- #1
nawidgc
- 25
- 0
I have a curve defined by following parametric equation:
\begin{equation}
\gamma(\theta) = 1 + 0.5 \times \cos (N \theta) (\cos(\theta),\sin(\theta)), 0 \leq \theta \leq 2 \pi \
\end{equation}
I need to calculate the length of the curve between say θ = 0 to θ = 1.0
Formula for calculating the arc length of a curve in polar form is well known (see http://mathworld.wolfram.com/ArcLength.html Eq. 17 therein). Using this formula
\begin{equation}
s = \int\limits_{\theta = 0}^{\theta = 1.0} \sqrt{r^2 +\left(\frac{dr}{d\theta}\right)^2}
\end{equation}
where for N = 10, r is given as
\begin{equation}
r = 1 + 0.5 \times \cos(10 \theta)
\end{equation}
and
\begin{equation}
\frac{dr}{d\theta} = -0.5 \times \sin(10\theta)
\end{equation}
The problem is I can't evaluate the integral in Eq. 17 from the Mathworks link for the above equation explicitly. Obviously one can use Matlab ( with trapz command ) to evaluate the integral for given limits but I want an explicit expression for the indefinite integral in terms of generic θ1 and θ2 ( I have hundreds of such integrals to evaluate and difference between θ1 and θ2 is not constant) How do I evaluate the integral? Is there a substitution possible?
Many thanks for help.
\begin{equation}
\gamma(\theta) = 1 + 0.5 \times \cos (N \theta) (\cos(\theta),\sin(\theta)), 0 \leq \theta \leq 2 \pi \
\end{equation}
I need to calculate the length of the curve between say θ = 0 to θ = 1.0
Formula for calculating the arc length of a curve in polar form is well known (see http://mathworld.wolfram.com/ArcLength.html Eq. 17 therein). Using this formula
\begin{equation}
s = \int\limits_{\theta = 0}^{\theta = 1.0} \sqrt{r^2 +\left(\frac{dr}{d\theta}\right)^2}
\end{equation}
where for N = 10, r is given as
\begin{equation}
r = 1 + 0.5 \times \cos(10 \theta)
\end{equation}
and
\begin{equation}
\frac{dr}{d\theta} = -0.5 \times \sin(10\theta)
\end{equation}
The problem is I can't evaluate the integral in Eq. 17 from the Mathworks link for the above equation explicitly. Obviously one can use Matlab ( with trapz command ) to evaluate the integral for given limits but I want an explicit expression for the indefinite integral in terms of generic θ1 and θ2 ( I have hundreds of such integrals to evaluate and difference between θ1 and θ2 is not constant) How do I evaluate the integral? Is there a substitution possible?
Many thanks for help.