geospatial-analytics-ai999x — offline, verified geospatial analytics solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Geospatial · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

geospatial-analytics-ai999x is an offline, deterministic Python solver that answers geospatial analytics questions with verified results from 99 certified capabilities, gives 615 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

geospatial-analytics-ai999x is a compact, deterministic geospatial analytics engine for Python. It reads a question in plain English (or a typed call), picks a certified geospatial analytics capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified geospatial analytics capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 615 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: ellipsoidal geodesy, coordinates and map projections, spatial statistics and geostatistics, remote sensing, terrain analysis, and urban and network analysis.

Model overview

Property	Value
Package	geospatial-analytics-ai999x 1.0.0 (import geospatial_analytics_ai999x)
Type	Deterministic neuro-symbolic geospatial analytics engine (rule-based reading + certified symbolic capabilities)
Capabilities	99 certified procedures
Knowledge pack	version 1.0.0
Golden answers	615 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install geospatial-analytics-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/geospatial-analytics-ai999x/resolve/main/dist/geospatial_analytics_ai999x-1.0.0-py3-none-any.whl

import geospatial_analytics_ai999x as geospatial_analytics

geospatial_analytics.ask("Geodesic destination: lat1 = 52.2, lon1 = 0.12, azimuth = 45, distance = 100 km, ellipsoid = WGS84").show()
# -> SOLVED — lat2 = 52.8308, lon2 = 1.16915, azimuth2 = 45.8325  (geo_geodesic_destination)
geospatial_analytics.ask("UTM forward: lat = 51.5, lon = -0.13, ellipsoid = WGS84").show()
# -> SOLVED — easting = 699196, northing = 5.70934e+06, k = 1.00009, gamma = 2.24682  (geo_utm_forward)
geospatial_analytics.ask("Geodesic polygon area: lats = [48.0, 49.5, 47.2], lons = [2.0, 4.5, 6.1], ellipsoid = GRS80").show()
# -> SOLVED — area = 3.36697e+10, perimeter = 850958  (geo_geodesic_polygon_area)
geospatial_analytics.ask("Vegetation indices: red = 0.08, nir = 0.45, blue = 0.04, L = 0.5").show()
# -> SOLVED — NDVI = 0.698113, EVI = 0.567485, SAVI = 0.538835, OSAVI = 0.536232  (geo_vegetation_indices)
geospatial_analytics.solve("geo_morans_i", y=[8, 6, 6, 3, 2, 1], W=[[0, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 1, 0]], transform="row").show()
# -> SOLVED — I = 0.65625, EI = -0.2, VI_norm = 0.140952, z_norm = 2.28068  (geo_morans_i)
geospatial_analytics.ask("Who created you?").message
# -> geospatial-analytics-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: geospatial-analytics-ai999x "<question>", geospatial-analytics-ai999x --capabilities. The outputs above were produced by running this exact code on geospatial-analytics-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

99 certified procedures. A few of them:

• Procedures: Albers equal-area conic projection: coordinates and scale factors; Arc length along a parallel of latitude; Area and perimeter of a geodesic polygon on the ellipsoid; Area of a latitude-longitude cell on the ellipsoid; At-sensor brightness temperature from thermal radiance; Auxiliary latitudes: geocentric, reduced, conformal, authalic and rectifying; Azimuthal equidistant projection (ellipsoidal, geodesic); Bivariate Moran's I (spatial cross-correlation); Closest facility for every node of a network; D8 flow accumulation of a small DEM; D8 flow direction of every cell of a small DEM; Decimal degrees to degrees, minutes and seconds; Decode a geohash: cell centre and size; Degrees, minutes and seconds to decimal degrees; Destination along a rhumb line (loxodrome) on the ellipsoid; Discrete Frechet distance between two polylines; Distance-band spatial weights; Douglas-Peucker line simplification; Empirical (Matheron) semivariogram in distance bins; Equirectangular (equidistant cylindrical / plate carree) projection; Fit an exponential or Gaussian variogram model (nugget, partial sill, range); Geary's C spatial autocorrelation with normal and randomization inference; Geocentric (ECEF) to geodetic coordinates; Geodesic destination point on the ellipsoid (direct problem) …

geospatial_analytics_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 99 formulas

Show every formula, by topic

Cartography

Ground distance from a map distance and a representative fraction

$$D=d\times SD\; =\; d\; \backslash times\; S$$

Map scale (representative fraction) from a map and a ground distance

$$S=\frac{D}{d},RF=\frac{1}{S}S\; =\; \backslash frac\{D\}\{d\},\backslash quad\; RF\; =\; \backslash frac\{1\}\{S\}$$

Coordinate notation

Decimal degrees to degrees, minutes and seconds

$$d=\lfloor \mathrm{\mid }x\mathrm{\mid }\rfloor ,m=\lfloor 60(\mathrm{\mid }x\mathrm{\mid }-d)\rfloor ,s=3600(\mathrm{\mid }x\mathrm{\mid }-d-\frac{m}{60})d\; =\; \backslash lfloor\backslash vert\; x\backslash vert\; \backslash rfloor,\backslash quad\; m\; =\; \backslash lfloor60(\backslash vert\; x\backslash vert\; -\; d)\backslash rfloor,\backslash quad\; s\; =\; 3600\backslash left(\backslash vert\; x\backslash vert\; -\; d\; -\; \backslash tfrac\{m\}\{60\}\backslash right)$$

Decode a geohash: cell centre and size

$$[{\phi }_{min},{\phi }_{max}]\times [{\lambda }_{min},{\lambda }_{max}]\text{ by alternate bisection}[\backslash varphi\_\{min\},\; \backslash varphi\_\{max\}]\; \backslash times\; [\backslash lambda\_\{min\},\; \backslash lambda\_\{max\}]\; \backslash \; \backslash text\{by\; alternate\; bisection\}$$

Degrees, minutes and seconds to decimal degrees

$$\phi =\pm (d+\frac{m}{60}+\frac{s}{3600})\backslash varphi\; =\; \backslash pm\backslash left(d\; +\; \backslash frac\{m\}\{60\}\; +\; \backslash frac\{s\}\{3600\}\backslash right)$$

Geohash of a point and its cell

$$\text{bits}={\lambda }_1{\phi }_1{\lambda }_2{\phi }_2\dots ,\text{character}=\text{base32}(5\text{ bits})\backslash text\{bits\}\; =\; \backslash lambda\_1\backslash varphi\_1\backslash lambda\_2\backslash varphi\_2\backslash ldots,\backslash quad\; \backslash text\{character\}\; =\; \backslash text\{base32\}(5\backslash \; \backslash text\{bits\})$$

MGRS grid reference of a point (1 m)

$$\text{MGRS}=zB{C}_{100k}{R}_{100k}\lfloor E\rfloor {10}^5\lfloor N\rfloor {10}^5\backslash text\{MGRS\}\; =\; z\backslash ,B\backslash ,C\_\{100k\}R\_\{100k\}\backslash ,\backslash lfloor\; E\backslash rfloor\; \backslash bmod\; 10^5\backslash ,\backslash lfloor\; N\backslash rfloor\; \backslash bmod\; 10^5$$

