Tangent and normal vectors for an ellipse

**psd** · October 28th 2009, 02:35 AM

I need to find unit normal and tangent vectors for an ellipse with major axis ( $x$ ) of length $a$ and minor axis ( $y$ ) of length $b$ .

My initial guesses are
$\hat{N}=b\cos{\theta}\hat{i}+a\sin{\theta}\hat{j}$
and
$\hat{T}=-a\sin{\theta}\hat{i}+b\cos{\theta}\hat{j}$

These are orthogonal, but lead to incorrect results later so I suspect they're wrong.

Anyone?

**tonio** · October 28th 2009, 02:51 AM

Originally Posted by psd

I need to find unit normal and tangent vectors for an ellipse with major axis ( $x$ ) of length $a$ and minor axis ( $y$ ) of length $b$ .

My initial guesses are
$\hat{N}=b\cos{\theta}\hat{i}+a\sin{\theta}\hat{j}$
and
$\hat{T}=-a\sin{\theta}\hat{i}+b\cos{\theta}\hat{j}$

These are orthogonal, but lead to incorrect results later so I suspect they're wrong.

Anyone?

"Guesses"? Why guesses? Since your ellipse is $a\cos t\, i+b\sin t\, j\,,\; t\in [0,2\pi]$ , derivating you get a tangent vector and thus a perpendicular vector to this first one is a normal to the ellipse...but you're forgetting, I'm afraid, that these vectors have to be of length 1 !

Tonio

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