ARC LENGTH, PARAMETRIC CURVES

ARC LENGTH, PARAMETRIC CURVES

Arc Length, Parametric Curves

Parametric Curves. A parametric curve can be thought
of as the trajectory of a point that moves trough the plane with coordinates
(x, y) = (f(t), g(t)), where f(t) and g(t) are functions of the
parameter t. For each value of t we get a point of the curve. Example:
A parametric equation for a circle of radius 1 and center (0, 0) is:
x = cos t, y = sin t .
The equations x = f(t), y = g(t) are called parametric equations.

Given a parametric curve, sometimes we can eliminate t and obtain
an equivalent non-parametric equation for the same curve. For instance
t can be eliminated from x = cos t, y = sin t by using the trigonometric
relation cos2 t + sin2 t = 1, which yields the (non-parametric) equation
for a circle of radius 1 and center (0, 0):
x2 + y2 = 1 .

Example: Find a non-parametric equation for the following parametric
curve:
x = t2 − 2t, y = t + 1 .
Answer: We eliminate t by isolating it from the second equation:
t = (y − 1) ,
and plugging it in the first equation:
x = (y − 1)2 − 2(y − 1) .
i.e.:
x = y2 − 4y + 3 ,
which is a parabola with horizontal axis.