Coordinate systems and conversions

Auxiliary latitudes: geocentric, reduced, conformal, authalic and rectifying

$$\tan \psi =(1-e^2)\tan \phi ,\tan \beta =(1-f)\tan \phi ,\chi =\mathrm{gd}{\psi }_{iso},\xi =\arcsin \frac{q}{q_p},\mu =\frac{\pi }{2}\frac{m(\phi )}{m(\pi \mathrm{/}2)}\backslash tan\backslash psi\; =\; (1-e^2)\backslash tan\backslash varphi,\backslash \; \backslash tan\backslash beta\; =\; (1-f)\backslash tan\backslash varphi,\backslash \; \backslash chi\; =\; \backslash operatorname\{gd\}\backslash psi\_\{iso\},\backslash \; \backslash xi\; =\; \backslash arcsin\backslash frac\{q\}\{q\_p\},\backslash \; \backslash mu\; =\; \backslash frac\{\backslash pi\}\{2\}\backslash frac\{m(\backslash varphi)\}\{m(\backslash pi/2)\}$$

Geocentric (ECEF) to geodetic coordinates

$$\phi =\mathrm{atan2}\text{\hspace{0.17em}⁣}(Z+e^{2}N\sin \phi ,\sqrt{X^2+Y^2}),h=p\cos \phi +Z\sin \phi -\frac{a^2}{N}\backslash varphi\; =\; \backslash operatorname\{atan2\}\backslash !\backslash left(Z\; +\; e^2N\backslash sin\backslash varphi,\backslash \; \backslash sqrt\{X^2+Y^2\}\backslash right),\backslash quad\; h\; =\; p\backslash cos\backslash varphi\; +\; Z\backslash sin\backslash varphi\; -\; \backslash frac\{a^2\}\{N\}$$

Geodetic to geocentric (ECEF) coordinates

$$X=(N+h)\cos \phi \cos \lambda ,Y=(N+h)\cos \phi \sin \lambda ,Z=(N(1-e^2)+h)\sin \phi X\; =\; (N+h)\backslash cos\backslash varphi\backslash cos\backslash lambda,\backslash \; Y\; =\; (N+h)\backslash cos\backslash varphi\backslash sin\backslash lambda,\backslash \; Z\; =\; \backslash left(N(1-e^2)+h\backslash right)\backslash sin\backslash varphi$$

Geodetic to local east-north-up (ENU) coordinates

Local east-north-up (ENU) to geodetic coordinates

$$\mathbf{X}={\mathbf{X}}_0+R^T({\phi }_0,{\lambda }_0)[enu{]}^T\backslash mathbf\{X\}\; =\; \backslash mathbf\{X\}\_0\; +\; R^\{T\}(\backslash varphi\_0,\backslash lambda\_0)\backslash ,[e\backslash \; n\backslash \; u]^\{T\}$$

Look angles: azimuth, elevation and range from an observer to a target

$$A=\mathrm{atan2}(e,n),\epsilon =\mathrm{atan2}\text{\hspace{0.17em}⁣}(u,\sqrt{e^2+n^2})A\; =\; \backslash operatorname\{atan2\}(e,\; n),\backslash quad\; \backslash varepsilon\; =\; \backslash operatorname\{atan2\}\backslash !\backslash left(u,\; \backslash sqrt\{e^2+n^2\}\backslash right)$$

UTM zone, latitude band and central meridian

$$z=\lfloor \frac{\lambda +{180}^{\circ }}{6^{\circ }}\rfloor +1,{\lambda }_0=6z-{183}^{\circ }z\; =\; \backslash left\backslash lfloor\backslash frac\{\backslash lambda\; +\; 180^\backslash circ\}\{6^\backslash circ\}\backslash right\backslash rfloor\; +\; 1,\backslash quad\; \backslash lambda\_0\; =\; 6z\; -\; 183^\backslash circ$$

Ellipsoidal geodesy

Arc length along a parallel of latitude

$$s=N(\phi )\cos \phi \mathrm{\mid }\mathrm{\Delta }\lambda \mathrm{\mid },N=\frac{a}{\sqrt{1-e^2{\sin }^2\phi }}s\; =\; N(\backslash varphi)\backslash cos\backslash varphi\backslash ,\backslash vert\; \backslash Delta\backslash lambda\backslash vert\; ,\backslash quad\; N\; =\; \backslash frac\{a\}\{\backslash sqrt\{1-e^2\backslash sin^2\backslash varphi\}\}$$

Area and perimeter of a geodesic polygon on the ellipsoid

$$S=-\oint \frac{a^2}{2}[q(\phi )-q({\phi }_c)]d\lambda ,q(\phi )=(1-e^2)[\frac{\sin \phi }{1-e^2{\sin }^2\phi }+\frac{\mathrm{artanh}(e\sin \phi )}{e}]S\; =\; -\backslash oint\; \backslash frac\{a^2\}\{2\}\backslash left[q(\backslash varphi)\; -\; q(\backslash varphi\_c)\backslash right]d\backslash lambda,\backslash quad\; q(\backslash varphi)\; =\; (1-e^2)\backslash left[\backslash frac\{\backslash sin\backslash varphi\}\{1-e^2\backslash sin^2\backslash varphi\}\; +\; \backslash frac\{\backslash operatorname\{artanh\}(e\backslash sin\backslash varphi)\}\{e\}\backslash right]$$

Area of a latitude-longitude cell on the ellipsoid

$$A=\frac{a^2}{2}[q({\phi }_2)-q({\phi }_1)]({\lambda }_2-{\lambda }_1)A\; =\; \backslash frac\{a^2\}\{2\}\backslash left[q(\backslash varphi\_2)\; -\; q(\backslash varphi\_1)\backslash right](\backslash lambda\_2\; -\; \backslash lambda\_1)$$

Destination along a rhumb line (loxodrome) on the ellipsoid

$$m_2=m_1+s\cos \alpha ,\mathrm{\Delta }\lambda =\tan \alpha ({\psi }_2-{\psi }_1)m\_2\; =\; m\_1\; +\; s\backslash cos\backslash alpha,\backslash quad\; \backslash Delta\backslash lambda\; =\; \backslash tan\backslash alpha\backslash ,(\backslash psi\_2\; -\; \backslash psi\_1)$$

Geodesic destination point on the ellipsoid (direct problem)

$$\frac{s}{b}={\int }_{{\sigma }_1}^{{\sigma }_2}\sqrt{1+k^2{\sin }^2\sigma }d\sigma ,{\lambda }_{12}={\omega }_{12}-f\sin {\alpha }_0{\int }_{{\sigma }_1}^{{\sigma }_2}\frac{2-f}{1+(1-f)\sqrt{1+k^2{\sin }^2\sigma }}d\sigma \backslash frac\{s\}\{b\}\; =\; \backslash int\_\{\backslash sigma\_1\}^\{\backslash sigma\_2\}\backslash sqrt\{1\; +\; k^2\backslash sin^2\backslash sigma\}\backslash ,d\backslash sigma,\backslash quad\; \backslash lambda\_\{12\}\; =\; \backslash omega\_\{12\}\; -\; f\backslash sin\backslash alpha\_0\backslash int\_\{\backslash sigma\_1\}^\{\backslash sigma\_2\}\backslash frac\{2-f\}\{1\; +\; (1-f)\backslash sqrt\{1+k^2\backslash sin^2\backslash sigma\}\}\backslash ,d\backslash sigma$$

