| \chapter{An exact closed-loop rotation writer} |
| \section{A short path built from a rotating commutator} |
| Work first in an embedded copy of $\SO(3)$ in $\SO(d)$. Choose elementary |
| rotations $X,Z$ sharing one coordinate direction and satisfying |
| $\norm X_Q=\norm Z_Q=1$, $Q(X,Z)=0$. Conjugation by $\exp(rX)$ rotates |
| $Z$ in a two-dimensional plane of the Lie algebra. In particular, |
| \begin{equation}\label{eq:adjointcircle} |
| Q\bigl(Z,\Ad_{\exp(rX)}Z\bigr)=\cos r. |
| \end{equation} |
| For $0<r<\pi$, define the based loop |
| \begin{equation}\label{eq:circleloop} |
| \gamma_r(t)=a\exp(tZ)a^{-1}\exp(-tZ),\qquad |
| a=\exp(rX),\quad 0\le t\le2\pi. |
| \end{equation} |
| The path begins and ends at the identity because $\exp(2\pi Z)=\Id$. |
| Although its coordinate formula contains factors making full rotations, the |
| actual curve $\gamma_r(t)$ is short when $r$ is small. The geometric budget |
| is the length of this curve, not the sum of lengths of its displayed factors |
| viewed as separate sequential operations. |
|
|
| Direct differentiation gives |
| \begin{equation}\label{eq:circlebody} |
| \gamma_r^{-1}\dot\gamma_r |
| =\Ad_{\exp(tZ)}C,\qquad C=\Ad_aZ-Z. |
| \end{equation} |
| The $Q$-norm of the right side is independent of $t$. By |
| \eqref{eq:adjointcircle}, |
| \begin{equation}\label{eq:circlelength} |
| \norm C_Q^2=2-2\cos r=4\sin^2(r/2),\qquad |
| \ell(\gamma_r)=4\pi\sin(r/2). |
| \end{equation} |
| This is an exact length formula in the fixed soul metric. |
|
|
| \section{Solving the transport exactly} |
| The transport equation along the curve is |
| \[ |
| \dot U=-\lambda\exp(tZ)C\exp(-tZ)U. |
| \] |
| Set $U(t)=\exp(tZ)V(t)$. The transformed equation has constant coefficient: |
| \[ |
| \dot V=-(Z+\lambda C)V. |
| \] |
| At $t=2\pi$, the first factor returns to the identity. Therefore |
| \begin{equation}\label{eq:circleH} |
| H_\lambda(\gamma_r)=\exp(-2\pi W),\qquad |
| W=(1-\lambda)Z+\lambda\Ad_aZ. |
| \end{equation} |
| Within the active three-dimensional subspace, $W$ is a simple skew matrix. |
| Its $Q$-norm is |
| \begin{equation}\label{eq:Wnorm} |
| s=\norm W_Q=\sqrt{1-4\coup\sin^2(r/2)}. |
| \end{equation} |
| For $0<r<\pi$ and $0<\lambda<1$, this is positive. Write $J=W/s$; |
| then $J$ is a unit elementary-type planar generator and $\exp(2\pi J)=\Id$. |
| Consequently |
| \begin{equation}\label{eq:positiveangle} |
| H_\lambda(\gamma_r)=\exp(\beta J),\qquad |
| \beta=2\pi(1-s). |
| \end{equation} |
|
|
| \begin{theorem}[Exact circle writer]\label{thm:writer} |
| Let $d\ge3$, $0<\lambda<1$, and $0<L<4\pi$. There is an absolutely |
| continuous, in fact smooth, based loop of length exactly $L$ whose normal |
| holonomy is a planar rotation through |
| \begin{equation}\label{eq:writer} |
| \beta_\lambda(L)=2\pi\left(1- |
| \sqrt{1-\frac{\coup L^2}{4\pi^2}}\right). |
| \end{equation} |
| The oriented output plane can be chosen arbitrarily. On the principal |
| one-circle branch, the inverse length formula is |
| \begin{equation}\label{eq:writerinverse} |
| L=\sqrt{\frac{\beta(4\pi-\beta)}{\coup}}, |
| \qquad 0<\beta<4\pi\min(\lambda,1-\lambda). |
| \end{equation} |
| \end{theorem} |
| \begin{proof} |
| Choose $r=2\arcsin(L/(4\pi))$ in \eqref{eq:circleloop}. Equations |
| \eqref{eq:circlelength}--\eqref{eq:positiveangle} give the angle. Every unit |
| simple skew matrix in $\so(d)$ is conjugate under $\SO(d)$ to any prescribed |
| unit planar generator. Conjugating the entire base loop by the same group |
| element preserves its length and conjugates its holonomy, proving the plane |
| assertion. Squaring the relation between $\beta$ and $s$ gives |
| $\coup L^2=\beta(4\pi-\beta)$. The stated branch is exactly the image of |
| $0<L<4\pi$ before its boundary. |
| \end{proof} |
|
|
| The inverse formula is a constructive cost, not a universal isoholonomic |
| minimum at arbitrary angles. In particular, when $\lambda$ is very small, |
| a fixed non-small rotation may not lie on the one-circle branch. Repeating |
| circles is possible, but that repetition must be charged its full length. |
