Reference D. Eberly

Author: agarnung

_posts/2025-01-26-torus-fit.md

@@ -170,7 +170,7 @@ It is confirmed that, even with much fewer data, the result is more satisfactory
 ### Then... when it could work well?
 - In **small regions near the toroid**. If the points are close to a flat surface or a small portion of the toroid, the linear approximation can work reasonably well because the geometry is less curved. So it would work well in simplified models, too.
 
-The workaround _does the trick_; it allows us to fit a set of points in 3D space to an off-centered torus efficiently to a certain degree of accurateness. Fitting a torus whose front symmetry plane has an arbitrary orientation complicates the problem significantly; the equation becomes even more complicated to express, even approximately, in a closed form. While the orientation can be integrated as a separate step by first fitting a plane as the torus's front symmetry plane, along with centering it using centroid approximation of the points, encapsulating the optimization entirely in a rigorous formulation is open for further discussion, such as the one presented here. For further exploration of this topic, refer to [7].
+The workaround _does the trick_; it allows us to fit a set of points in 3D space to an off-centered torus efficiently to a certain degree of accurateness. Fitting a torus whose front symmetry plane has an arbitrary orientation complicates the problem significantly; the equation becomes even more complicated to express, even approximately, in a closed form. While the orientation can be integrated as a separate step by first fitting a plane as the torus's front symmetry plane, along with centering it using centroid approximation of the points, encapsulating the optimization entirely in a rigorous formulation is open for further discussion, such as the one presented here. For further exploration of this topic, refer to [6].
 
 For better accuracy, it is encomended to use **non-linear fitting methods** (e.g., Levenberg-Marquardt or Gauss-Newton), which can handle the true non-linear nature of the toroid and provide a more robust and precise solution. The linear approximation, although efficient, should only be used in very specific, simple cases. Additionally, RANSAC-type iterations applied to either the nonlinear or the linearized method could strengthen the results, as the torus is a geometric entity that lends itself to being associated with a notion of distance/error that can be leveraged.
 
@@ -187,5 +187,3 @@ For better accuracy, it is encomended to use **non-linear fitting methods** (e.g
 [5] https://mathworld.wolfram.com/Torus.html
 
 [6] https://www.geometrictools.com/Documentation/TorusFitting.pdf
-
-[7] ...