Meridian arc length between two latitudes

$$m(\phi )=\frac{a}{1+n}[(1+\frac{n^2}{4}+\frac{n^4}{64})\phi -\frac{3}{2}(n-\frac{n^3}{8})\sin 2\phi +\frac{15}{16}(n^2-\frac{n^4}{4})\sin 4\phi -\frac{35}{48}{n}^3\sin 6\phi +\frac{315}{512}{n}^4\sin 8\phi ]m(\backslash varphi)\; =\; \backslash frac\{a\}\{1+n\}\backslash left[\backslash left(1+\backslash tfrac\{n^2\}\{4\}+\backslash tfrac\{n^4\}\{64\}\backslash right)\backslash varphi\; -\; \backslash tfrac32\backslash left(n-\backslash tfrac\{n^3\}\{8\}\backslash right)\backslash sin2\backslash varphi\; +\; \backslash tfrac\{15\}\{16\}\backslash left(n^2-\backslash tfrac\{n^4\}\{4\}\backslash right)\backslash sin4\backslash varphi\; -\; \backslash tfrac\{35\}\{48\}n^3\backslash sin6\backslash varphi\; +\; \backslash tfrac\{315\}\{512\}n^4\backslash sin8\backslash varphi\backslash right]$$

Midpoint of a geodesic on the ellipsoid

$$P_{mid}=\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}(P_1,{\alpha }_1,s_{12}\mathrm{/}2)P\_\{mid\}\; =\; \backslash mathrm\{direct\}\backslash left(P\_1,\backslash \; \backslash alpha\_1,\backslash \; s\_\{12\}/2\backslash right)$$

Radii of curvature of the ellipsoid at a latitude

$$M=\frac{a(1-e^2)}{(1-e^2{\sin }^2\phi {)}^{3\mathrm{/}2}},N=\frac{a}{\sqrt{1-e^2{\sin }^2\phi }},R_G=\sqrt{MN}M\; =\; \backslash frac\{a(1-e^2)\}\{(1-e^2\backslash sin^2\backslash varphi)^\{3/2\}\},\backslash quad\; N\; =\; \backslash frac\{a\}\{\backslash sqrt\{1-e^2\backslash sin^2\backslash varphi\}\},\backslash quad\; R\_G\; =\; \backslash sqrt\{MN\}$$

Rhumb-line (loxodrome) distance and bearing on the ellipsoid

$$\tan \alpha =\frac{\mathrm{\Delta }\lambda }{\mathrm{\Delta }\psi },s=\frac{\mathrm{\Delta }m}{\cos \alpha },\psi =\mathrm{arsinh}(\tan \phi )-e\mathrm{artanh}(e\sin \phi )\backslash tan\backslash alpha\; =\; \backslash frac\{\backslash Delta\backslash lambda\}\{\backslash Delta\backslash psi\},\backslash quad\; s\; =\; \backslash frac\{\backslash Delta\; m\}\{\backslash cos\backslash alpha\},\backslash quad\; \backslash psi\; =\; \backslash operatorname\{arsinh\}(\backslash tan\backslash varphi)\; -\; e\backslash operatorname\{artanh\}(e\backslash sin\backslash varphi)$$

Facility location

Closest facility for every node of a network

$$f^{\ast }(i)=\arg \underset{f\in F}{\min }d(i,f)f^*(i)\; =\; \backslash arg\backslash min\_\{f\backslash in\; F\}\; d(i,\; f)$$

Maximal covering location problem (MCLP) on a network

$$\underset{S\subseteq C,\mathrm{\mid }S\mathrm{\mid }=p}{\max }\sum _i{w}_i\mathbf{1}[\underset{j\in S}{\min }{d}_{ij}\le s]\backslash max\_\{S\backslash subseteq\; C,\backslash vert\; S\backslash vert\; =\; p\}\backslash \; \backslash sum\_i\; w\_i\backslash ,\backslash mathbf\; 1\backslash left[\backslash min\_\{j\backslash in\; S\}d\_\{ij\}\backslash le\; s\backslash right]$$

p-median facility location on a network

$$\underset{S\subseteq C,\mathrm{\mid }S\mathrm{\mid }=p}{\min }\sum _i{w}_i\underset{j\in S}{\min }{d}_{ij}\backslash min\_\{S\backslash subseteq\; C,\; \backslash vert\; S\backslash vert\; =\; p\}\backslash \; \backslash sum\_i\; w\_i\backslash min\_\{j\backslash in\; S\}\; d\_\{ij\}$$

Geostatistics

Empirical (Matheron) semivariogram in distance bins

$$\widehat{\gamma }(h_k)=\frac{1}{2N_k}\sum _{(i,j)\in k}(z_i-z_j{)}^2\backslash hat\backslash gamma(h\_k)\; =\; \backslash frac\{1\}\{2N\_k\}\backslash sum\_\{(i,j)\backslash in\; k\}(z\_i\; -\; z\_j)^2$$

Fit an exponential or Gaussian variogram model (nugget, partial sill, range)

$$\gamma (h)=c_0+c(1-e^{-3h\mathrm{/}a})\text{ or }{c}_0+c(1-e^{-(7h\mathrm{/}4a{)}^2}),\underset{c_0,c,a}{\min }\sum _k{({\widehat{\gamma }}_k-\gamma (h_k))}^2\backslash gamma(h)\; =\; c\_0\; +\; c\backslash left(1\; -\; e^\{-3h/a\}\backslash right)\backslash \; \backslash text\{or\}\backslash \; c\_0\; +\; c\backslash left(1\; -\; e^\{-(7h/4a)^2\}\backslash right),\backslash quad\; \backslash min\_\{c\_0,\; c,\; a\}\backslash sum\_k\backslash left(\backslash hat\backslash gamma\_k\; -\; \backslash gamma(h\_k)\backslash right)^2$$

Inverse distance weighted (IDW) interpolation at a point

$$\widehat{z}(\mathbf{p})=\frac{\sum _i{d}_i^{-p}{z}_i}{\sum _i{d}_i^{-p}}\backslash hat\; z(\backslash mathbf\; p)\; =\; \backslash frac\{\backslash sum\_i\; d\_i^\{-p\}z\_i\}\{\backslash sum\_i\; d\_i^\{-p\}\}$$

Ordinary kriging estimate and kriging variance at a point

Polynomial trend surface (first or second order) by least squares

$$z=b_0+b_{1}x+b_{2}y[+b_3{x}^2+b_{4}xy+b_5{y}^2]z\; =\; b\_0\; +\; b\_1x\; +\; b\_2y\backslash \; \backslash left[+\backslash ,\; b\_3x^2\; +\; b\_4xy\; +\; b\_5y^2\backslash right]$$

Map projections

Albers equal-area conic projection: coordinates and scale factors

$$n=\frac{m_1^2-m_2^2}{q_2-q_1},\rho =\frac{a\sqrt{C-nq}}{n},x=FE+\rho \sin \theta ,y=FN+{\rho }_0-\rho \cos \theta ,k=\frac{\rho n}{am}=\frac{1}{h}n\; =\; \backslash frac\{m\_1^2\; -\; m\_2^2\}\{q\_2\; -\; q\_1\},\backslash quad\; \backslash rho\; =\; \backslash frac\{a\backslash sqrt\{C\; -\; nq\}\}\{n\},\backslash quad\; x\; =\; FE\; +\; \backslash rho\backslash sin\backslash theta,\backslash \; y\; =\; FN\; +\; \backslash rho\_0\; -\; \backslash rho\backslash cos\backslash theta,\backslash \; k\; =\; \backslash frac\{\backslash rho\; n\}\{am\}\; =\; \backslash frac1h$$

