Geodesics (and mean) of SpecialEuclidean Group/Manifold #1944

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1. dehann

   added this to the v0.38.0 milestone on Jul 9, 2026
2. dehann

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   design

   inference

   manifolds

   on Jul 9, 2026
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   pre-docs in-situFilter for content that should end up in the Documentation; likely captured as in-situ discussionFilter for content that should end up in the Documentation; likely captured as in-situ discussion

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   Affie

   on Jul 9, 2026
6. Affie commented on Jul 10, 2026

   

   Affie

   on Jul 10, 2026

   Last edited by Affie

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   Pink (q) and green (p) shows the x,y components of 2 normal distributions on SE(2), blue is the brute force product. The orange line is the geodesic p->q on TranslationGroup(2) × SpecialOrthogonalGroup(2) while the pink line is the geodesic of p->q on SpecialEuclideanGroup(2).

   The green distribution mode/mean m (gaussian) corresponds with the MAP estimate and not the exponential barycenter that will fall on the pink line. m will correspond to the Karcher mean when using TranslationGroup(2) × SpecialOrthogonalGroup(2). On SpecialEuclideanGroup(2), m is a left invariant mean only.
   See "Xavier Pennec, Vincent Arsigny. Exponential Barycenters of the Canonical Cartan Connection and Invariant
   Means on Lie Groups" for more detail
7. Affie commented on Jul 10, 2026

   

   Affie

   on Jul 10, 2026

   Last edited by Affie

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   The exact plotting script does not exist any more, but the core is captured in a new test (PR soon)

   Here are the plotting functions.

   cols = Makie.wong_colors()
   contour(xs, ys, sum(pdf_pqs; dims=3)[:, :, 1]; color = cols[1], linewidth = 2, levels = 15, axis =(aspect=DataAspect(),))
   contour!(xs, ys, sum(pdf_ps; dims=3)[:, :, 1]; color = cols[2], alpha = 0.5)
   contour!(xs, ys, sum(pdf_qs; dims=3)[:, :, 1]; color = cols[3], alpha = 0.5)
   
   t_geo = range(0.0, 1.0; length = 100)
   pq_geo = map(t_geo) do t
       exp(G, p, t * log(G, p, q))
   end
   pq_geo_x = [first(g.x)[1] for g in pq_geo]
   pq_geo_y = [first(g.x)[2] for g in pq_geo]
   lines!(pq_geo_x, pq_geo_y; color = cols[4], linewidth = 3) # magenta
   
   #untested to plot the Riemannian geodesic (straight line), we use the base manifold
   M = base_manifold(G)
   t_geo = range(0.0, 1.0; length = 100)
   pq_geo = map(t_geo) do t
       exp(M, p, t * log(M, p, q))
   end
   pq_geo_x = [first(g.x)[1] for g in pq_geo]
   pq_geo_y = [first(g.x)[2] for g in pq_geo]
   lines!(pq_geo_x, pq_geo_y; color = cols[5], linewidth = 3) 
   
8. dehann commented on Jul 10, 2026

   

   dehann

   on Jul 10, 2026

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   Thanks, Alberto is going to help us review these
9. Affie

   closed this as completedin #1947on Jul 15, 2026