Re: [eigen] Why does Eigen::Quaterniond::angularDistance() method return the absolute angle?
• To: eigen <eigen@xxxxxxxxxxxxxxxxxxx>
• Subject: Re: [eigen] Why does Eigen::Quaterniond::angularDistance() method return the absolute angle?
• From: Gael Guennebaud <gael.guennebaud@xxxxxxxxx>
• Date: Mon, 13 Aug 2018 23:53:48 +0200
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Again, in 3D if you have a rotation of angle 'a' and axis `v`, then the pair (-a,-v) represents the same rotation, and this the same for the quaternions q and -q. This is why it is enough to restrict the angle to the [0,pi) range, which is what angularDistance does. Another example (following yours):

say we take:

q2.coeffs() = -q1.coeffs()

then q2 and q1 represents the exact same rotation, and it thus makes sense that:

q0.angularDistance(q1) == q0.angularDistance(q2).

Same for:

AngleAxisd(q1).angle() == AngleAxisd(q2).angle()
AngleAxisd(q1).axis() == AngleAxisd(q2).axis()

gael

On Thu, Aug 9, 2018 at 11:16 AM Dluhosch, Michael <michael.dluhosch@xxxxxxxxxx> wrote:

Ok, I can see your point regarding angular distance. The issue I'm facing happens for really large angles in my application I have following 2 quaternions:

q0=

[0]:0.42320069127859006
[1]:-0.52593954473002769
[2]:0.45987326974154225
[3]:0.5768928375077278

and q1=

[0]:-0.50596192324633082
[1]:0.59255532789450238
[2]:-0.38450528029832376
[3]:-0.49501152009970484

a = q0.angularDistance(q1) returns 0.30801826469545818 ( ~17.6 deg )

while
q = q0.conjugate() * q1;

b = 2 * std::atan2( q.vec().norm(), std::fabs(q.w()) ) returns 5.9751670424841281 (342.4 deg)

For my understanding if you wanted to ensure (-pi,pi] the angle should be -17.6 deg

Maybe the atan2 when used with proper quaternions can never return negative values? So the times 2 is used to ensure [0,pi) ?

I did not find the derivation of this formula and even if probably I wouldn't understand it. But at least also here it is written without the absolute value:

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Recovering_the_axis-angle_representation

Michael

From: Gael Guennebaud <gael.guennebaud@xxxxxxxxx>
Sent: Wednesday, August 8, 2018 10:13 AM
To: eigen
Subject: Re: [eigen] Why does Eigen::Quaterniond::angularDistance() method return the absolute angle?

One reasons is that a distance is expected to be symmetric and positive. Another one is that the "sign" is relative to the axis of rotation, so returning a "signed" angular distance alone does not seem to make much sense. Perhaps you want:

AngleAxisd aa(q1*q2.conjugate());

and if you really don't care about the axis defining the sign:

double a = AngleAxisd(q1*q2.conjugate()).angle();

gael

On Tue, Aug 7, 2018 at 4:59 PM Dluhosch, Michael <michael.dluhosch@xxxxxxxxxx> wrote:

Hi All,

This is my first mailling list mail. So if I could improve something tell me.

The Issue I face currently with the Eigen::Quaterniond class is that my application relies on the signed angle between two quaternions. Is there a really easy way to get that sign information I'm not aware of?

I had a look in the history why there is the numext::abs(d.w()) inside the atan2 call and found this:

http://eigen.tuxfamily.org/bz/show_bug.cgi?id=824#c1

This is not really an explanation so I wanted to ask for the reasoning behind the abs.

I would appreciate if you could change this method or add another one which gives the signed angle.

Regards,

Michael

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