Timeline for How to sample uniformly points around a neighborhood of a point lying on a n-sphere?
Current License: CC BY-SA 3.0
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| Nov 16, 2017 at 19:23 | comment | added | whuber♦ |
@Benjamin, all orthogonal transformations are generated by reflections. The reflection that sends a point $p$ to a point $q$ is the reflection through the hyperplane that bisects line segment $pq$ perpendicularly. Equivalently, you can decompose any point $x$ into a component along vector $v=pq$ and a component perpendicular to it. All you have to do is negate the $pq$ component; equivalently, subtract twice the $pq$ component from $x$. That is what the last line of reflect does to each row of x.
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| Nov 16, 2017 at 17:31 | comment | added | beangoben | sorry, should have clarified that I meant orthogonal transformations that map one point to another. Was able to understand the arguments and code on sampling. But on the reflection transformation, was not clear where the formulas came from, also since I not so acostumed to R code. I have python code for the previous snippets that I intend to add to your answer once I verify they work. | |
| Nov 16, 2017 at 15:27 | comment | added | whuber♦ | @BenjaminSanchezLengeling Various threads discuss generating orthogonal transformations, such as stats.stackexchange.com/questions/215497. What "particular transformation" are you referring to here? | |
| Nov 16, 2017 at 9:10 | comment | added | beangoben | Any sources on constructing orthogonal transformations? or where this particular transformation is from? | |
| Oct 30, 2017 at 21:21 | vote | accept | beangoben | ||
| Oct 29, 2017 at 20:36 | history | edited | whuber♦ | CC BY-SA 3.0 |
added 712 characters in body
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| Oct 29, 2017 at 20:18 | history | answered | whuber♦ | CC BY-SA 3.0 |