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From "Luc Maisonobe (JIRA)" <j...@apache.org>
Subject [jira] [Commented] (MATH-639) numerical problems in rotation creation
Date Fri, 05 Aug 2011 14:19:27 GMT

    [ https://issues.apache.org/jira/browse/MATH-639?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13079989#comment-13079989
] 

Luc Maisonobe commented on MATH-639:
------------------------------------

The fact that c = <k | (u1^u2)> is really wrong is due to the fact the rotation axis
is almost exactly in the (u1, u2) plane. In fact it is only 1.833e-8 degrees (i.e. 3.199e-10
radians) out of the plane!

> numerical problems in rotation creation
> ---------------------------------------
>
>                 Key: MATH-639
>                 URL: https://issues.apache.org/jira/browse/MATH-639
>             Project: Commons Math
>          Issue Type: Bug
>    Affects Versions: 2.2
>         Environment: Linux
>            Reporter: Luc Maisonobe
>            Assignee: Luc Maisonobe
>             Fix For: 3.0
>
>
> building a rotation from the following vector pairs leads to NaN:
> u1 = -4921140.837095533, -2.1512094250440013E7, -890093.279426377
> u2 = -2.7238580938724895E9, -2.169664921341876E9, 6.749688708885301E10
> v1 = 1, 0, 0
> v2 = 0, 0, 1
> The constructor first changes the (v1, v2) pair into (v1', v2') ensuring the following
scalar products hold:
>  <v1'|v1'> == <u1|u1>
>  <v2'|v2'> == <u2|u2>
>  <u1 |u2>  == <v1'|v2'>
> Once the (v1', v2') pair has been computed, we compute the cross product:
>   k = (v1' - u1)^(v2' - u2)
> and the scalar product:
>   c = <k | (u1^u2)>
> By construction, c is positive or null and the quaternion axis we want to build is q
= k/[2*sqrt(c)].
> c should be null only if some of the vectors are aligned, and this is dealt with later
in the algorithm.
> However, there are numerical problems with the vector above with the way these computations
are done, as shown
> by the following comparisons, showing the result we get from our Java code and the result
we get from manual
> computation with the same formulas but with enhanced precision:
> commons math:   k = 38514476.5,            -84.,                           -1168590144
> high precision: k = 38514410.36093388...,  -0.374075245201180409222711..., -1168590152.10599715208...
> and it becomes worse when computing c because the vectors are almost orthogonal to each
other, hence inducing additional cancellations. We get:
> commons math    c = -1.2397173627587605E20
> high precision: c =  558382746168463196.7079627...
> We have lost ALL significant digits in cancellations, and even the sign is wrong!

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