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I have a star-shaped geometry described by following parametric equation:

\begin{equation}

\gamma(\theta) = 1 + 0.5 \times \cos (10 \theta) (\cos(\theta),\sin(\theta), 0 \leq \theta \leq 2 \pi \

\end{equation}

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 θ 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 -

θ ( lobe_number ) = 2 \pi - lobe_number × Segtheta, ... (2)

where 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 (see attached figures). How do I find the theta values at the midpoints? I know I can always check the (x,y) data and do a tan inverse but I need an equation which gives me these values exactly / analytically.

Many thanks for help.

\begin{equation}

\gamma(\theta) = 1 + 0.5 \times \cos (10 \theta) (\cos(\theta),\sin(\theta), 0 \leq \theta \leq 2 \pi \

\end{equation}

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 θ 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 -

θ ( lobe_number ) = 2 \pi - lobe_number × Segtheta, ... (2)

where 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 (see attached figures). How do I find the theta values at the midpoints? I know I can always check the (x,y) data and do a tan inverse but I need an equation which gives me these values exactly / analytically.

Many thanks for help.