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Diffy
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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:
K(t) = \frac{||r\prime(t) \times r\prime\prime(t)||}{||r\prime(t)||^3}
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:
r(t) =\left( \begin{array}a a\cos(t) \\ b \sin(t) \\0 \end{array}<br /> \right)
Now I am running into a problem when trying to calulate K(t)
The numerator actually works out very nice:
\pm 2ab
the denominator I can't figure out. I get stuck at this:
(\sqrt{a^2\sin(t)^2 + b^2 \cos(t)^2})^3
Anyone see a way to reduce this?
The formula is as follows:
K(t) = \frac{||r\prime(t) \times r\prime\prime(t)||}{||r\prime(t)||^3}
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:
r(t) =\left( \begin{array}a a\cos(t) \\ b \sin(t) \\0 \end{array}<br /> \right)
Now I am running into a problem when trying to calulate K(t)
The numerator actually works out very nice:
\pm 2ab
the denominator I can't figure out. I get stuck at this:
(\sqrt{a^2\sin(t)^2 + b^2 \cos(t)^2})^3
Anyone see a way to reduce this?
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