Curves

Definition: A curve is simply a continuous map from an interval in R to R² or R³. A curve sigma(t): I -> R³ is graphed by plotting all the points in its image, sigma(I) = { sigma(t): t in I }. Some people call paths what we call curves, and call curves what we call the image of a curve; we use the words path and curve interchangably.

Here are some examples of curves:

Lines. Here the interval I is taken to be all of R, and the curve is l(t) = OP + t v. If you want a ray instead of a line, make I into [0,infinity), and if you want a line segment let I be a closed interval [a,b].
A circle. I = [0,2 pi] and sigma(t) = ( cos t, sin t ). A circle is a closed curve because sigma(0) = sigma(2 pi). cos t and sin t are called the component functions of the curve. A subarc of the circle, I = [a,b] with 0 < a < b < 2 pi would not be closed because the ends of the subarc, sigma(a) and sigma(b), would not be equal.
A helix. sigma(t) = ( cos t, sin t, t ) and I = R.
The graph of a continuous function f(t) can be made into a curve by setting sigma(t) = ( t, f(t) ).

There is an index of planar curves online.

Exercises:

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Last modified 1 July 1997