Nomenclature
A single reference for every symbol used in the mathematics across this site — 227 distinct symbols
over 3,532 equations on 64 pages. Organised by domain rather than alphabetically, because the same
letter means different things in different fields and a flat A-to-Z table would quietly tell you the wrong
thing. If you arrived here from a link on another page, the section you want is probably the one named
after that page's subject.
Read §2 first if you are looking up a single letter. Several symbols are badly overloaded across
disciplines — $R$ is a rotation matrix in geometry and a measurement-noise covariance in estimation;
$P$ is a projection matrix, a state covariance, and a probability. The meaning is set by context, and
§2 lists every collision explicitly.
1 · Notation conventions
Most of the information in a symbol on this site is carried by its decoration, not its letter. These
conventions are used consistently, so once you know them you can read an unfamiliar equation without
looking anything up.
1.1 Decorations
| Form | Name | Meaning | Example |
|---|
| $x$ | plain | The true (unknowable) value |
$\mathbf x$ — true state |
| $\hat x$ | hat | An estimate of the quantity |
$\hat{\mathbf x}$ — state estimate |
| $\tilde x$ | tilde | Measured / sensed, i.e. truth plus error. Also used for
homogeneous coordinates in multi-view geometry |
$\tilde{\boldsymbol\omega}$ — gyro output; $\tilde{\mathbf x}$ — $(u,v,1)^\top$ |
| $\bar x$ | bar | Mean or time-average over an interval |
$\bar{\mathbf f}^b$ — averaged specific force |
| $\dot x,\ \ddot x$ | dot | First and second time derivative |
$\dot{\mathbf v}$ — acceleration |
| $\delta x$ | delta (lower) | Error state: estimate minus truth. The quantity an
error-state filter actually propagates | $\delta\mathbf r$ — position error |
| $\Delta x$ | Delta (upper) | Increment over a finite interval — an integral, not
a rate | $\Delta\boldsymbol\theta$ — integrated angle |
| $\mathbf x$ | bold lower | Vector | $\mathbf v$ — velocity |
| $\mathbf X$, $A$ | bold/plain upper | Matrix | $K$ — intrinsics |
| $\hat{\mathbf u}$ | hatted bold | Unit vector (a direction) |
$\hat{\mathbf e}_1$ — basis direction |
| $\mathcal X$ | calligraphic | A set, space, or loss functional |
$\mathcal L$ — loss; $\mathcal C_{\text{free}}$ — free space |
| $x^-,\ x^+$ | super minus/plus | Before / after a measurement update |
$P^-$, $P^+$ |
| $[\mathbf a\times]$ | cross matrix | Skew-symmetric matrix with
$[\mathbf a\times]\mathbf b = \mathbf a\times\mathbf b$ | — |
1.2 Frame superscripts and subscripts
A vector is meaningless without the frame it is expressed in. A superscript names that frame; a
rotation matrix $C^a_b$ (equivalently $R^a_b$) takes a vector from frame $b$ to frame $a$,
so adjacent indices cancel: $C^n_b C^b_e = C^n_e$.
| Index | Frame | Definition |
|---|
| $i$ | Inertial (ECI) | Non-rotating, Earth-centred. Newton's laws hold here |
| $e$ | Earth-fixed (ECEF) | Rotates with the Earth at $\omega_{ie}$ |
| $n$ | Navigation / local-level | North-East-Down or Up-East-North at the current
position |
| $b$ | Body / sensor | Fixed to the vehicle (or to the IMU case) |
| $c$ | Camera | $+z$ along the optical axis, $+x$ right, $+y$ down |
| $w$ | World / global | The fixed map frame in SLAM and in nuScenes |
| $\omega_{ie}$ | — | Read as "rate of $e$ with respect to $i$" — the Earth rate |
2 · Overloaded symbols
These are the symbols where the same glyph carries genuinely different meanings in different sections of
the site. This is not sloppiness on any one page — each usage is the standard convention in its own
field — but it is the single most likely source of confusion when reading across domains.
