- 441

- 0

## Main Question or Discussion Point

The motivation behind my question stems from my own curiosity. There was recently a post in this forum titled "The Widest Point on an ellipse" (or something to that effect). In any event, I misread the title, as "The wildest". I got to thinking, and remembered from vector calculus there existed a formula to measure the severity of a curve at any point.

The formula is as follows:

[tex]K(t) = \frac{||r\prime(t) \times r\prime\prime(t)||}{||r\prime(t)||^3}[/tex]

Where r(t) is our curve in parametric form.

I wanted to use this to figure out in general , what points on an ellipse are the wildest, ie at which points can we find a maximum value for K?

To keep things simple I assumed the ellipse I would look at would be soley in the xy plane. the equation I decided on is the following:

[tex]r(t) =\left( \begin{array}a a\cos(t) \\ b \sin(t) \\0 \end{array}

\right) [/tex]

Now I am running into a problem when trying to calulate K(t)

The numerator actually works out very nice:

[tex] \pm 2ab[/tex]

the denominator I can't figure out. I get stuck at this:

[tex](\sqrt{a^2\sin(t)^2 + b^2 \cos(t)^2})^3[/tex]

Anyone see a way to reduce this?

The formula is as follows:

[tex]K(t) = \frac{||r\prime(t) \times r\prime\prime(t)||}{||r\prime(t)||^3}[/tex]

Where r(t) is our curve in parametric form.

I wanted to use this to figure out in general , what points on an ellipse are the wildest, ie at which points can we find a maximum value for K?

To keep things simple I assumed the ellipse I would look at would be soley in the xy plane. the equation I decided on is the following:

[tex]r(t) =\left( \begin{array}a a\cos(t) \\ b \sin(t) \\0 \end{array}

\right) [/tex]

Now I am running into a problem when trying to calulate K(t)

The numerator actually works out very nice:

[tex] \pm 2ab[/tex]

the denominator I can't figure out. I get stuck at this:

[tex](\sqrt{a^2\sin(t)^2 + b^2 \cos(t)^2})^3[/tex]

Anyone see a way to reduce this?

Last edited: