Derivatives and Logarithms of 3D Transforms

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105 points by nosferalatu123 a month ago


ndriscoll - a month ago

In addition to the differential equation, you can also tweak the definition of the exponential function as a limit used with e.g. compound interest:

    exp(tA) = lim n->infinity (I+tA/n)^n
    = lim n->infinity (I+tA/n)...(I+tA/n) (n times)
So you can interpret A (or log T) as a direction to move from the identity, and exp does infinite iterated compositions of an infinitesimal shift away from the identity in that direction.
MITSardine - a month ago

Cool article. Regarding section "The exponential map and logarithm map", if you're interested in computing the matrix exponential, there is the classic: C Moler, C Van Loan, "Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later". Also, using the series expansion is not necessarily unrobust as long as you don't stop at a fixed number of iterations but instead go on as long as terms have a norm greater than some tolerance. Scaling and squaring can be used to remain always in a given range of norms (less than 1, say).

Regarding Pitfall #3, the interpolation scheme exp(tlog(A) + (1-t)log(B)) is shortest path in a sense, just not with the usual matrix norms. See V Arsigny et al., "Log‐Euclidean metrics for fast and simple calculus on diffusion tensors". I can't help but find it more elegant than exp(log(BA^{-1})t)A which could just as well have been exp(log(A^{-1}B)t)A, or even Aexp(log(A^{-1}B)t), right? It also fixes the "no more than two transforms", as you can put any convex combination in exp(sum_i x_i log(A_i)).

lenkite - a month ago

Just amazed that applets are still being used and are still working. Many folks here are younger than Java Applet technology.

moktonar - a month ago

What if you spherically interpolate the basis vectors and linearly interpolate the translation vector?

xeonmc - a month ago

essentially you decompose the transformation into (axial translation) + (screw rotation) + (oriented orthogonal stretch) and each of them are just straightforward interpolation: axial is linear, screw is angular, and stretch is exponential.

imtringued - a month ago

See also:

Screw Theory: https://en.wikipedia.org/wiki/Screw_theory