User:Egm6341.s11.team1.arm/HW6

Problem 7: Methods of computing cicumferences
 Solved without assistance 

Given: Bifolium and ellipse
Bifolium

Ellipse

Arc length

Ellipse circumference

Find: Circumference using different methods
Compute the circumference of a bifolium and ellipse using the following methods: a) Composite trapezoidal rule b) Romberg's method c) Clenshaw-Curtis quadrature (Trefethen's code) d) Clenshaw-Curtis quadrature (Winckel's code) Compute the circumference of the ellipse using both the arc length method and the elliptic integral method.

Composite Trapezoidal Method
The MATLAB code below was developed to calculate the circumference of the bifolium and the given ellipse. The circumference of the ellipse was calculated two ways: the arc length method (Eq. 7.3), and the elliptic integral method (Eq. 7.4).

Romberg's method
The MATLAB code below was developed to calculate the circumference of the bifolium and the given ellipse. The circumference of the ellipse was calculated two ways: the arc length method (Eq. 7.3), and the elliptic integral method (Eq. 7.4).

bifolium
solution here

ellipse, where $$ a=10 \ $$ and $$ b=3 \ $$ using $$ r( \theta\ )=\frac{a(1-e^2)}{1-e \cos \theta\ } \ $$ and $$ C = \int_{ \theta\ =0}^{2 \pi\ } dl \ $$
solution here

ellipse where $$ a=10 \ $$ and $$ b=3 \ $$ using $$ E(e) = \int_{0}^{\frac{ \pi\ }{2}} \left [ 1-e^2 \sin \theta\ \right ] ^{\frac{1}{2}}, d \theta\ \ \ $$ and $$ C = 4AE(e) \ $$
solution here

bifolium
solution here

ellipse, where $$ a=10 \ $$ and $$ b=3 \ $$ using $$ r( \theta\ )=\frac{a(1-e^2)}{1-e \cos \theta\ } \ $$ and $$ C = \int_{ \theta\ =0}^{2 \pi\ } dl \ $$
solution here

ellipse where $$ a=10 \ $$ and $$ b=3 \ $$ using $$ E(e) = \int_{0}^{\frac{ \pi\ }{2}} \left [ 1-e^2 \sin \theta\ \right ] ^{\frac{1}{2}}, d \theta\ \ \ $$ and $$ C = 4AE(e) \ $$
solution here