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15.1 Parametric Representation of a Curve and its Intrinsic Properties

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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.