Star Geometry: 10 Lobed Star Shape Described by Parametric Equation

[nawidgc]

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.

 

[phyzguy]

The midpoints will be at theta values which maximize r. Write r^2 = x^2 + y^2 as a function of theta and then solve for the points where dr/d(theta) = 0. This should give the midpoints.

Are you sure these are not at the values 2*pi*k/10?

 

[nawidgc]

The midpoints will be at theta values which maximize r. Write r^2 = x^2 + y^2 as a function of theta and then solve for the points where dr/d(theta) = 0. This should give the midpoints.

Are you sure these are not at the values 2*pi*k/10?

r being the distance between any two points on the parametric curve, right?

 

[nawidgc]

Are you sure these are not at the values 2*pi*k/10?

The line through 2*pi*k/10 appears to be passing through a point slightly off ( to left) the mid point of the lobe.

 

[phyzguy]

r being the distance between any two points on the parametric curve, right?

No, r being the distance from the origin.

 

[phyzguy]

The line through 2*pi*k/10 appears to be passing through a point slightly off ( to left) the mid point of the lobe.

Not when I plot it. You've just distorted the plot by plotting it with unequal X and Y scales. If you plot it with equal scales, you'll see that theta = 2 pi k/10 does bisect the lobes.

Do you have Mathematica? I've uploaded a notebook showing this.

 

[nawidgc]

Not when I plot it. You've just distorted the plot by plotting it with unequal X and Y scales. If you plot it with equal scales, you'll see that theta = 2 pi k/10 does bisect the lobes.

Do you have Mathematica? I've uploaded a notebook showing this.

So silly of me not to notice it. Yes, I did have a distorted scale. An equal scale does remove the confusion. Thanks a lot!