Azimuthal equidistant projection (ellipsoidal, geodesic)

$$x=FE+s_{12}\sin {\alpha }_1,y=FN+s_{12}\cos {\alpha }_{1}x\; =\; FE\; +\; s\_\{12\}\backslash sin\backslash alpha\_1,\backslash quad\; y\; =\; FN\; +\; s\_\{12\}\backslash cos\backslash alpha\_1$$

Equirectangular (equidistant cylindrical / plate carree) projection

$$x=R(\lambda -{\lambda }_0)\cos {\phi }_{ts},y=R(\phi -{\phi }_0)x\; =\; R(\backslash lambda-\backslash lambda\_0)\backslash cos\backslash varphi\_\{ts\},\backslash quad\; y\; =\; R(\backslash varphi-\backslash varphi\_0)$$

Gnomonic projection (sphere)

$$x=\frac{R\cos \phi \sin \mathrm{\Delta }\lambda }{\cos c},y=\frac{R(\cos {\phi }_0\sin \phi -\sin {\phi }_0\cos \phi \cos \mathrm{\Delta }\lambda )}{\cos c}x\; =\; \backslash frac\{R\backslash cos\backslash varphi\backslash sin\backslash Delta\backslash lambda\}\{\backslash cos\; c\},\backslash quad\; y\; =\; \backslash frac\{R\backslash left(\backslash cos\backslash varphi\_0\backslash sin\backslash varphi\; -\; \backslash sin\backslash varphi\_0\backslash cos\backslash varphi\backslash cos\backslash Delta\backslash lambda\backslash right)\}\{\backslash cos\; c\}$$

Inverse Albers equal-area conic projection

$$q=\frac{C-{\rho }^2{n}^2\mathrm{/}{a}^2}{n},\phi :q(\phi )=q,\lambda ={\lambda }_0+\frac{\theta }{n}q\; =\; \backslash frac\{C\; -\; \backslash rho^2n^2/a^2\}\{n\},\backslash quad\; \backslash varphi:\backslash \; q(\backslash varphi)\; =\; q,\backslash quad\; \backslash lambda\; =\; \backslash lambda\_0\; +\; \backslash frac\{\backslash theta\}\{n\}$$

Inverse Lambert conformal conic projection (two standard parallels)

$$\rho =\mathrm{sgn}(n)\sqrt{x^{\mathrm{\prime }2}+({\rho }_0-y^{\mathrm{\prime }}{)}^2},t={\left(\frac{\rho }{aF}\right)}^{1\mathrm{/}n},\lambda ={\lambda }_0+\frac{\theta }{n}\backslash rho\; =\; \backslash operatorname\{sgn\}(n)\backslash sqrt\{x\text{'}^2\; +\; (\backslash rho\_0\; -\; y\text{'})^2\},\backslash quad\; t\; =\; \backslash left(\backslash frac\{\backslash rho\}\{aF\}\backslash right)^\{1/n\},\backslash quad\; \backslash lambda\; =\; \backslash lambda\_0\; +\; \backslash frac\{\backslash theta\}\{n\}$$

Inverse Mercator projection (ellipsoidal)

$$\psi =\frac{y}{k_{0}a},{\tau }^{\mathrm{\prime }}=\sinh \psi ,\tau =\tan \phi :\tau \sqrt{1+{\sigma }^2}-\sigma \sqrt{1+{\tau }^2}={\tau }^{\mathrm{\prime }}\backslash psi\; =\; \backslash frac\{y\}\{k\_0a\},\backslash quad\; \backslash tau\text{'}\; =\; \backslash sinh\backslash psi,\backslash quad\; \backslash tau\; =\; \backslash tan\backslash varphi:\backslash \; \backslash tau\backslash sqrt\{1+\backslash sigma^2\}\; -\; \backslash sigma\backslash sqrt\{1+\backslash tau^2\}\; =\; \backslash tau\text{'}$$

Lambert azimuthal equal-area projection (ellipsoidal)

$$B=R_q\sqrt{\frac{2}{1+\sin {\beta }_1\sin \beta +\cos {\beta }_1\cos \beta \cos \mathrm{\Delta }\lambda }},x=BD\cos \beta \sin \mathrm{\Delta }\lambda ,y=\frac{B}{D}(\cos {\beta }_1\sin \beta -\sin {\beta }_1\cos \beta \cos \mathrm{\Delta }\lambda )B\; =\; R\_q\backslash sqrt\{\backslash frac\{2\}\{1\; +\; \backslash sin\backslash beta\_1\backslash sin\backslash beta\; +\; \backslash cos\backslash beta\_1\backslash cos\backslash beta\backslash cos\backslash Delta\backslash lambda\}\},\backslash \; x\; =\; BD\backslash cos\backslash beta\backslash sin\backslash Delta\backslash lambda,\backslash \; y\; =\; \backslash frac\{B\}\{D\}\backslash left(\backslash cos\backslash beta\_1\backslash sin\backslash beta\; -\; \backslash sin\backslash beta\_1\backslash cos\backslash beta\backslash cos\backslash Delta\backslash lambda\backslash right)$$

Lambert conformal conic projection (two standard parallels): coordinates, scale factor and convergence

$$n=\frac{\ln m_1-\ln m_2}{\ln t_1-\ln t_2},\rho =aF{t}^n,x=FE+\rho \sin \theta ,y=FN+{\rho }_0-\rho \cos \theta ,k=\frac{\rho n}{am}n\; =\; \backslash frac\{\backslash ln\; m\_1\; -\; \backslash ln\; m\_2\}\{\backslash ln\; t\_1\; -\; \backslash ln\; t\_2\},\backslash \; \backslash rho\; =\; aFt^n,\backslash \; x\; =\; FE\; +\; \backslash rho\backslash sin\backslash theta,\backslash \; y\; =\; FN\; +\; \backslash rho\_0\; -\; \backslash rho\backslash cos\backslash theta,\backslash \; k\; =\; \backslash frac\{\backslash rho\; n\}\{am\}$$

Lambert cylindrical equal-area projection (sphere)

$$x=R(\lambda -{\lambda }_0)\cos {\phi }_s,y=\frac{R\sin \phi }{\cos {\phi }_s}x\; =\; R(\backslash lambda-\backslash lambda\_0)\backslash cos\backslash varphi\_s,\backslash quad\; y\; =\; \backslash frac\{R\backslash sin\backslash varphi\}\{\backslash cos\backslash varphi\_s\}$$

Latitude and longitude to UTM coordinates (with scale factor and convergence)

$$E=500000+0.9996A\eta ,N=N_0+0.9996A\xi E\; =\; 500\backslash ,000\; +\; 0.9996\backslash ,A\backslash eta,\backslash quad\; N\; =\; N\_0\; +\; 0.9996\backslash ,A\backslash xi$$

