How to Compute Lengths of Arcs on Parametric Curves

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How to Compute Lengths of Arcs on Parametric
Curves

• Parametric Curves
• Lemniscate
• Length of Parametric Curves
• Lengths of a Circle, an Ellipse and a Lemniscate

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Lengths of Parametric Curves

• Definition of Parametric Curves
• Example Lemniscate
• Approximating Parametric Curves
• Lengths of Approximations
• Length Formula
• Examples

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Parametric Curves
Definition
Let C be a curve in the plane. A
parameterization of the curve C is a mapping
p such that p(I)C.
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Lemniscate
Curves in the plane are usually defined by giving
an equation that the points of the curve satisfy.
For example x2/a2y2/b21 is an equation for an
ellipse. The mapping p(t)(a cos(t),b sin(t)) is
a parameterization of this ellipse. In general
it is a very difficult question to find a
parameterization for a general curve in the
plane. If a parameterization can be found, then
it is a powerful tool when one wants to study the
properties of the curve.
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Plotting the Lemniscate
From this plot one can get the idea that the
lemniscate has two components, and does not pass
through the origin. This is not correct, since
the point (0,0) clearly satisfies the equation
defining the lemniscate.
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Approximating Parametric Curves
The pictures above show a polygonarc
approximation of the lemniscate with n10, n20
and n30.
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Lengths of Approximations
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Length Formula
This formula is valid assuming that the
parameterization p covers the curve C only
once (excluding possibly finitely many points
which can be covered many times).
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Circumference of a Circle
Using the parametric representation of a circle
of radius r, the computation of the length of the
curve in question was simpler than the
computation based on the representation of upper
half of the circle as a graph of a function.
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Length of an Ellipse
This integral can be computed in terms of special
functions only. Numerical integration gives the
approximation 15.865 for the length of this
ellipse.
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Length of a Lemniscate
These computations are technical and can be best
done with Maple. Parametric representation of the
lemniscate is the best way to compute its length.