3 ms·

Derivatives and Logarithms of 3D Transforms

105 points by nosferalatu123 2y ago

moktonar 2y ago
What if you spherically interpolate the basis vectors and linearly interpolate the translation vector?
xeonmc 2y 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.
- nosferalatu123 2y ago
  (author here) Be careful because (ignoring scale for to keep things simple) a transformation T can be decomposed into a translation and rotation, such that T=Rotation * Translation, but that translation is not along the axis of rotation. You probably want to instead interpolate with a screw motion, which is a rotation about an axis along with a translation along that axis (a helical curve; a curve along the outside of a cylinder). The screw motion is what you get when you interpolate with exp(log(T)*t).
  - xeonmc 2y ago
    Yes, hence I wrote (axial translation)+(screw rotation) instead of just (translation)+(rotation).
    - nosferalatu123 2y ago
      Yep!
lenkite 2y ago
Just amazed that applets are still being used and are still working. Many folks here are younger than Java Applet technology.
- nextts 2y ago
  An applet? I thought they didn't work? Or is this Java on WASM?
- billfruit 2y ago
  Is that an applet? What type of libraries could have been used to build that type of interactive 3D graphics
  - nosferalatu123 2y ago
    It's using the three.js library and a bit of javascript I wrote.
MITSardine 2y 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)).
ndriscoll 2y 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.
imtringued 2y ago
See also: Screw Theory: https://en.wikipedia.org/wiki/Screw_theory https://en.wikipedia.org/wiki/Screw_theory

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

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.

nosferalatu123 2y ago
(author here) Be careful because (ignoring scale for to keep things simple) a transformation T can be decomposed into a translation and rotation, such that T=Rotation * Translation, but that translation is not along the axis of rotation. You probably want to instead interpolate with a screw motion, which is a rotation about an axis along with a translation along that axis (a helical curve; a curve along the outside of a cylinder). The screw motion is what you get when you interpolate with exp(log(T)*t).
- xeonmc 2y ago
  Yes, hence I wrote (axial translation)+(screw rotation) instead of just (translation)+(rotation).
  - nosferalatu123 2y ago
    Yep!

lenkite 2y ago

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

nextts 2y ago
An applet? I thought they didn't work? Or is this Java on WASM?
billfruit 2y ago
Is that an applet? What type of libraries could have been used to build that type of interactive 3D graphics
- nosferalatu123 2y ago
  It's using the three.js library and a bit of javascript I wrote.

MITSardine 2y 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)).

ndriscoll 2y 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.

imtringued 2y ago

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