Star Geometry: 10 Lobed Star Shape Described by Parametric Equation

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In summary, the points where the lobes are "exactly" bisected will be at theta values which maximize r.
  • #1
nawidgc
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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.
 

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  • #2
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?
 
  • #3
phyzguy said:
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?
 
  • #4
phyzguy said:
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.
 
  • #5
nawidgc said:
r being the distance between any two points on the parametric curve, right?

No, r being the distance from the origin.
 
  • #6
nawidgc said:
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.
 

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  • #7
phyzguy said:
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!
 

1. What is a 10 lobed star shape?

A 10 lobed star shape is a geometric figure with 10 points or lobes, resembling a star. It is created by connecting 10 points on a circle with straight lines.

2. How is the 10 lobed star shape described?

The 10 lobed star shape is described by a parametric equation, which is a set of equations that express the coordinates of the points on the shape in terms of one or more parameters.

3. What is the significance of a parametric equation in star geometry?

A parametric equation allows us to describe and study the properties of geometric shapes, such as the 10 lobed star, in a systematic and precise manner. It also allows us to easily manipulate and transform the shape using different values for the parameters.

4. Can a 10 lobed star shape be created in real life?

Yes, a 10 lobed star shape can be created in real life using various methods such as drawing it on paper, constructing it with a compass and straight edge, or using computer software. It can also be found in nature, such as in the shape of certain flowers or sea stars.

5. What are some real-world applications of the 10 lobed star shape?

The 10 lobed star shape has been used in various architectural and artistic designs, such as in the construction of domes and stained glass windows. It is also seen in the design of some logos and symbols. Additionally, the 10 lobed star has been studied in mathematics and physics for its unique properties and applications in areas such as symmetry and wave interference.

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