Maths - AxisAngle to Matrix - Forum discussion

By: s_ludwig ( Sven Ludwig )
tilde matrix squared and the outer product  
2003-08-31 20:17

Hi,

I have an additional comment to
http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToMatrix/index.htm

The square of the tilde matrix of a vector x,
[~x]^2
becomes its outer product minus the 3x3 identity matrix if the vector is of unit norm:
(x * x^T) - I

Therefore, the equation for building a rotation martix from Axis Angle can also be written as

R = I * cos(angle) + (1 - cos(angle)) * (axis * axis^T) + sin(angle) * [~axis]

as it is for example stated in the book "Introductory Techniques for 3D Computer Vision" by Trucco and Verri.

Best Regards,

Sven

By: martinbaker ( Martin Baker )
RE: tilde matrix squared and the outer product  
2003-09-01 09:51

Hi Sven,

From what you say the Trucco and Verri book looks useful (although I had a look on Amazon and it is quite expensive).

I haven't yet filled in the gaps in this proof, I need to think about it some more.

Thanks very much for telling me about this.

Martin

By: s_ludwig ( Sven Ludwig )
RE: tilde matrix squared and the outer produc  
2003-09-01 14:37

Hi,

yes, the book is certainly a very good book, but about rotations in general it talks only in an appendix section. It covers topics of computer vision and is among very few others one of the first books regarded as a textbook for this area.

Sven