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 @page alignment_composite_jacobians Composite alignment Jacobians
 
 # Composite alignment Jacobians
 
 This page documents the composite (hierarchy) Jacobians, which relate the
 alignment parameters of a component (e.g. a sensor) to those of a rigid
 structure it belongs to (e.g. a stave, a layer or a half-barrel). It covers the
 parameter conventions they rely on, the derivation, the implementation, and what
 is still missing.
 
 [TOC]
 
 ## Motivation
 
 ACTS computes derivatives of the residuals w.r.t. the alignment parameters of
 single surfaces. Aligning every sensor individually does not scale to a real
 tracker: the first steps of a real alignment are done on larger rigid bodies.
 
 Aligning a rigid structure requires the chain rule from the sensor alignment
 parameters, for which ACTS computes derivatives, to the structure parameters:
 
 @f[
 \frac{\partial r}{\partial \mathbf{A}_\mathrm{structure}}
   = \frac{\partial r}{\partial \mathbf{a}_\mathrm{sensor}}
     \cdot \frac{\partial \mathbf{a}_\mathrm{sensor}}{\partial \mathbf{A}_\mathrm{structure}}
 @f]
 
 The second factor is the *composite* or *hierarchy Jacobian*. The derivatives
 of a structure are the sum of this product over all its sensors on the track.
 
 ## ACTS alignment parameter convention
 
 In short, **current ACTS surface alignment parameters are mixed**: global-frame
 translations of the center, and local-frame rotations about the center.
 
 ACTS is expected to move to fully local-frame parameters. The implementation is
 structured so that only one function changes when that happens (see
 @ref alignment_composite_convention_adapter).
 
 ## Derivation
 
 ### Local-frame parameters
 
 For an object with local-to-global transform @f$(R, \mathbf{c})@f$, the
 *local-frame* alignment parameters are @f$(\delta\mathbf{t}, \delta\boldsymbol{\omega})@f$:
 a translation along the object's own local axes, and small rotation angles about
 its own local axes, pivoting on its local origin. The moved object is
 
 @f[
 R' = R\,(1 + [\delta\boldsymbol{\omega}]_\times), \qquad
 \mathbf{c}' = \mathbf{c} + R\,\delta\mathbf{t},
 @f]
 
 i.e. @f$T' = T \cdot \Delta@f$ with @f$\Delta = (1 + [\delta\boldsymbol{\omega}]_\times,\ \delta\mathbf{t})@f$.
 This is the convention used by CMS and Belle II for hierarchical alignment.
 
 ### Composite to component
 
 Let the composite (structure) have transform @f$(R_c, \mathbf{c}_c)@f$ and
 parameters @f$(\delta\mathbf{T}, \delta\mathbf{W})@f$, and the component
 (sensor) have @f$(R_s, \mathbf{c}_s)@f$. A rigid motion of the composite moves
 the global point @f$\mathbf{x}@f$ by
 @f$R_c\,\delta\mathbf{T} + (R_c\,\delta\mathbf{W}) \times (\mathbf{x} - \mathbf{c}_c)@f$
 and rotates by the global rotation vector @f$R_c\,\delta\mathbf{W}@f$.
 With the lever arm @f$\mathbf{d} = \mathbf{c}_s - \mathbf{c}_c@f$ (global
 coordinates), the component therefore moves by
 
 @f[
 \delta\mathbf{c}_s = R_c\,\delta\mathbf{T} - [\mathbf{d}]_\times R_c\,\delta\mathbf{W},
 @f]
 @f[
 R_s' = (1 + [R_c\,\delta\mathbf{W}]_\times)\,R_s
      = R_s\,(1 + [R_s^T R_c\,\delta\mathbf{W}]_\times).
 @f]
 
 Expressing @f$\delta\mathbf{c}_s@f$ along the component axes
 (@f$\delta\mathbf{t}_s = R_s^T\,\delta\mathbf{c}_s@f$) and reading off
 @f$\delta\boldsymbol{\omega}_s = R_s^T R_c\,\delta\mathbf{W}@f$ gives
 
 @f[
 \frac{\partial(\delta\mathbf{t}, \delta\boldsymbol{\omega})_s}
      {\partial(\delta\mathbf{T}, \delta\mathbf{W})_c}
 = \begin{pmatrix}
     R_\mathrm{rel}^T & -R_s^T\,[\mathbf{d}]_\times\,R_c \\
     0                & R_\mathrm{rel}^T
   \end{pmatrix},
 \qquad R_\mathrm{rel} = R_c^T R_s .
 @f]
 
 ### Relation to the CMS form
 
 The CMS form of this matrix is
 
 @f[
 \begin{pmatrix} R^{-1} & (\mathbf{T} \times \mathbf{a}_s)\cdot\mathbf{b} \\ 0 & R^{-1} \end{pmatrix},
 @f]
 
 with @f$R@f$ the component rotation relative to the composite, @f$\mathbf{T}@f$
 the component position relative to the composite, @f$\mathbf{a}_s@f$ the
 component axes and @f$\mathbf{b}@f$ the composite axes. The two agree:
 
 - the diagonal blocks are both @f$R_\mathrm{rel}^T = R_s^T R_c@f$;
 - element @f$(i, j)@f$ of the top-right block is
   @f$-\hat{a}_i \cdot (\mathbf{d} \times \hat{b}_j) = \hat{b}_j \cdot (\mathbf{d} \times \hat{a}_i)
   = (\mathbf{d} \times \hat{a}_i)\cdot\hat{b}_j@f$, the CMS expression with
   @f$\mathbf{T} = \mathbf{d}@f$, including the sign.
 
