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Topic: star-shaped geometry - finding theta at midpoint of the lobes
Replies: 2   Last Post: Sep 27, 2012 12:38 PM

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ganesh

Posts: 37
Registered: 2/15/06
star-shaped geometry - finding theta at midpoint of the lobes
Posted: Sep 22, 2012 7:27 AM
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I have a star-shaped geometry described by following parametric equation:

?(?)=1+0.5×cos(10?)(cos(?),sin(?),0???2?

Thus, $\gamma (1)$ = x - coordinate and $\gamma (2)$ = y - coordinate of the point on the star - shaped geometry.

When plotted, one can see that the number 10 in above equation results in 10 lobes. So this is a 10 lobed star. The question is how to find the ? values for the points where the lobes are "exactly" bisected. I tried to plot above equation for a total 10 values of calculated as follows -

\begin{equation}
\theta ( n ) = 2 \pi - n × Segtheta
\end{equation}

where n is the lobe number and Segtheta is the angle between the lines bisecting the lobes exactly. Clearly, in this case, Segtheta = 2 \pi / 10, 10 being the total number of lobes. I am surprised to see that these points do not lie on the line bisecting the lobes (I can't attach the figures here, but one can plot in Matlab / Octave). How do I find the theta values at these midpoints of the lobes? I know I can always check the (x,y) data on the plot and do a tan inverse but I need an equation which gives me these values exactly / analytically. I intend to work with star-shapes with at least 100 lobes.
Many thanks for help.



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