| Symbol | In geometry / vision | In estimation | Elsewhere |
|---|
| $R$ | Rotation matrix | Measurement noise covariance |
Earth radius (Schuler); reward (RL) |
| $P$ | Projection matrix $K[R\,|\,\mathbf t]$ |
State covariance | Probability; precision |
| $Q$ | Orthogonal factor in QR | Process noise covariance |
Query matrix in attention |
| $K$ | Camera intrinsics | Kalman gain |
Key matrix in attention; number of clusters |
| $H$ | Homography | Measurement Jacobian |
Hessian; entropy |
| $F$ | Fundamental matrix | Dynamics Jacobian $\;\partial\dot{\mathbf x}/\partial\mathbf x$ |
Force |
| $E$ | Essential matrix | Expectation $\mathbb E[\cdot]$ |
Energy; error functional |
| $\mathbf t$ | Translation vector | — | $t$ = time |
| $\sigma$ | Singular value | Standard deviation |
Sigmoid; SH coefficient scale |
| $\lambda$ | Eigenvalue; depth scale | — |
Weight decay; damping (LM); wavelength |
| $\Sigma$ | Gaussian covariance (splatting) | Covariance |
Summation $\sum$ when used as an operator |
| $\alpha$ | Opacity (splatting) | — |
Learning rate; accelerometer bias drift; significance level |
| $\mathbf x$ | Image point | State vector | Network input |
| $d$ | Disparity (stereo) | — | Dimension; distance; depth |
| $N$ | Number of points | — | Gaussian distribution $\mathcal N$; batch count |
| $C$ | Direction cosine matrix $C^a_b$ | — |
Matching cost (stereo); channel count; colour |
| $T$ | Homogeneous transform $T^a_b$ | — |
Temperature; time span; transmittance; matrix transpose $(\cdot)^\top$ |
| $\mathbf f$ | — | — |
Specific force (inertial); focal length $f$; a function |
3 · Frames, position and motion
| Symbol | Meaning | Units | Where |
|---|
| $\mathbf r$, $\mathbf p$ | Position vector | m |
IMU, SLAM |
| $\mathbf v$ | Velocity | m/s | IMU |
| $\mathbf a$ | Acceleration | m/s² | IMU |
| $\mathbf X$, $\mathbf X_w$ | 3-D world point | m |
Camera, SfM |
| $\mathbf x$, $(u,v)$ | Image point / pixel coordinates | px |
Camera |
| $Z$ | Depth along the optical axis | m |
Stereo |
| $\boldsymbol\omega$ | Angular rate vector | rad/s |
Attitude |
| $\omega_{ie}$ | Earth rotation rate ($7.292\times10^{-5}$) | rad/s |
Ground align |
| $L$, $\varphi$ | Latitude | rad | Ground align |
| $g$ | Gravity magnitude (≈9.81) | m/s² | Schuler |
| $R_E$ | Earth radius (≈6371 km) | m | Schuler |
| $\kappa$ | Path curvature (signed, + is left) | 1/m |
OpenEMMA, RRT |
4 · Attitude and rotations
| Symbol | Meaning | Notes |
|---|
| $C^a_b$, $R^a_b$ | Direction cosine / rotation matrix, $b\!\to\!a$ |
Orthonormal, $\det = +1$; $(C^a_b)^{-1} = (C^a_b)^\top = C^b_a$ |
| $\mathbf q$ | Attitude quaternion |
4 components, norm 1. nuScenes stores $(w,x,y,z)$; many libraries use $(x,y,z,w)$ |
| $\boldsymbol\phi$, $\boldsymbol\theta$ | Rotation vector / axis-angle |
Direction is the axis, magnitude the angle |
| $\Delta\boldsymbol\theta$ | Integrated angle increment over $\Delta t$ |
What a strapdown gyro actually outputs — an integral, not a rate |
| $\delta\boldsymbol\psi$ | Attitude error (tilt + heading) |
$\delta\psi_N,\delta\psi_E$ are tilt; $\delta\psi_D$ is heading |
| $\psi,\theta,\phi$ | Yaw, pitch, roll | Euler angles; order matters |
| $\exp$, $\log$ | $SO(3)$ exponential / logarithm |
Maps rotation vector $\leftrightarrow$ matrix |
5 · Inertial navigation
| Symbol | Meaning | Units |
|---|
| $\mathbf f^b$ | Specific force (what an accelerometer senses) | m/s² |
| $\Delta\mathbf v$ | Integrated velocity increment | m/s |
| $\mathbf b_a$, $\mathbf b_g$ | Accelerometer / gyro bias | m/s², rad/s |
| $S_a$, $S_g$ | Scale-factor error matrix (diagonal) | ppm |
| $M_a$, $M_g$ | Misalignment / non-orthogonality (off-diagonal) | rad |
| $\omega_s$ | Schuler frequency $\sqrt{g/R_E}$ | rad/s |
| $T_s$ | Schuler period (≈84.4 min) | s |
