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docs: Add a white paper for line surface jacobian (#2657)
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This PR adds a white paper for line surface jacobian

Abstract:
```
Line surface used for wire measurements or perigee surface is an unique type of surface where the intersections of a track is not defined on a physical plane. 
Due to this trait, the underlying mathematics gets quite complicated especially with coordinate transform jacobian. 
In this white paper, we attempt to derive the jacobians for both of local-to-global and global-to-local transforms.
```

~~The whitepaper contains the calculation of local-to-global
transformation jacobian but it doesn't include global-to-local one,
which I left for future authors (I didn't manage to solve it)~~

Update: global to local jacobian calculation is updated.

Co-authored-by: kodiakhq[bot] <49736102+kodiakhq[bot]@users.noreply.github.com>
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beomki-yeo and kodiakhq[bot] committed Dec 14, 2023
1 parent 1b29c4c commit bbbbf6b
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4 changes: 4 additions & 0 deletions Core/src/Surfaces/LineSurface.cpp
Original file line number Diff line number Diff line change
Expand Up @@ -227,6 +227,10 @@ Acts::BoundToFreeMatrix Acts::LineSurface::boundToFreeJacobian(
jacToGlobal(eFreeDir2, eBoundTheta) = -sinTheta;
jacToGlobal(eFreeQOverP, eBoundQOverP) = 1;

// For the derivative of global position with bound angles, refer the
// following white paper:
// https://acts.readthedocs.io/en/latest/white_papers/line-surface-jacobian.html

// the projection of direction onto ref frame normal
double ipdn = 1. / direction.dot(rframe.col(2));
// build the cross product of d(D)/d(eBoundPhi) components with y axis
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9 changes: 9 additions & 0 deletions docs/white_papers.toml
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Expand Up @@ -34,3 +34,12 @@ pdf_url = "https://github.com/felix-russo/billoir-covariances/releases/download/
authors = [ "Felix Russo",]
title = "Cross-Covariance Matrices in the Billoir Vertex Fit"
description = "In this white paper we derive\n\\begin{itemize}\n \\item the 4x4 covariance matrix of the vertex position,\n \\item the 4x3 cross-covariance matrix between the vertex position and the $i$-th particle momentum,\n \\item and the 3x3 cross-covariance matrix between the $i$-th and the $j$-th particle momentum.\n\\end{itemize}"
[[white_papers]]
repository = "https://github.com/beomki-yeo/line_surface"
slug = "line-surface-jacobian"
pdf_url = "https://github.com/beomki-yeo/line_surface/releases/download/v7.0/main.pdf"

[white_papers.metadata]
authors = [ "Beomki Yeo",]
title = "Line Surface Jacobian"
description = "Line surface used for wire measurements or perigee surface is an unique type of surface where the intersections of a track is not defined on a physical plane. Due to this trait, the underlying mathematics gets quite complicated especially with coordinate transform jacobian whose reference was missing in the source code. (Please let the authors know if anyone finds it). In this white paper, thus we attempt to derive the jacobians for both local-to-global and global-to-local transforms."
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