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A straight line with slope r through the point (x0,
y0) can be represented parametrically as y - y0
= rt, x - x0 = rt. In two dimensions it 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 r 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 general curve need not be linear.
Suppose we have a curve represented parametrically.
Here’s an example: x = cos t, y = sin t, z = t.
You can imagine that the parameter t represents the time variable
and these equations describe the motion of some particle in
time. These particular equations describe the curve known
as the "helix".
The equations contain two kinds of information: information
about the curve: the "speed" of the motion along
the particle’s orbit, and information about the orbit or curve
itself.
We want to extract from these equations the intrinsic properties
of the curve it represents. we further want to know how to
compute them.
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