Spherical Images of Space Curves

In the previous laboratory, we introduced three unit vectors associated with a point on a smooth space curve: the unit tangent, the principal normal, and the binormal. As the parameter varies over the domain of the original curve, these three vectors trace out curves on the unit sphere. In this section we relate the properties of these spherical image curves to the properties of the original curve.

Need to put an APPLET here! The Tangential Indicatrix of a Curve Need to put an IMG here!

Two display windows appear, one with the curve and the other with the &tindic;. The curve can be defined by the user or chosen from a number of standard space curves, including Helix , SpaceCardioid and TwistedCubic (which is the default curve). The TapeDeck indicates the position of the marker WHee! on the curve and the indicatrix. The two display windows are attached in a special way. When The &tindic; is rotated, The Curve will rotate in a similar manner. Trying to rotate The Curve produces no effect. The reason for this is so that one can rotate the curve and indicatrix at the same time, and see the connections between the two.

This demo allows one to look at a curve and its &tindic;.

Need to put an APPLET here! The Normal Indicatrix of a Curve Need to put an IMG here!

The &NormIndic; of a curve is, analogously to the &tindic;, the curve traced out on the unit sphere by P(t) .

Need to put an APPLET here! The Binormal Indicatrix of a Curve Need to put an IMG here!

The Binormal Indicatrix of a curve is the curve drawn on the unit sphere by B(t) .