Latitude and longitude to Web Mercator (EPSG:3857)

$$x=R\lambda ,y=R\ln \tan (\frac{\pi }{4}+\frac{\phi }{2}),R=6378137\mathrm{m}x\; =\; R\backslash lambda,\backslash quad\; y\; =\; R\backslash ln\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}\; +\; \backslash frac\{\backslash varphi\}\{2\}\backslash right),\backslash quad\; R\; =\; 6378137\backslash \; \backslash mathrm\{m\}$$

Mercator projection (ellipsoidal): coordinates and scale factor

$$x=k_{0}a(\lambda -{\lambda }_0),y=k_{0}a[\mathrm{arsinh}(\tan \phi )-e\mathrm{artanh}(e\sin \phi )],k=\frac{k_0\sqrt{1-e^2{\sin }^2\phi }}{\cos \phi }x\; =\; k\_0a(\backslash lambda-\backslash lambda\_0),\backslash quad\; y\; =\; k\_0a\backslash left[\backslash operatorname\{arsinh\}(\backslash tan\backslash varphi)\; -\; e\backslash operatorname\{artanh\}(e\backslash sin\backslash varphi)\backslash right],\backslash quad\; k\; =\; \backslash frac\{k\_0\backslash sqrt\{1-e^2\backslash sin^2\backslash varphi\}\}\{\backslash cos\backslash varphi\}$$

Mollweide equal-area projection (sphere)

$$2\theta +\sin 2\theta =\pi \sin \phi ,x=\frac{2\sqrt{2}}{\pi }R\mathrm{\Delta }\lambda \cos \theta ,y=\sqrt{2}R\sin \theta 2\backslash theta\; +\; \backslash sin2\backslash theta\; =\; \backslash pi\backslash sin\backslash varphi,\backslash quad\; x\; =\; \backslash frac\{2\backslash sqrt2\}\{\backslash pi\}R\backslash ,\backslash Delta\backslash lambda\backslash cos\backslash theta,\backslash quad\; y\; =\; \backslash sqrt2R\backslash sin\backslash theta$$

Orthographic projection (sphere)

$$x=R\cos \phi \sin \mathrm{\Delta }\lambda ,y=R(\cos {\phi }_0\sin \phi -\sin {\phi }_0\cos \phi \cos \mathrm{\Delta }\lambda )x\; =\; R\backslash cos\backslash varphi\backslash sin\backslash Delta\backslash lambda,\backslash quad\; y\; =\; R\backslash left(\backslash cos\backslash varphi\_0\backslash sin\backslash varphi\; -\; \backslash sin\backslash varphi\_0\backslash cos\backslash varphi\backslash cos\backslash Delta\backslash lambda\backslash right)$$

Polar stereographic projection (standard parallel): coordinates and scale factor

$$\rho =a{m}_c\frac{t}{t_c},t=e^{-\psi },x=FE+\rho \sin \mathrm{\Delta }\lambda ,y=FN\mp \rho \cos \mathrm{\Delta }\lambda ,k=\frac{\rho }{am}\backslash rho\; =\; a\backslash ,m\_c\backslash frac\{t\}\{t\_c\},\backslash quad\; t\; =\; e^\{-\backslash psi\},\backslash quad\; x\; =\; FE\; +\; \backslash rho\backslash sin\backslash Delta\backslash lambda,\backslash \; y\; =\; FN\; \backslash mp\; \backslash rho\backslash cos\backslash Delta\backslash lambda,\backslash quad\; k\; =\; \backslash frac\{\backslash rho\}\{am\}$$

Sinusoidal equal-area projection (ellipsoidal)

$$x=\frac{a\cos \phi }{\sqrt{1-e^2{\sin }^2\phi }}(\lambda -{\lambda }_0),y=m(\phi )x\; =\; \backslash frac\{a\backslash cos\backslash varphi\}\{\backslash sqrt\{1-e^2\backslash sin^2\backslash varphi\}\}(\backslash lambda\; -\; \backslash lambda\_0),\backslash quad\; y\; =\; m(\backslash varphi)$$

Tissot's indicatrix: scale factors and distortion of a projection at a point

$$a^{\mathrm{\prime }}=\sqrt{h^2+k^2+2hk\sin {\theta }^{\mathrm{\prime }}},b^{\mathrm{\prime }}=\sqrt{h^2+k^2-2hk\sin {\theta }^{\mathrm{\prime }}},a=\frac{a^{\mathrm{\prime }}+b^{\mathrm{\prime }}}{2},b=\frac{a^{\mathrm{\prime }}-b^{\mathrm{\prime }}}{2},\omega =2\arcsin \frac{a-b}{a+b},s=hk\sin {\theta }^{\mathrm{\prime }}a\text{'}\; =\; \backslash sqrt\{h^2+k^2+2hk\backslash sin\backslash theta\text{'}\},\backslash \; b\text{'}\; =\; \backslash sqrt\{h^2+k^2-2hk\backslash sin\backslash theta\text{'}\},\backslash \; a\; =\; \backslash tfrac\{a\text{'}+b\text{'}\}\{2\},\backslash \; b\; =\; \backslash tfrac\{a\text{'}-b\text{'}\}\{2\},\backslash \; \backslash omega\; =\; 2\backslash arcsin\backslash frac\{a-b\}\{a+b\},\backslash \; s\; =\; hk\backslash sin\backslash theta\text{'}$$

Transverse Mercator projection: coordinates, scale factor and grid convergence

$$\xi ={\xi }^{\mathrm{\prime }}+\sum _{j=1}^6{\alpha }_j\sin 2j{\xi }^{\mathrm{\prime }}\cosh 2j{\eta }^{\mathrm{\prime }},\eta ={\eta }^{\mathrm{\prime }}+\sum _{j=1}^6{\alpha }_j\cos 2j{\xi }^{\mathrm{\prime }}\sinh 2j{\eta }^{\mathrm{\prime }},x=FE+k_{0}A\eta ,y=FN+k_{0}A(\xi -{\xi }_0)\backslash xi\; =\; \backslash xi\text{'}\; +\; \backslash sum\_\{j=1\}^\{6\}\backslash alpha\_j\backslash sin2j\backslash xi\text{'}\backslash cosh2j\backslash eta\text{'},\backslash quad\; \backslash eta\; =\; \backslash eta\text{'}\; +\; \backslash sum\_\{j=1\}^\{6\}\backslash alpha\_j\backslash cos2j\backslash xi\text{'}\backslash sinh2j\backslash eta\text{'},\backslash quad\; x\; =\; FE\; +\; k\_0A\backslash eta,\backslash \; y\; =\; FN\; +\; k\_0A(\backslash xi\; -\; \backslash xi\_0)$$

UTM coordinates to latitude and longitude

$${\xi }^{\mathrm{\prime }}=\xi -\sum _{j=1}^6{\beta }_j\sin 2j\xi \cosh 2j\eta ,{\eta }^{\mathrm{\prime }}=\eta -\sum _{j=1}^6{\beta }_j\cos 2j\xi \sinh 2j\eta \backslash xi\text{'}\; =\; \backslash xi\; -\; \backslash sum\_\{j=1\}^\{6\}\backslash beta\_j\backslash sin2j\backslash xi\backslash cosh2j\backslash eta,\backslash quad\; \backslash eta\text{'}\; =\; \backslash eta\; -\; \backslash sum\_\{j=1\}^\{6\}\backslash beta\_j\backslash cos2j\backslash xi\backslash sinh2j\backslash eta$$

