15.1 Parametric Representation of a Curve and its Intrinsic Properties

A straight line with slope
$r$
through the point
$({x}_{0},{y}_{0})$
can be represented parametrically as
$y-{y}_{0}=rt,x-{x}_{0}=t$
.
In three dimensions you will also have
$z-{z}_{0}=st$
.

In two dimensions a line can be described as the solution to one linear equation, and in three dimensions as the solution to two such equations.

A curve similarly can be represented parametrically by expressing the components of a vector from the origin to a point
$P$
with coordinates
$x,y$
and
$z$
on it, as functions of a parameter
$t$
, or by solutions to one or two equations depending on the dimension of space.

The difference is that a typical curve is not a line.

Suppose we have a curve represented parametrically.

Here's an example:
$x=\mathrm{cos}t,y=\mathrm{sin}t,z=t$
.

These particular equations describe the curve known as the "helix".

You can imagine, if it pleases you, that the parameter
$t$
represents the time variable and these equations describe the motion of some particle in time.

The equations contain two kinds of information: information about
motion
along the curve:
that is, the "speed" of the motion along the particle's orbit, and information about the orbit
or curve itself.

We want to extract from our information about the curve, the intrinsic properties of the curve it represents. We further want to know how to compute them.