Curves on Hypersurfaces

Classic curve on parametrized surfaces. The curves are defined via ordinary differential equations on the surfaces. These equations are numerically integrated in JavaView. Select one of various parametrized hypersurfaces that are pre-defined in applet PaSurface and choose among four types of curves on these surfaces: geodesics, shadow lines, asymptotic lines, and principal curvature lines.

The curve are described by differential equations that are integrated with a fourth-order Runge-Kutta method with constant step size. There are three scrollbars that allow you to change the length, discretization and the initial direction of the curve. The direction is given by an angle in the tangent space at the selected point.


Press key i and pick a point on the surface. The selected curve is integrated with a step size of length/(discr-1) to a maximum of the given length, or until the edge of the surface is reached.

The last two scrollbars 'thickness' and 'tube discr' control the shape of the tube around the curve.

You like motion? Press the button 'animate direction' and start the animation which varies the initial direction of the current curve at selected initial point. Look and enjoy (if your computer is fast).

© 1996-2012 Last modified: 29.10.2012 --- Konrad Polthier --- Freie Universität Berlin, Germany