Web Mercator (EPSG:3857) to latitude and longitude

$$\phi =\arctan \sinh \frac{y}{R},\lambda =\frac{x}{R}\backslash varphi\; =\; \backslash arctan\backslash sinh\backslash frac\{y\}\{R\},\backslash quad\; \backslash lambda\; =\; \backslash frac\{x\}\{R\}$$

Web-map tile (slippy map / XYZ) and quadkey containing a point

$$x_t=\lfloor 2^z\frac{\lambda +{180}^{\circ }}{{360}^{\circ }}\rfloor ,y_t=\lfloor 2^z(\frac{1}{2}-\frac{\mathrm{arsinh}(\tan \phi )}{2\pi })\rfloor x\_t\; =\; \backslash left\backslash lfloor\; 2^z\backslash ,\backslash frac\{\backslash lambda\; +\; 180^\backslash circ\}\{360^\backslash circ\}\backslash right\backslash rfloor,\backslash quad\; y\_t\; =\; \backslash left\backslash lfloor\; 2^z\backslash left(\backslash frac12\; -\; \backslash frac\{\backslash operatorname\{arsinh\}(\backslash tan\backslash varphi)\}\{2\backslash pi\}\backslash right)\backslash right\backslash rfloor$$

Network accessibility

Gravity-based (Hansen) accessibility on a road network

$$A_i=\sum _{j\mathrm{\ne }i}{O}_{j}f(d_{ij}),f(d)=d^{-\beta }\text{ or }{e}^{-\beta d}A\_i\; =\; \backslash sum\_\{j\backslash ne\; i\}\; O\_j\backslash ,f(d\_\{ij\}),\backslash quad\; f(d)\; =\; d^\{-\backslash beta\}\backslash \; \backslash text\{or\}\backslash \; e^\{-\backslash beta\; d\}$$

Network connectivity indices: cyclomatic number, alpha, beta and gamma

$$\mu =e-v+p,\alpha =\frac{e-v+p}{2v-5},\beta =\frac{e}{v},\gamma =\frac{e}{3(v-2)}\backslash mu\; =\; e\; -\; v\; +\; p,\backslash quad\; \backslash alpha\; =\; \backslash frac\{e\; -\; v\; +\; p\}\{2v\; -\; 5\},\backslash quad\; \backslash beta\; =\; \backslash frac\{e\}\{v\},\backslash quad\; \backslash gamma\; =\; \backslash frac\{e\}\{3(v-2)\}$$

Network detour index (circuity) between two nodes

$$DI=\frac{d_{net}(o,t)}{\parallel {\mathbf{x}}_o-{\mathbf{x}}_t\parallel }DI\; =\; \backslash frac\{d\_\{net\}(o,\; t)\}\{\backslash lVert\backslash mathbf\; x\_o\; -\; \backslash mathbf\; x\_t\backslash rVert\}$$

Network service area (isochrone) of a facility

$$SA(s,c)=\{v:d(s,v)\le c\}SA(s,\; c)\; =\; \backslash \{v\; :\; d(s,\; v)\; \backslash le\; c\backslash \}$$

Shimbel accessibility index of every node

$$A_i=\sum _j{d}_{ij},D(G)=\sum _i{A}_{i}A\_i\; =\; \backslash sum\_j\; d\_\{ij\},\backslash quad\; D(G)\; =\; \backslash sum\_i\; A\_i$$

Two-step floating catchment area (2SFCA) accessibility

$$R_j=\frac{S_j}{\sum _{k:d_{kj}\le d_0}{P}_k},A_i=\sum _{j:d_{ij}\le d_0}{R}_{j}R\_j\; =\; \backslash frac\{S\_j\}\{\backslash sum\_\{k:\; d\_\{kj\}\backslash le\; d\_0\}P\_k\},\backslash quad\; A\_i\; =\; \backslash sum\_\{j:\; d\_\{ij\}\backslash le\; d\_0\}R\_j$$

Planar vector analysis

Discrete Frechet distance between two polylines

$$c_{i,j}=\max (d(P_i,Q_j),\min (c_{i-1,j},c_{i-1,j-1},c_{i,j-1}))c\_\{i,j\}\; =\; \backslash max\backslash left(d(P\_i,\; Q\_j),\backslash \; \backslash min(c\_\{i-1,j\},\backslash \; c\_\{i-1,j-1\},\backslash \; c\_\{i,j-1\})\backslash right)$$

Douglas-Peucker line simplification

$$d_{max}=\underset{k}{\max }\mathrm{dist}(P_k,\overline{P_i{P}_j})>\epsilon \Rightarrow \text{keep }{P}_{k}d\_\{max\}\; =\; \backslash max\_k\; \backslash operatorname\{dist\}(P\_k,\; \backslash overline\{P\_iP\_j\})\; >\; \backslash varepsilon\; \backslash Rightarrow\; \backslash text\{keep\; \}\; P\_k$$

Shape compactness indices of a polygon (Polsby-Popper, Schwartzberg, convex hull, Reock)

$$PP=\frac{4\pi A}{P^2},S=\frac{P}{2\sqrt{\pi A}},C_H=\frac{A}{A_{hull}},R=\frac{A}{\pi r_{mec}^2}PP\; =\; \backslash frac\{4\backslash pi\; A\}\{P^2\},\backslash quad\; S\; =\; \backslash frac\{P\}\{2\backslash sqrt\{\backslash pi\; A\}\},\backslash quad\; C\_H\; =\; \backslash frac\{A\}\{A\_\{hull\}\},\backslash quad\; R\; =\; \backslash frac\{A\}\{\backslash pi\; r\_\{mec\}^2\}$$

Point pattern analysis

Kernel density estimate of a point pattern at a location

$$\widehat{f}(\mathbf{p})=\frac{1}{n}\sum _i{K}_h(\parallel \mathbf{p}-{\mathbf{x}}_i\parallel ),K_h^{gauss}=\frac{e^{-u^2\mathrm{/}2}}{2\pi h^2},K_h^{epa}=\frac{2(1-u^2)}{\pi h^2},K_h^{unif}=\frac{1}{\pi h^2}\backslash hat\; f(\backslash mathbf\; p)\; =\; \backslash frac1n\backslash sum\_i\; K\_h(\backslash lVert\backslash mathbf\; p\; -\; \backslash mathbf\; x\_i\backslash rVert),\backslash quad\; K^\{gauss\}\_h\; =\; \backslash frac\{e^\{-u^2/2\}\}\{2\backslash pi\; h^2\},\backslash \; K^\{epa\}\_h\; =\; \backslash frac\{2(1-u^2)\}\{\backslash pi\; h^2\},\backslash \; K^\{unif\}\_h\; =\; \backslash frac\{1\}\{\backslash pi\; h^2\}$$

Mean centre and standard distance of a point pattern

$$SD=\sqrt{\frac{\sum _i(x_i-\bar{x}{)}^2+(y_i-\bar{y}{)}^2}{n}}SD\; =\; \backslash sqrt\{\backslash frac\{\backslash sum\_i\; (x\_i\; -\; \backslash bar\; x)^2\; +\; (y\_i\; -\; \backslash bar\; y)^2\}\{n\}\}$$

