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1. Space Curves and Their RepresentationBy a space curve , we mean a continuous mapping of an interval into ordinary three-dimensional Euclidean space. The representation of space curves is identical to that of plane curves with the addition of a third component to all of the vectors. See the section called Plane Curves and Their Representation in Lab 1. | |
Demonstration 1: Curves in Three Dimensions This demo displays space curves described by coordinate functions. The display window shows the coordinate axes and the graph of the curve. The default function is a "twisted cubic" with coordinate functions y(t)=t2 z(t)=t3 You can enter your own coordinate functions in the type-ins in the control panel. The left- and right-hand endpoints of the domain are the pair of numbers in the domain type-in. This demo allows the user to enter the coordinate functions for a space curve. | |
Rotate the graph and look at it down each axis to see what makes this curve different from cubic and quadratic plane curves. | |
Try entering sums of sines and cosines of the same periods for each of the coordinate functions, e.g. for x(t), enter cos(t)+.5*cos(7*t)+.3*sin(17*t), for y(t), enter sin(t)+.5*sin(7*t)+.3*cos(17*t) and for z(t), enter some combination of signs and cosines with the same coefficients. |