Using analytical geometry, derive the expression for the perimeter of a deltoid curve (red colored).

A deltoid can be represented (up to rotation and translation) by the following parametric equations \[x=(b-a)\cos (t) + a \cos \bigg(\frac{b-a}{a} t\bigg)\] \[y=(b-a)\sin (t) - a \sin \bigg(\frac{b-a}{a} t\bigg)\] where $a$ is the radius of the rolling circle, $b$ is the radius of the circle within which the aforementioned circle is rolling.

Using analytical geometry, derive the expression for perimeter of the deltoid curve (red colored portion in the figure). It's 16 $a$ . How?

#Geometry

Easy Math Editor

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Comments

Ummmm, this is just a simple application of calculating arc-length of a function when it is given parametrically.......
Here............see this wiki page

Aaghaz Mahajan - 2 years, 5 months ago

As mentioned in the question, I specifically wanted an answer using analytical geometry. The calculus based solution is not the one I am looking for.

K V Shenoy - 2 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$