| $\sigma_{\text{ARW}}$ | Angle random walk (gyro white noise) | °/√h |
| $\sigma_{\text{VRW}}$ | Velocity random walk (accel white noise) | m/s/√h |
| $\sigma_{\text{BI}}$ | Bias instability (Allan-variance floor) | °/h |
| $\tau$ | Allan averaging time; also Gauss–Markov correlation time | s |
| $\beta$ | Gauss–Markov inverse correlation time $1/\tau$ | 1/s |
| RPE | Radial position error $\|\delta\mathbf r\|$ | m |
6 · Estimation and filtering
| Symbol | Meaning | Dimension |
|---|
| $\mathbf x$, $\hat{\mathbf x}$ | State vector; state estimate | $n$ |
| $\delta\mathbf x$ | Error state (what the filter propagates) | $n$ |
| $P$ | State covariance | $n\times n$ |
| $Q$ | Process-noise covariance | $n\times n$ |
| $R$ | Measurement-noise covariance | $m\times m$ |
| $K$ | Kalman gain | $n\times m$ |
| $H$ | Measurement Jacobian $\partial\mathbf h/\partial\mathbf x$ | $m\times n$ |
| $F$ | Dynamics Jacobian (continuous) | $n\times n$ |
| $\Phi$ | State transition matrix (discrete), $\approx e^{F\Delta t}$ | $n\times n$ |
| $\mathbf z$ | Measurement vector | $m$ |
| $\mathbf y$ | Innovation / residual $\mathbf z - \mathbf h(\hat{\mathbf x})$ | $m$ |
| $S$ | Innovation covariance $HPH^\top + R$ | $m\times m$ |
| $\mathbf w$, $\boldsymbol\nu$ | Process noise; measurement noise | — |
| $\mathcal N(\boldsymbol\mu,\Sigma)$ | Gaussian with that mean and covariance | — |
| $\mathbb E[\cdot]$ | Expectation | — |
| $U$, $D$ | Factors in UD (square-root) covariance form | $n\times n$ |
| $w^{(i)}$ | Particle weight (particle filter / FastSLAM) | — |
7 · Camera, multi-view geometry and stereo
| Symbol | Meaning | Notes |
|---|
| $K$ | Intrinsic matrix | $[[f_x,s,c_x],[0,f_y,c_y],[0,0,1]]$ |
| $f_x$, $f_y$ | Focal length in pixels | Equal for square pixels |
| $c_x$, $c_y$ | Principal point | px |
| $[R\,|\,\mathbf t]$ | Extrinsics: world $\to$ camera | $3\times4$ |
| $P$ | Full projection matrix $K[R\,|\,\mathbf t]$ | $3\times4$ |
| $\tilde{\mathbf x}$ | Homogeneous image point $(u,v,1)^\top$ |
Equality is up to scale, written $\sim$ |
| $k_1,k_2,k_3$ | Radial distortion coefficients | — |
| $p_1,p_2$ | Tangential distortion coefficients | — |
| $F$ | Fundamental matrix, $\tilde{\mathbf x}'^\top F\tilde{\mathbf x} = 0$ |
Rank 2, 7 DoF |
| $E$ | Essential matrix $= K'^\top F K$ | 5 DoF |
| $H$ | Homography (planar scene or pure rotation) | $3\times3$, 8 DoF |
| $B$ | Stereo baseline | m |
| $d$ | Disparity $u_L - u_R$ | px; $Z = fB/d$ |
| $\mathbf e$, $\mathbf e'$ | Epipoles | — |
| $\rho$ | Inverse depth $1/Z$ | 1/m |
8 · Optimisation and robust fitting
| Symbol | Meaning |
|---|
| $J$ | Jacobian $\partial\mathbf r/\partial\boldsymbol\theta$ |
| $\mathbf r$ | Residual vector |
| $\mathcal L$, $E$ | Loss / cost functional |
| $\boldsymbol\theta$ | Parameter vector being optimised |
| $\mathbf g$ | Gradient $\nabla_{\boldsymbol\theta}\mathcal L$ |
| $\mathbf H$ | Hessian $\nabla^2\mathcal L$; or $J^\top J$ (Gauss–Newton approximation) |
| $\lambda$ | Levenberg–Marquardt damping; or regularisation weight |
| $\varepsilon$ | Inlier threshold (RANSAC); convergence tolerance |
| $w$ | Inlier fraction (RANSAC); or a weight |
| $s$ | Minimal sample size (RANSAC: 2 for a line, 3 for PnP) |
| $N_{\text{iter}}$ | $\log(1-p)/\log(1-w^s)$ — iterations for success probability $p$ |
| $\Lambda$ | Information matrix $= \Sigma^{-1}$ (pose graphs) |
| $\boxplus$ | Manifold increment ("boxplus") — retraction onto $SO(3)$ or $SE(3)$ |
9 · Deep learning and training
| Symbol | Meaning | Typical value |
|---|
| $\eta$, $\alpha$ | Learning rate | $10^{-4}$–$10^{-3}$ (AdamW) |
| $\beta_1,\beta_2$ | Adam moment decay rates | 0.9, 0.999 |
| $\mathbf m_t$, $\mathbf v_t$ | Adam first / second moment estimates | — |
| $\epsilon$ | Numerical floor in the Adam denominator | $10^{-8}$ |
| $\gamma$ | Momentum coefficient; or LR decay factor | 0.9 |
