4.1: 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 vectors. 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.
This demonstration draws the parameterized curve X(t) in red and its Frenet frame at the point X(t0). Use the tapedeck for t0 to see the frame travel along the curve. In a second window, labeled "Spherical Images", the tangential indicatrix, principal normal indicatrix, and binormal indicatrix will be traced out on the unit sphere.
1. Show that the tangential indicatrix of a closed curve is also a closed curve.
Need to put an APPLET here!
The Tangential Indicatrix of a Curve
Need to put an IMG here!2. For the curve given in the demo, 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;. Try rotating the curve in such a way that the &tindic;
appears to cross the origin in the two dimensional projection of the screen.
Look at the curve - it should now have at least one cusp when looked at in
the projection of the screen. You can show that the cusp and the point of
the &tindic; which crosses the origin correspond by moving the tapedeck so
that
WHee!
lies on the origin. This is because when the &tindic; crosses the
origin, the viewing screen in perpendicular to the direction of the
tangent at the point, so the image of the tangent line becomes a point
and the projection of the curve into the viewing plane has a
singularity at that point. In most cases when we project into the
normal plane at a point, the image of the curve will have a cusp at
the point.
If the original curve is closed, then the tangential indicatrix will be a
closed curve on the unit sphere, and in general this curve will divide
the sphere into two or more regions. If we project into a plane that is not
perpendicular to any tangent line, then we obtain a smooth projected
curve with a certain number of double points. If we rotate the curve, then the
number of double points can change when the the image of the tangential
indicatrix passes over the center of the unit sphere. What happens in
such a case?
What do the tangential images of the circular helix and the exponential
helix have in common? If possible, find values for b
and c in the twisted cubic X(t)=[t, bt2,
ct3] such that the tangential indicatrix is contained in a small circle on the unit sphere.
The &NormIndic; of a curve is, analogously to the &tindic;, the curve traced out on the unit sphere by P(t) .
What happens to the projection of the curve when the viewing point crosses
the &nindic;?
The Binormal Indicatrix of a curve is the curve drawn on the unit sphere by B(t) .
What happens to the projection of the curve when the point crosses
the &bindic;?
What happens at the projected image of a point on the curve when the
binormal indicatrix at the corresponding point is perpendicular to the
viewing direction (so that the image of the binormal at that point is on
the circumference of the projection of the unit sphere into the viewing
plane)?
What happens when the binormal image curve has a cusp? An
inflection point? What does it mean when the binormal image has a
double point?
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