Projecting differential element onto another
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If I have a differential element $dE_1$ with normal $n_1$ and a differential element $dE_2$ with normal $n_2$, is the projected area of the first onto the second $(hat{r} cdot n_1)(hat{r} cdot n_2)dudv$? Where $hat{r}$ is a unit vector pointing from the first to the second. I came up with this just based on intuition, I have no idea where to find more on it, so even references are highly appreciated. Specifically a more rigourous and formal treatment of this problem.
Edit: the normals can be considered to be $n_1 = frac{partial x_1}{partial u} times frac{partial x_1}{partial v}$, $n_2 = frac{partial x_2}{partial u} times frac{partial x_2}{partial v}$, where $dE_1$ is a surface element of $S_1$ parametrised through $x_1(u,v)$ and $dE_2$ is a surface element of $S_2$ parametrised through $x_2(u,v)$.
multivariable-calculus differential-geometry
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$begingroup$
If I have a differential element $dE_1$ with normal $n_1$ and a differential element $dE_2$ with normal $n_2$, is the projected area of the first onto the second $(hat{r} cdot n_1)(hat{r} cdot n_2)dudv$? Where $hat{r}$ is a unit vector pointing from the first to the second. I came up with this just based on intuition, I have no idea where to find more on it, so even references are highly appreciated. Specifically a more rigourous and formal treatment of this problem.
Edit: the normals can be considered to be $n_1 = frac{partial x_1}{partial u} times frac{partial x_1}{partial v}$, $n_2 = frac{partial x_2}{partial u} times frac{partial x_2}{partial v}$, where $dE_1$ is a surface element of $S_1$ parametrised through $x_1(u,v)$ and $dE_2$ is a surface element of $S_2$ parametrised through $x_2(u,v)$.
multivariable-calculus differential-geometry
$endgroup$
add a comment |
$begingroup$
If I have a differential element $dE_1$ with normal $n_1$ and a differential element $dE_2$ with normal $n_2$, is the projected area of the first onto the second $(hat{r} cdot n_1)(hat{r} cdot n_2)dudv$? Where $hat{r}$ is a unit vector pointing from the first to the second. I came up with this just based on intuition, I have no idea where to find more on it, so even references are highly appreciated. Specifically a more rigourous and formal treatment of this problem.
Edit: the normals can be considered to be $n_1 = frac{partial x_1}{partial u} times frac{partial x_1}{partial v}$, $n_2 = frac{partial x_2}{partial u} times frac{partial x_2}{partial v}$, where $dE_1$ is a surface element of $S_1$ parametrised through $x_1(u,v)$ and $dE_2$ is a surface element of $S_2$ parametrised through $x_2(u,v)$.
multivariable-calculus differential-geometry
$endgroup$
If I have a differential element $dE_1$ with normal $n_1$ and a differential element $dE_2$ with normal $n_2$, is the projected area of the first onto the second $(hat{r} cdot n_1)(hat{r} cdot n_2)dudv$? Where $hat{r}$ is a unit vector pointing from the first to the second. I came up with this just based on intuition, I have no idea where to find more on it, so even references are highly appreciated. Specifically a more rigourous and formal treatment of this problem.
Edit: the normals can be considered to be $n_1 = frac{partial x_1}{partial u} times frac{partial x_1}{partial v}$, $n_2 = frac{partial x_2}{partial u} times frac{partial x_2}{partial v}$, where $dE_1$ is a surface element of $S_1$ parametrised through $x_1(u,v)$ and $dE_2$ is a surface element of $S_2$ parametrised through $x_2(u,v)$.
multivariable-calculus differential-geometry
multivariable-calculus differential-geometry
edited Jan 18 at 13:43
lightxbulb
asked Jan 17 at 23:58
lightxbulblightxbulb
945311
945311
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