Previous: Introduction


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.

Demonstration 1: Curves in Three Dimensions
ERROR: Java not enabled.

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

    x(t)=t
    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.


Next: Velocity Vectors and Speed