Nearest-neighbour index (Clark-Evans ratio) of a point pattern

$$R=\frac{{\bar{d}}_{obs}}{0.5\mathrm{/}\sqrt{n\mathrm{/}A}},z=\frac{{\bar{d}}_{obs}-{\bar{d}}_{exp}}{0.26136\mathrm{/}\sqrt{n^2\mathrm{/}A}}R\; =\; \backslash frac\{\backslash bar\; d\_\{obs\}\}\{0.5/\backslash sqrt\{n/A\}\},\backslash quad\; z\; =\; \backslash frac\{\backslash bar\; d\_\{obs\}\; -\; \backslash bar\; d\_\{exp\}\}\{0.26136/\backslash sqrt\{n^2/A\}\}$$

Quadrat analysis: chi-square test of complete spatial randomness

$${\chi }^2=\sum _c\frac{(O_c-E{)}^2}{E},E=\frac{n}{n_x{n}_y},VMR=\frac{{\chi }^2}{n_x{n}_y-1}\backslash chi^2\; =\; \backslash sum\_\{c\}\backslash frac\{(O\_c\; -\; E)^2\}\{E\},\backslash quad\; E\; =\; \backslash frac\{n\}\{n\_xn\_y\},\backslash quad\; VMR\; =\; \backslash frac\{\backslash chi^2\}\{n\_xn\_y\; -\; 1\}$$

Ripley's K and L functions of a point pattern

Standard deviational ellipse of a point pattern

$$s_{x,y}=\sqrt{\frac{2n}{n-2}{\lambda }_{1,2}},{\lambda }_{1,2}=\frac{s_{xx}+s_{yy}}{2}\pm \sqrt{\frac{(s_{xx}-s_{yy}{)}^2}{4}+s_{xy}^2}s\_\{x,y\}\; =\; \backslash sqrt\{\backslash frac\{2n\}\{n-2\}\backslash lambda\_\{1,2\}\},\backslash quad\; \backslash lambda\_\{1,2\}\; =\; \backslash frac\{s\_\{xx\}+s\_\{yy\}\}\{2\}\; \backslash pm\; \backslash sqrt\{\backslash frac\{(s\_\{xx\}-s\_\{yy\})^2\}\{4\}\; +\; s\_\{xy\}^2\}$$

Weighted mean centre and weighted standard distance

$${\bar{x}}_w=\frac{\sum w_i{x}_i}{\sum w_i},S{D}_w=\sqrt{\frac{\sum w_i[(x_i-{\bar{x}}_w{)}^2+(y_i-{\bar{y}}_w{)}^2]}{\sum w_i}}\backslash bar\; x\_w\; =\; \backslash frac\{\backslash sum\; w\_ix\_i\}\{\backslash sum\; w\_i\},\backslash quad\; SD\_w\; =\; \backslash sqrt\{\backslash frac\{\backslash sum\; w\_i\backslash left[(x\_i\; -\; \backslash bar\; x\_w)^2\; +\; (y\_i\; -\; \backslash bar\; y\_w)^2\backslash right]\}\{\backslash sum\; w\_i\}\}$$

Remote sensing

At-sensor brightness temperature from thermal radiance

$$T_B=\frac{K_2}{\ln (\frac{K_1}{L_{\lambda }}+1)}T\_B\; =\; \backslash frac\{K\_2\}\{\backslash ln\backslash left(\backslash frac\{K\_1\}\{L\_\backslash lambda\}\; +\; 1\backslash right)\}$$

Ground sample distance and swath of a nadir-looking camera

$$GSD=\frac{pH}{f},W=GSD\times NGSD\; =\; \backslash frac\{p\backslash ,H\}\{f\},\backslash quad\; W\; =\; GSD\; \backslash times\; N$$

Thematic map accuracy: overall accuracy, kappa, producer's and user's accuracy

$$OA=\frac{\sum _i{n}_{ii}}{N},\kappa =\frac{p_o-p_e}{1-p_e},P{A}_j=\frac{n_{jj}}{n_{+j}},U{A}_i=\frac{n_{ii}}{n_{i+}}OA\; =\; \backslash frac\{\backslash sum\_i\; n\_\{ii\}\}\{N\},\backslash quad\; \backslash kappa\; =\; \backslash frac\{p\_o\; -\; p\_e\}\{1\; -\; p\_e\},\backslash quad\; PA\_j\; =\; \backslash frac\{n\_\{jj\}\}\{n\_\{+j\}\},\backslash quad\; UA\_i\; =\; \backslash frac\{n\_\{ii\}\}\{n\_\{i+\}\}$$

Top-of-atmosphere reflectance from at-sensor radiance

$${\rho }_{TOA}=\frac{\pi L_{\lambda }{d}^2}{ESU{N}_{\lambda }\cos {\theta }_s},\frac{1}{d^2}=1.00011+0.034221\cos \mathrm{\Gamma }+0.00128\sin \mathrm{\Gamma }+0.000719\cos 2\mathrm{\Gamma }+0.000077\sin 2\mathrm{\Gamma }\backslash rho\_\{TOA\}\; =\; \backslash frac\{\backslash pi\; L\_\backslash lambda\; d^2\}\{ESUN\_\backslash lambda\backslash cos\backslash theta\_s\},\backslash quad\; \backslash frac\{1\}\{d^2\}\; =\; 1.00011\; +\; 0.034221\backslash cos\backslash Gamma\; +\; 0.00128\backslash sin\backslash Gamma\; +\; 0.000719\backslash cos2\backslash Gamma\; +\; 0.000077\backslash sin2\backslash Gamma$$

Vegetation indices: NDVI, EVI, SAVI, OSAVI, MSAVI and EVI2 from band reflectances

$$NDVI=\frac{{\rho }_N-{\rho }_R}{{\rho }_N+{\rho }_R},EVI=2.5\frac{{\rho }_N-{\rho }_R}{{\rho }_N+6{\rho }_R-7.5{\rho }_B+1},SAVI=(1+L)\frac{{\rho }_N-{\rho }_R}{{\rho }_N+{\rho }_R+L}NDVI\; =\; \backslash frac\{\backslash rho\_\{N\}\; -\; \backslash rho\_R\}\{\backslash rho\_N\; +\; \backslash rho\_R\},\backslash \; EVI\; =\; 2.5\backslash frac\{\backslash rho\_N\; -\; \backslash rho\_R\}\{\backslash rho\_N\; +\; 6\backslash rho\_R\; -\; 7.5\backslash rho\_B\; +\; 1\},\backslash \; SAVI\; =\; (1+L)\backslash frac\{\backslash rho\_N\; -\; \backslash rho\_R\}\{\backslash rho\_N\; +\; \backslash rho\_R\; +\; L\}$$

Water, moisture, burn and built-up indices: NDWI, MNDWI, NDMI, NBR and NDBI

$$ND(a,b)=\frac{{\rho }_a-{\rho }_b}{{\rho }_a+{\rho }_b}ND(a,\; b)\; =\; \backslash frac\{\backslash rho\_a\; -\; \backslash rho\_b\}\{\backslash rho\_a\; +\; \backslash rho\_b\}$$

