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Star shaped object

  1. Sep 22, 2012 #1
    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.
     

    Attached Files:

  2. jcsd
  3. Sep 22, 2012 #2

    phyzguy

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    Science Advisor

    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?
     
  4. Sep 22, 2012 #3
    r being the distance between any two points on the parametric curve, right?
     
  5. Sep 22, 2012 #4
    The line through 2*pi*k/10 appears to be passing through a point slightly off ( to left) the mid point of the lobe.
     
  6. Sep 22, 2012 #5

    phyzguy

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    No, r being the distance from the origin.
     
  7. Sep 22, 2012 #6

    phyzguy

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    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.
     

    Attached Files:

    • Star.nb
      Star.nb
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  8. Sep 22, 2012 #7
    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!
     
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