 This form is only correct when **both** levels use local-frame parameters. It
 does not apply directly to current ACTS surface parameters, whose translations
 are global. Porting CMS or Belle II code without accounting for this gives a
 wrong top-left block. On a pure translation test with @f$R_c \approx 1@f$ that
 mistake can look correct.
 
 @anchor alignment_composite_convention_adapter
 ### Convention adapter
 
 The ACTS parameters of a surface relate to its local-frame parameters by
 @f$\delta\mathbf{c} = R_s\,\delta\mathbf{t}@f$, with the rotations unchanged:
 
 @f[
 \frac{\partial\,\mathbf{a}_\mathrm{ACTS}}{\partial(\delta\mathbf{t}, \delta\boldsymbol{\omega})}
 = \begin{pmatrix} R_s & 0 \\ 0 & 1 \end{pmatrix}.
 @f]
 
 The Jacobian used on the derivatives ACTS provides is therefore
 
 @f[
 \frac{\partial\,\mathbf{a}_\mathrm{ACTS}}{\partial(\delta\mathbf{T}, \delta\mathbf{W})}
 = \begin{pmatrix} R_s & 0 \\ 0 & 1 \end{pmatrix}
   \begin{pmatrix} R_\mathrm{rel}^T & -R_s^T [\mathbf{d}]_\times R_c \\ 0 & R_\mathrm{rel}^T \end{pmatrix}
 = \begin{pmatrix} R_c & -[\mathbf{d}]_\times R_c \\ 0 & R_s^T R_c \end{pmatrix}.
 @f]
 
 The adapter is only needed when chaining to structure parameters:
 
 @f[
 \frac{\partial r}{\partial(\delta\mathbf{T}, \delta\mathbf{W})}
 = \frac{\partial r}{\partial \mathbf{a}_\mathrm{ACTS}}
   \begin{pmatrix} R_s & 0 \\ 0 & 1 \end{pmatrix}
   \frac{\partial(\delta\mathbf{t}, \delta\boldsymbol{\omega})_s}
        {\partial(\delta\mathbf{T}, \delta\mathbf{W})_c}.
 @f]
 
 An alignment of single surfaces uses the derivatives
 @f$\partial r/\partial \mathbf{a}_\mathrm{ACTS}@f$ as they are: the solver then
 returns ACTS parameters, which is the convention corrections are applied in, and
 no adapter enters. Applying the adapter to the surface derivatives alone (the
 surface as its own structure) would instead give parameters in the surface's
 local frame.
 
 When ACTS moves to local-frame parameters, the adapter becomes the identity and
 the composite Jacobian is used unchanged.
 
 ## Implementation
 
 `Acts/Surfaces/detail/AlignmentHelper.hpp` provides two free functions next to
 `rotationToLocalAxesDerivative`:
 
 ```cpp
 namespace Acts::detail {
 
 /// d(dt, dw)_component / d(dT, dW)_composite, both in local-frame parameters
 AlignmentMatrix compositeToComponentJacobian(
     const Transform3& compositeTransform, const Transform3& componentTransform);
 
 /// d(ACTS alignment parameters) / d(local-frame parameters) = diag(R, 1)
 AlignmentMatrix localFrameToAlignmentParametersJacobian(
     const Transform3& surfaceTransform);
 
 }  // namespace Acts::detail
 ```
 
 Both take local-to-global transforms. `localFrameToAlignmentParametersJacobian`
 is the only place that encodes the ACTS convention.
 
 ## Limitations and open items
 
 - **Structure frames.** The Jacobian needs the frame of the composite: its
   origin (the pivot of the rotations, entering through the lever arm) and its
   orientation @f$R_c@f$. The choice is up to the user; a common one is the
   global orientation centred on the structure, so that the translations are
   global shifts.
 - **Several levels.** The Jacobians chain: for a sensor in a stave in a layer,
   the Jacobian w.r.t. the layer is the product of the sensor-to-stave and
   stave-to-layer Jacobians, both in local-frame parameters. No helper for
   multi-level chains is provided yet.
 - **Degrees of freedom at several levels.** When both a structure and its
   components float, the structure motion is degenerate with the common motion
   of its components. The usual way out (CMS, Belle II) is 6 linear constraints
   per structure, @f$\sum_{s \in \mathrm{structure}} J_s^T\, \mathbf{a}_s = 0@f$,
   so that the component corrections carry no net rigid-body motion of their
   structure. Since the component and structure parameters are in different
   frames, a component degree of freedom maps to a linear combination of
   structure ones; the degeneracy is not visible in the alignment masks alone.
 - **Linearisation point.** The Jacobians depend on the transforms they are
   evaluated with. Evaluating them on the nominal geometry is fine for small
   misalignments; an iterative alignment with large corrections would have to
   re-evaluate them on the updated alignment.

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