User:Egm6341.s10.team3.ynahn81/HW5-11

= (11) Derivation of the formula to obtain arc length of ellipse from cosine raw = Ref: Lecture notes [[media:Egm6341.s10.mtg31.djvu|p.31-1]]

Problem Statement
Derive the following formula for arc length of ellipse from cosine raw.


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Arclength(PQ) = \int_{\theta_{P}}^{\theta_{Q}}d\theta\left[r^2+\left(\frac{dr}{d\theta}\right)^{2}\right]^{\frac{1}{2}} $$ 
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 * $$\displaystyle
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Solution


Recall the cosine raw:


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\overline{AB}^2=\overline{OA}^2+\overline{OB}^2-2\overline{OA}\,\overline{OB}\cos d\theta $$ $$ 
 * $$\displaystyle
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 * $$\displaystyle (Eq. 1)
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To simply the derivation, let's use the following notations.

$$\displaystyle \overline{AB} = dl $$   $$\displaystyle \overline{OA} = r(\theta) = r $$   $$\displaystyle \overline{OB} = r(\theta+d\theta) \approx r + dr $$ 

Also, from the Taylor series expansion of $$\displaystyle \cos d\theta $$,
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\cos d\theta = \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}d\theta^{2n} = 1 - \frac{d\theta^2}{2!} + \frac{d\theta^4}{4!} - \cdots, $$  we can use the following approximated expression since the angle $$\displaystyle d\theta $$ is extremely small,
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\cos d\theta \approx 1 - \frac{d\theta^2}{2} $$ 
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As a consequence, (Eq.1) can be expressed as below,


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\begin{align} dl^2 &= r(\theta)^2+r(\theta+d\theta)^2-2r(\theta)r(\theta+d\theta)\cos d\theta\\ &\approx r^2 + (r+dr)^2 -2r(r+dr)\left(1-\frac{1}{2}d\theta^2\right)\\ &=2r^2+2rdr+dr^2-(2r^2+2rdr)+(r^2+rdr)d\theta^2\\ &=dr^2+r^2d\theta^2+\underbrace{rdrd\theta^2}_{\approx 0}\\ &\approx d\theta^2\left[r^2+\left(\frac{dr}{d\theta}\right)^2\right] \end{align} $$ 
 * $$\displaystyle
 * $$\displaystyle
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Therefore,
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dl = d\theta\left[r^2+\left(\frac{dr}{d\theta}\right)^2\right]^{\frac{1}{2}} $$ 
 * $$\displaystyle
 * $$\displaystyle
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Finally, the arc length PQ can be obtained by


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$$\displaystyle Arclength(PQ)=\int_{\theta_{P}}^{\theta_{Q}}dl = \int_{\theta_{P}}^{\theta_{Q}}d\theta\left[r^2+\left(\frac{dr}{d\theta}\right)^2\right]^{\frac{1}{2}} $$
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Above result is exactly same with (2) [[media:Egm6341.s10.mtg30.djvu|p.30-3]].