# Thread: Tangent and normal vectors for an ellipse

1. ## Tangent and normal vectors for an ellipse

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

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

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

Anyone?

2. Originally Posted by psd
I need to find unit normal and tangent vectors for an ellipse with major axis ($\displaystyle x$) of length $\displaystyle a$ and minor axis ($\displaystyle y$) of length $\displaystyle b$.

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

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

Anyone?
"Guesses"? Why guesses? Since your ellipse is $\displaystyle a\cos t\, i+b\sin t\, j\,,\; t\in [0,2\pi]$, 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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### unit tangent vector formula for ellipse

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