Year 12s: what will be the first way you research your future options? >>

Converting from parametric

How would you find the intrinsic equation of the curve defined by the parametric equations

x=3ap^2, y=2ap^3

, a is constant.
I have

\tan \psi =p

but don't really know where to go from there.

(edited 7 years ago)

Reply 1

Farhan.Hanif93

Original post by Ano123

How would you find the intrinsic equation of the curve defined by the parametric equations

x=3ap^2, y=2ap^3

, a is constant.
I have

\tan \psi =p

but don't really know where to go from there.

So the task is to determine the specified arclength

s(p)

satisfying

\dfrac{ds}{dx} = \sqrt{1+ \left( \dfrac{dy}{dx} \right) ^2 }

and an initial condition of your convenience.

By two uses of the chain rule, observe that this can be rewritten in the form:

\dfrac{ds}{dp} = \dfrac{dx}{dp}\sqrt{1+\left( \dfrac{dy}{dx} \right)^2} = \sqrt{\left( \dfrac{dx}{dp} \right)^2 + \left(\dfrac{dy}{dp}\right)^2}

Can you finish it from there?

Reply 2

Ano123

Original post by Farhan.Hanif93

So the task is to determine the specified arclength

s(p)

satisfying

\dfrac{ds}{dx} = \sqrt{1+ \left( \dfrac{dy}{dx} \right) ^2 }

and an initial condition of your convenience.

By two uses of the chain rule, observe that this can be rewritten in the form:

\dfrac{ds}{dp} = \dfrac{dx}{dp}\sqrt{1+\left( \dfrac{dy}{dx} \right)^2} = \sqrt{\left( \dfrac{dx}{dp} \right)^2 + \left(\dfrac{dy}{dp}\right)^2}

Can you finish it from there?

The task is to find the arc length

s

in terms of the angle of the tangent

\psi

from whatever direction.

Reply 3

Farhan.Hanif93

Original post by Ano123

The task is to find the arc length

s

in terms of the angle of the tangent

\psi

from whatever direction.

Right, so what's the issue? If you follow through the above post, you'll see that this is easily obtained by plugging in

\tan \psi = p

after finding

s(p)

Reply 4

Ano123

Original post by TeeEm

http://www.madasmaths.com/archive/maths_booklets/advanced_topics/intrinsic_coordinates.pdf

Thanks, this is perfect.

Quick Reply

Latest

Articles for you

Writing a psychology personal statement: expert advice from universities

How to look after yourself and live independently at uni

Employability questions you should ask before choosing a university

Seven things people are always getting wrong about results day and Clearing

Converting from parametric

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you