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7. Parallel CurvesAs in the plane (cf. Lab2, Parallel Curves ), one of the most important topics in the theory of space curves is the study of curves and surfaces parallel to a curve. We can generalize the planar situation in several ways. We can move a curve away from itself along the principal normal to get a new curve, a "principal parallel curve", or we can move off the curve in some other normal direction, for example the binormal, to obtain a "binormal parallel curve". Another possibility is to consider all the points in space at a fixed distance from a given curve, obtaining a parallel tube around the curve. We will study this possibility in Lab 4a. | |
Demonstration 10: Normal and Binormal Parallel curves The number of steps in the second interval is the number of parallel curves that will be drawn in each direction (the direction of the principal normal and the direction of the binormal). You can make use of the connect checkbox to toggle between viewing a strip or viewing many curves. You can also fill in both sets of parallel curves to make them surfaces. (Stuff like this is described in the tutorial.) | |
One interesting phenomenon which can be observed using this demo is the interlocking of parallel curves. This is when a closed space curve and one of its parallel curves are linked together such that they cannot be pulled apart. For example, for small values of r, the principal parallel curves of the space cardioid and the space cardioid itself are linked (the space cardioid can be displayed by typing SpaceCardioid in the curve type-in). This is perhaps most easily observed by looking at a strip between the original curve and a parallel curve. The same phenomenon occurs for the binormal parallel curves. A curve with this property is said to be self-linked. |