| It is not legitimate to extrapolate a small-loop square-root cost to an |
| arbitrary fixed target while taking $\lambda$ to zero. |
|
|
| \section{The sharp deficit follows} |
| Conjugate the output plane to contain the marked ray $u$. Then |
| \begin{equation}\label{eq:writerdeficit} |
| d_\lambda(\gamma)=1-\cos\beta_\lambda(L) |
| =2\sin^2\bigl(\beta_\lambda(L)/2\bigr). |
| \end{equation} |
| The stable algebraic form |
| \begin{equation}\label{eq:rationalizedbeta} |
| \beta_\lambda(L)= |
| \frac{\coup L^2}{2\pi\left(1+\sqrt{1-\coup L^2/(4\pi^2)}\right)} |
| \end{equation} |
| avoids cancellation in numerical evaluation. It also shows directly that |
| \begin{equation}\label{eq:betaexpansion} |
| \beta_\lambda(L)=\frac{\coup L^2}{4\pi} |
| +O(\coup^2L^4) |
| \end{equation} |
| uniformly for $0<\lambda<1$ and $L$ in a fixed sufficiently small interval. |
| Squaring the sine expansion gives |
| \[ |
| d_\lambda(\gamma)=\frac{\coup^2L^4}{32\pi^2} |
| \bigl(1+O(L^2)\bigr). |
| \] |
| Together with \eqref{eq:Dupper}, this proves |
| \begin{theorem}[Sharp uniform scalar law]\label{thm:scalarsharp} |
| For fixed $d\ge3$, |
| \begin{equation}\label{eq:scalarsharp} |
| \sup_{0<\lambda<1} |
| \left|\frac{32\pi^2D_\lambda(L)}{\coup^2L^4}-1\right| |
| \le C_dL |
| \end{equation} |
| for $0<L<L_0(d)$. In particular, the leading constant $1/(32\pi^2)$ is |
| optimal, including in joint limits where $\lambda$ approaches a pure-gauge |
| endpoint as $L$ shrinks. |
| \end{theorem} |
|
|
| \section{Exact regular-simplex preparation} |
| Let $z_1,\ldots,z_d\in u^\perp$ be unit vectors satisfying |
| \begin{equation}\label{eq:simplex} |
| \sum_i z_i=0,\qquad |
| \ip{z_i}{z_j}=-\frac1{d-1}\quad(i\ne j),\qquad |
| \sum_i z_iz_i^{\mathsf T}=\frac d{d-1}P_{u^\perp}. |
| \end{equation} |
| Conjugate $d$ copies of the circle writer so that their outputs are |
| \begin{equation}\label{eq:exactframe} |
| v_i=c\,u+s_\beta z_i,\qquad |
| c=\cos\beta,\quad s_\beta=\sin\beta, |
| \quad\beta=\beta_\lambda(L). |
| \end{equation} |
| Every preparation loop has length $L$. The Gram matrix is |
| \begin{equation}\label{eq:exactGram} |
| G_V=\frac d{d-1}s_\beta^2\Id+ |
| \left(c^2-\frac{s_\beta^2}{d-1}\right)\mathbf1\mathbf1^{\mathsf T}. |
| \end{equation} |
| Its eigenvalues are |
| \begin{equation}\label{eq:exactGramspectrum} |
| dc^2\quad\text{once},\qquad |
| \frac d{d-1}s_\beta^2\quad\text{$d-1$ times}. |
| \end{equation} |
| No asymptotic approximation is needed for these formulas. |
|
|
| When $0<\beta\le\arccos(1/\sqrt d)$, the first eigenvalue is the largest, |
| and the Gram condition number is |
| \begin{equation}\label{eq:conditionexact} |
| \operatorname{cond}_2(G_V)=(d-1)\cot^2\beta. |
| \end{equation} |
| At the cap threshold $\cos^2\beta=1/d$, the frame is orthonormal. Whether |
| that threshold belongs to the single-circle branch depends on $\lambda$; |
| it always does at $\lambda=1/2$. The local regime of this manuscript instead |
| has $\beta$ small, so the transverse directions are necessarily much weaker |
| than the common direction. |
|
|
| \section{Why the construction is useful beyond a numerical example} |
| The circle writer performs four independent tasks. It gives an exact |
| differential-geometric realization of small rotations, it identifies the |
| correct length normalization, it proves attainment of the sharp local |
| constant, and it produces a complete family of known Gram spectra for testing |
| inverse algorithms. These roles are logically separate. A code check of the |
| spectrum is not a proof that the quotient has nonnegative curvature; that |
| curvature statement came from the quotient construction. Conversely, |
| nonnegative curvature alone would not provide the exact angle formula. |
|
|
| The orientation conjugations used in an optimal design may depend on a known |
| marked model and on the connection parameter. The design theorem is an |
| optimization statement over available loops, not a claim that an uncalibrated |
| instrument automatically knows the best loop. Finite scalar-history discovery |
| and noisy calibration are treated separately in Chapters~8 and~9. |
|
|
