Timeline for Étendue measure of the set of lines between two Euclidean balls
Current License: CC BY-SA 4.0
22 events
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| Jan 28 at 17:11 | vote | accept | Gro-Tsen | ||
| Jan 28 at 13:40 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 28 at 13:38 | comment | added | Iosif Pinelis | Spelled out what the function $g$ is. | |
| Jan 28 at 4:09 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 28 at 4:02 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 21 at 3:29 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 14:31 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 14:27 | comment | added | Iosif Pinelis | @Gro-Tsen : As is now shown, the étendue measure is the ordinary integral of an expression involving incomplete beta functions; please see the latter edit of this answer. Perhaps, the result of this repeated integration can be expressed in known special functions (if not elementary ones), but that does not seem very likely now to me. | |
| Jan 18 at 14:25 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 14:15 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 14:03 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 13:56 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 13:34 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 13:19 | comment | added | Gro-Tsen | (Sorry, $1/AB^2$ should be $1/AB^{m-1}$. I was thinking about the case $m=3$.) | |
| Jan 18 at 13:10 | comment | added | Gro-Tsen | Forget the word “Grassmannian” then. 😅 What I mean is this: take $A$ on $S_1$ and $B$ on $S_2$. Now take an infinitesimal surface of area $δ$ on $S_1$ around $A$ and one of area $ε$ on $S_2$ around $B$: the set of lines through these two small surfaces has a certain étendue measure which should be $δε·h(A,B)$ for a certain density $h$ which has at least a factor $1/AB^2$ and some angles related to how the surfaces are oriented, I'm not sure what they are. Then the final result is (I think) $\frac{1}{4}$ times the integral of $h(A,B)$ on $S_1 × S_2$. | |
| Jan 18 at 13:07 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 13:03 | comment | added | Iosif Pinelis | @Gro-Tsen : This may be simpler, in principle, but I have never dealt with the Grassmannian; should probably learn that. | |
| Jan 18 at 13:00 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 12:59 | comment | added | Gro-Tsen | I wonder if it's not simpler to proceed as follows: letting $A$ a point on the sphere $S_1$ bounding $B_1$ and $B$ a point on the analogous $S_2$, the line $AB$ varies in the line Grassmannian as $A$ and $B$ vary on $S_1$ and $S_2$: the density of the étendue measure wrt the product measure on $S_1 × S_2$ is $1/AB^2$ times the product of cosines of various angles which should be expressible geometrically; then it remains to integrate this over $S_1 × S_2$ (and divide by $4$ because each line will be counted 4 times). | |
| Jan 18 at 12:57 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 12:44 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 18 at 12:37 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |