 # numerical problems in rotation creation

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#### Details

• Type: Bug
• Status: Closed
• Priority: Major
• Resolution: Fixed
• Affects Version/s: 2.2
• Fix Version/s:
• Labels:
None
• Environment:

Linux

#### Description

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!

#### People

• Assignee: Luc Maisonobe
Reporter: Luc Maisonobe