3.1: Space Curves and Their Representation

By 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.


This demo displays space curves described by coordinate functions x(t), y(t), and z(t). This demo shows the three coordinate axes and allows you to graph a three-dimensional curve X(t). The default curve is a "twisted cubic," given by X(t) = (x(t), y(t), z(t)) = (t, t^2, t^3).



1. Rotate the graph and look at it down each axis to see what makes this curve different from cubic and quadratic plane curves.

2. 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 sines and cosines with the same coefficients.
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