Curvature of an ellipse

**MajMinor** · October 15th 2012, 04:07 PM

I'm asked to calculate the curvature of an ellipse:

(x^2/a^2)+(y^2/b^2)=1

The professor is setting x(t) = acos(t) and y(t) = bsin(t)

but I don't know why.

I think I can calculate the curvature from there.

Any help is appreciated.

**TheEmptySet** · October 15th 2012, 04:36 PM

Originally Posted by MajMinor

I'm asked to calculate the curvature of an ellipse:

(x^2/a^2)+(y^2/b^2)=1

The professor is setting x(t) = acos(t) and y(t) = bsin(t)

but I don't know why.

I think I can calculate the curvature from there.

Any help is appreciated.

You can use this defintion to to calculate the curvature or a parametrically defined curve.

$\kappa = \frac{|x'y''-y'x''|}{((x')^2+(y')^2)^{\frac{3}{2}}}$

This gives
$\kappa = \frac{|(-a\sin(t))(-b\sin(t))-(b\cos(t))(-a\cos(t))|}{((-a\sin(t))^2+(b\cos(t))^2)^{\frac{3}{2}}}=\frac{2ab }{(a^2\sin(t)+b^2\cos(t))^{\frac{3}{2}}}$

**MajMinor** · October 15th 2012, 04:53 PM

Why is x(t)= acos(t) and y(t)=bsin(t), though?

**TheEmptySet** · October 15th 2012, 04:58 PM

Originally Posted by MajMinor

Why is x(t)= acos(t) and y(t)=bsin(t), though?

That is the parametric represention of an ellipse.

Note that

$\frac{x}{a}=\cos(t) \quad \frac{y}{b}=\sin(t)$

Now if square both equations and add them you get

$\left( \frac{x}{a} \right)^2=\cos^2(t) \quad \left( \frac{y}{b}\right)^2=\sin^2(t)$

$\ \frac{x^2}{a^2}+\frac{y^2}{b^2}=\cos^2(t) +\sin^2(t)=1$

**MajMinor** · October 15th 2012, 05:08 PM

Got it. Thanks a lot!

Thread: Curvature of an ellipse

LinkBack

Thread Tools

Display

Curvature of an ellipse

Re: Curvature of an ellipse

Re: Curvature of an ellipse

Re: Curvature of an ellipse

Re: Curvature of an ellipse

Similar Math Help Forum Discussions

curvature

curvature

Curvature

Calc of Vector Valued Func.: Curvature of Ellipse - help!

Curvature

Search Tags