plane, convex curves. They reach α = π/ α = 0 and at α = π/ ∑ ≤ /R the ODE has some resemblence to the ODE f = p − f of the sine function: it is not a Lipschitz ODE, with non-uniqueness alongthe constant solution α ( s ) = arcsin( √ ∑ · R ). As with the sine-ODE we candifferentiate the square of the ODE, cancel 2 α ( s ) and obtain a second orderLipschitz ODE: α ( s ) = − ( α ( s )) cos( α ( s )) /R . If we choose ∑ < /R , then the second order ODE forces α ( s ) to change signwhen α ( s ) reaches α max given by sin ( α max ) = ∑ · R <
1. The solution curvesoscillate around a parallel to the x-axis and look a bit like sin-curves.If we choose ∑ = 1 /R , then α max = π/
2. We see that the circles α ( s ) := π/ α (0) < π/ π/ π/ F ◦ c therefore spirals towards oneof the circle-latitudes of the cylinder! Convex curve in the plane,rolled onto a cylinder to aconstant curvature space curve.Periodic curve on cylinder, aconstant curvature space curve.Geodesic curvature changessign when crossing the drawnline.
2. Examples on Tori
The idea is the same as on the cylinder and works for many surfaces of revo-lution which also have a symmetry plane orthogonal to the rotation axis. A2urve on the surface has constant space curvature ∑ if its geodesic curvature ∑ g and its normal curvature ∑ n satisfy ∑ = ∑ g + ∑ n . Since the normal cur-vature depends only on the tangent of the curve there is again the easy casewhere we choose ∑ > max ∑ n and compute ∑ g ( c ( s )) = + p ∑ − ∑ n ( c ) toget a second order ODE for the curve: c ( s ) = ∑ n ( c ( s )) · N ( c ( s )) + ∑ g ( c ( s )) · ( c ( s ) × N ( c ( s )) . We start the integration on the equator, direction vertically up. Because thegeodesic curvature is bounded away from zero, the curve will turn until itmeets some meridian orthogonally. Reflection in the plane of this meridianand reflection in the equator plane complete the initial quarter arc to a closedcurve. On the torus one can start at the inner or the outer equator.Constant curvature space curve, symmetric to the equatorplane. The geodesic curvature is ∑ g ≥ c at thepoint where it crosses the equator. Therefore the integration of the ODEstarts on the equator, the initial direction c (0) is a free parameter, the spacecurvature is computed as ∑ = abs( ∑ n ( c (0))).Up from the equator the normal curvature decrease and therefore the geodesiccurvature increases. Again the angle of the curve against the meridians in-creases until it meets some meridian orthogonally. Reflection in the meridianplane continues the curve back to the equator and 180 degree rotation around3he torus normal gives the next half-wave. In general these curves do not close.One needs to adjust eiither c (0) or the size of the torus until an even numberof the initial half waves fits just once around. The curve closes smoothly,because the final tangent has the same angle with the equator as the initialtangent.Constant curvature space curve, oscillating around the outerequator. The geodesic curvature changes sign where the curvecrosses the equator.
3. Examples via the Frenet Equations