3.7: Parallel CurvesAs in the plane, 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". The principal parallel curve and binormal parallel curve at a distance d to X(t) can be written as:Xd(t) = X(t) + d*B(t) 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 4. The demonstration draws a curve X(t) in red, its principal parallel curve in green, and its binormal parallel curve in blue. The distance d can be changed using the tapedeck in the control panel. The tapedeck for t0 allows you to see the parallel curves being traced out by d*P(T) and d*B(t). 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. 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. |