User:Egm6341.s11.team4.hylon/hw3

=Problem 3.4 Literature search the history and application of bifolium, compute the area of one leaf= Refer to lecture slide [[media:nm1.s11.mtg15.djvu|15-2]] for the problem statement.

Given
The equation of the bifolium curve in polar coordinates:$$r(t)=2\sin (t){{\cos }^{2}}(t)$$, with $$t\in [0,\pi ]$$.

Objectives
1. Do literature research to find history and application(if any) of this classic curve

2. Compute the area in one leaf to $$10^{-6}$$ accuracy.

History and application of bifolium
The folium curve is first described by Johannes Kepler in 1609. And it has two general equation, with $$({{x}^{2}}+{{y}^{2}})({{x}^{2}}+bx+{{y}^{2}})=4axy$$ in Cartesian form and $$\rho =4a\cos t{{\sin }^{2}}t-b\cos t$$ in polar coordinates. It has the meaning of “leaf-shaped” in Latin. It has three forms, known as simple folium(or single folium), the bifolium(or doublefolium), and the trifolium, corresponding to the cases when $$b=4a$$, $$b=0$$ and $$b=a$$, respectively. In 1638, Rene Descartes first discussed the type with Cartesian equation of $${{x}^{3}}+{{y}^{3}}=3axy$$, which thereafter named the folium of Descartes. Rene Descartes has found the correct shape of the curve in the positive quadrant, but he wrongly view that this leaf shape was repeated in each quadrant like the four petals of a flower. The problem to determinate the tangent to the curve was proposed to Gilles de Roberval who, having made the same incorrect assumption, called the curve fleur de jasmine after the four-petal jasmine bloom, a name that was later dropped. The folium of Descartes has an asymptote $$x+y+a=0$$. See The universal book of mathematics, by David J. Darling, for more information.

The bifolium has been studied by Longchamps (1886) and Henri Brocard(1887).

For more information about bifolium, refer to wolfram_Bifolium and 2Dcurves_Bifolium.

Matlab Code for above plots:

Compute the area of one leaf
The exact value of one leaf: Integrate the curve equation with $$t\in (0,\frac{\pi }{2})$$, we get the exact value of area of one leaf:
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$$\displaystyle
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Area=\int\limits_{0}^{\frac{\pi }{2}}{2\sin (t){{\cos }^{2}}(t)}dt=\frac{\pi }{16}

$$
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Method 1: Using Composite Trapezoidal rule and Composite Simpson's rule


In this method, we have two approaches to calculate the area of one leaf, which are Composite Trapezoidal rule and Composite Simpson’s rule. In order to compute the area of one leaf, we divide the area between the upper leaf curve and the vertical axis into two parts B and C, as depicted in the figure above. We use the equation in Cartesian coordinates in this method, that is $$({{x}^{2}}+{{y}^{2}})=2{{x}^{2}}y$$. First, we use Matlab to find the right-most and top-most point of the folium curve of one leaf:

The right-most point is(0.52, 0.748410929036851).

The top-most point is (0.495377246516107,0.499957647395650).

Matlab Code: The equation in Cartesian coordinates is implicit. And it’s unable for us to express dependent variable y in terms of independent variable x. But we can express x in terms of y. For the reason of convenience, we choose to project in the y-direction to calculate the area of one leaf. Matlab Code for solving explicit express: Using Matlab, we can get following expressions for x of one leaf:
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$$\displaystyle
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\begin{align} & {{x}_{1}}=\sqrt{y+y\sqrt{(1-2y)}-{{y}^{2}}} \\ & {{x}_{2}}=\sqrt{y-y\sqrt{(1-2y)}-{{y}^{2}}} \\ \end{align}

$$ $${{x}_{1}}$$ is the equation for bottom curve, and $${{x}_{2}}$$ is the equation for the top curve. Next, we use two approaches to calculate the area of the two different parts.
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Using Composite Trapezoidal rule:

Matlab Code: The result:



Using Composite Simpson’s rule: Matlab Code: The result:



Method 2: Using sum of area of subdivided triangles


We use the equation in polar coordinates in this method, that is $$r=2\sin (t){{\cos }^{2}}(t)$$. Matlab Code:

The result:



Comment
As seen from above results using three different methods to compute the area of one leaf, we can conclude that:

Composite Simpson's rule converges much faster than Composite Trapezoidal rule;

The Sum of Triangle Area method converges much faster than both Composite Trapezoidal rule and Composite Simpson's rule.



Matlab Code:

Reference
1.Wolfram_Bifolium

2.Composite Trapezoidal rule

3.Composite Simpson's rule

Problem solved by Hailong Chen.