| $\lambda$ | Weight decay | 0.01–0.1 |
| $B$ | Batch size | — |
| $T$ | Total training steps; or softmax temperature | — |
| $Q,K,V$ | Query, key, value matrices in attention | — |
| $d_k$ | Key dimension — the $\sqrt{d_k}$ in scaled dot-product attention | 64 |
| $\sigma(\cdot)$ | Sigmoid; or a generic activation | — |
| $\mathcal D$ | Dataset | — |
| $r$ | LoRA rank | 8–64 |
| $\mathbf z$ | Latent vector / embedding | — |
| $p_{\text{drop}}$ | Dropout probability | 0.1 |
10 · Gaussian splatting
| Symbol | Meaning |
|---|
| $\boldsymbol\mu$ | Gaussian centre in world space |
| $\Sigma$ | 3-D covariance, factorised as $R S S^\top R^\top$ |
| $S$ | Diagonal scale matrix (the three axis lengths) |
| $\Sigma'$ | Projected 2-D covariance $J W \Sigma W^\top J^\top$ (EWA splatting) |
| $J$ | Jacobian of the perspective projection |
| $W$ | World-to-camera rotation |
| $\alpha_i$ | Opacity of Gaussian $i$ |
| $T_i$ | Accumulated transmittance $\prod_{j\lt i}(1-\alpha_j)$ |
| $c_i$ | View-dependent colour from spherical harmonics |
| $Y_{\ell m}$ | Spherical harmonic basis of degree $\ell$, order $m$ |
11 · Planning
| Symbol | Meaning |
|---|
| $\mathcal C$ | Configuration space |
| $\mathcal C_{\text{free}}$, $\mathcal C_{\text{obs}}$ | Free / obstacle subsets |
| $q$ | A configuration (a point in $\mathcal C$) |
| $\eta$ | RRT steering step size |
| $r_n$ | RRT* connection radius $\min\{\gamma(\log n/n)^{1/d},\eta\}$ |
| $c(q)$ | Cost-to-come at $q$ |
| $\zeta_d$ | Volume of the unit $d$-ball |
| $\mu(\cdot)$ | Lebesgue measure (volume) of a set |
12 · Metrics
| Symbol | Meaning |
|---|
| ADE / FDE | Average / final displacement error of a predicted trajectory |
| IoU | Intersection over union |
| mAP | Mean average precision |
| NDS | nuScenes detection score (a weighted blend of mAP and error terms) |
| AMOTA | Average multi-object tracking accuracy |
| PSNR / SSIM | Image reconstruction quality (splatting) |
| $\|\cdot\|_2$, $\|\cdot\|_1$ | Euclidean and absolute-sum norms |
| $\|\mathbf a\|_\Sigma$ | Mahalanobis norm $\sqrt{\mathbf a^\top\Sigma^{-1}\mathbf a}$ |
13 · Operators
| Symbol | Meaning |
|---|
| $(\cdot)^\top$ | Transpose |
| $(\cdot)^{-1}$, $(\cdot)^{+}$ | Inverse; Moore–Penrose pseudoinverse |
| $\otimes$ | Quaternion product; or Kronecker product |
| $\times$ | Cross product |
| $\odot$ | Elementwise (Hadamard) product |
| $\nabla$, $\partial$ | Gradient; partial derivative |
| $\operatorname{tr}$, $\det$ | Trace; determinant |
| $\sim$ | "Distributed as"; or equality up to scale (projective geometry) |
| $\propto$ | Proportional to |
| $\triangleq$ | Defined as |
| $I$ | Identity matrix |
| $\mathcal O(\cdot)$ | Asymptotic complexity |
14 · How this page is maintained
The symbol inventory is extracted from the pages, not written from memory, by
tools/extract_symbols.py. It pulls every inline and display math span out of
the HTML, tokenises the LaTeX, and reports each symbol with its use count and the pages it appears on.
That is how the §2 collision list was built, and it means a symbol introduced on a new page can be found
rather than forgotten:
python3 tools/extract_symbols.py # summary + top pages by math density
python3 tools/extract_symbols.py --full # every symbol with its pages
python3 tools/extract_symbols.py --min-pages 5 # only the widely-used ones
Why there are no per-equation links back here. The obvious implementation — wrapping every symbol
in every formula in an anchor — is not possible: the content inside a math delimiter is handed to MathJax
as LaTeX, and HTML tags placed inside it either break the parse or get typeset literally. Instead each
maths-bearing page carries a single link to the section of this page that covers its domain, which is
one click away from any symbol and cannot break the rendering.
