1. How to Compute Lengths of Arcs on Parametric Curves Mika Seppälä

2. Lengths of Parametric Curves • Definition of Parametric Curves • Example: Lemniscate • Approximating Parametric Curves • Lengths of Approximations • Length Formula • Examples Mika Seppälä: Lengths of Parametric Curves

3. A parametric representation of the circle x2+y2=1 with radius 1 is given by x(t)=cos(t), y(t)=sin(t). This mapping maps the t-axis onto the circle. The parameterization covers the unit circle infinitely many times. Restricting the parameter t to the interval [0,2p) one gets a parameterization of the circle which parameterization covers the points of the circle only once. Example 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. Mika Seppälä: Lengths of Parametric Curves

4. Example Lemniscate Curves in the plane are usually defined by giving an equation that the points of the curve satisfy. For example x2/a2+y2/b2=1 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. Mika Seppälä: Lengths of Parametric Curves

5. 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. Mika Seppälä: Lengths of Parametric Curves

6. Approximating Parametric Curves The pictures above show a polygonarc approximation of the lemniscate with n=10, n=20 and n=30. Mika Seppälä: Lengths of Parametric Curves

7. Lengths of Approximations Mika Seppälä: Lengths of Parametric Curves

8. 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). Mika Seppälä: Lengths of Parametric Curves

9. 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. Mika Seppälä: Lengths of Parametric Curves

10. 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. Mika Seppälä: Lengths of Parametric Curves

11. Lemniscate of Bernoulli 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. Mika Seppälä: Lengths of Parametric Curves