Spatial statistics

Bivariate Moran's I (spatial cross-correlation)

$$I_{xy}=\frac{\sum _i{z}_{x,i}\sum _j{w}_{ij}{z}_{y,j}}{n-1}I\_\{xy\}\; =\; \backslash frac\{\backslash sum\_i\; z\_\{x,i\}\backslash sum\_j\; w\_\{ij\}z\_\{y,j\}\}\{n-1\}$$

Distance-band spatial weights

Geary's C spatial autocorrelation with normal and randomization inference

$$C=\frac{(n-1)\sum _i\sum _j{w}_{ij}(y_i-y_j{)}^2}{2S_0\sum _i(y_i-\bar{y}{)}^2},V_N[C]=\frac{(2S_1+S_2)(n-1)-4S_0^2}{2(n+1)S_0^2}C\; =\; \backslash frac\{(n-1)\backslash sum\_i\backslash sum\_j\; w\_\{ij\}(y\_i\; -\; y\_j)^2\}\{2S\_0\backslash sum\_i\; (y\_i\; -\; \backslash bar\; y)^2\},\backslash quad\; V\_N[C]\; =\; \backslash frac\{(2S\_1\; +\; S\_2)(n-1)\; -\; 4S\_0^2\}\{2(n+1)S\_0^2\}$$

Getis-Ord general G (global high/low clustering)

$$G=\frac{\sum _i\sum _{j\mathrm{\ne }i}{w}_{ij}{y}_i{y}_j}{\sum _i\sum _{j\mathrm{\ne }i}{y}_i{y}_j},E[G]=\frac{S_0}{n(n-1)}G\; =\; \backslash frac\{\backslash sum\_i\backslash sum\_\{j\backslash neq\; i\}\; w\_\{ij\}y\_iy\_j\}\{\backslash sum\_i\backslash sum\_\{j\backslash neq\; i\}y\_iy\_j\},\backslash quad\; E[G]\; =\; \backslash frac\{S\_0\}\{n(n-1)\}$$

Global Moran's I spatial autocorrelation with normal and randomization inference

$$I=\frac{n}{S_0}\frac{\sum _i\sum _j{w}_{ij}{z}_i{z}_j}{\sum _i{z}_i^2},E[I]=-\frac{1}{n-1},V_N[I]=\frac{n^2{S}_1-n{S}_2+3S_0^2}{(n^2-1)S_0^2}-E[I{]}^{2}I\; =\; \backslash frac\{n\}\{S\_0\}\backslash frac\{\backslash sum\_i\backslash sum\_j\; w\_\{ij\}z\_iz\_j\}\{\backslash sum\_i\; z\_i^2\},\backslash quad\; E[I]\; =\; -\backslash frac\{1\}\{n-1\},\backslash quad\; V\_N[I]\; =\; \backslash frac\{n^2S\_1\; -\; nS\_2\; +\; 3S\_0^2\}\{(n^2-1)S\_0^2\}\; -\; E[I]^2$$

Join count statistics of a binary map (BB, BW, WW)

$$BB=\frac{1}{2}\sum _{i,j}{w}_{ij}{y}_i{y}_j,BW=\frac{1}{2}\sum _{i,j}{w}_{ij}(y_i-y_j{)}^2,WW=\frac{1}{2}\sum _{i,j}{w}_{ij}(1-y_i)(1-y_j)BB\; =\; \backslash tfrac12\backslash sum\_\{i,j\}\; w\_\{ij\}y\_iy\_j,\backslash \; BW\; =\; \backslash tfrac12\backslash sum\_\{i,j\}\; w\_\{ij\}(y\_i\; -\; y\_j)^2,\backslash \; WW\; =\; \backslash tfrac12\backslash sum\_\{i,j\}w\_\{ij\}(1-y\_i)(1-y\_j)$$

Local Getis-Ord Gi and Gi hot-spot statistics*

$$G_i^{\ast }=\frac{\sum _j{w}_{ij}{y}_j}{\sum _j{y}_j},Z_i=\frac{G_i-W_i\mathrm{/}N}{\sqrt{\frac{W_i(N-W_i)}{(N-1)N^2}\frac{s^2}{{\bar{y}}^2}}}G\_i^*\; =\; \backslash frac\{\backslash sum\_j\; w\_\{ij\}y\_j\}\{\backslash sum\_j\; y\_j\},\backslash quad\; Z\_i\; =\; \backslash frac\{G\_i\; -\; W\_i/N\}\{\backslash sqrt\{\backslash frac\{W\_i(N\; -\; W\_i)\}\{(N-1)N^2\}\backslash frac\{s^2\}\{\backslash bar\; y^2\}\}\}$$

Local Moran's I (LISA) and Moran scatterplot quadrants

$$I_i=(n-1)\frac{z_i\sum _j{w}_{ij}{z}_j}{\sum _k{z}_k^2}I\_i\; =\; (n-1)\backslash frac\{z\_i\backslash sum\_j\; w\_\{ij\}z\_j\}\{\backslash sum\_k\; z\_k^2\}$$

Spatial lag of a variable

$$[Wy{]}_i=\sum _j{w}_{ij}{y}_j[Wy]\_i\; =\; \backslash sum\_j\; w\_\{ij\}y\_j$$

k-nearest-neighbour spatial weights

$$w_{ij}=1⟺j\in N_k(i)w\_\{ij\}\; =\; 1\; \backslash iff\; j\; \backslash in\; N\_k(i)$$

Terrain analysis

D8 flow accumulation of a small DEM

$$A(c)=1+\sum _{u\to c}A(u)A(c)\; =\; 1\; +\; \backslash sum\_\{u\backslash to\; c\}A(u)$$

D8 flow direction of every cell of a small DEM

$$d^{\ast }=\arg \underset{k}{\max }\frac{z_0-z_k}{{\mathrm{\ell }}_k},{\mathrm{\ell }}_k\in \{\mathrm{\Delta },\sqrt{2}\mathrm{\Delta }\}d^*\; =\; \backslash arg\backslash max\_\{k\}\backslash frac\{z\_0\; -\; z\_k\}\{\backslash ell\_k\},\backslash quad\; \backslash ell\_k\; \backslash in\; \backslash \{\backslash Delta,\; \backslash sqrt2\backslash Delta\backslash \}$$

Hillshade (analytical relief shading) of a 3x3 elevation window

$$\cos i=\cos Z\cos S+\sin Z\sin S\cos (A-\alpha ),H=255\max (0,\cos i)\backslash cos\; i\; =\; \backslash cos\; Z\backslash cos\; S\; +\; \backslash sin\; Z\backslash sin\; S\backslash cos(A\; -\; \backslash alpha),\backslash quad\; H\; =\; 255\backslash max(0,\; \backslash cos\; i)$$

Line-of-sight visibility along a terrain profile (with Earth curvature and refraction)

Slope and aspect of a 3x3 elevation window (Horn's method)

$$\frac{\mathrm{\partial }z}{\mathrm{\partial }x}=\frac{(z_3+2z_6+z_9)-(z_1+2z_4+z_7)}{8\mathrm{\Delta }},\frac{\mathrm{\partial }z}{\mathrm{\partial }y}=\frac{(z_1+2z_2+z_3)-(z_7+2z_8+z_9)}{8\mathrm{\Delta }},S=\arctan \parallel \mathrm{\nabla }z\parallel \backslash frac\{\backslash partial\; z\}\{\backslash partial\; x\}\; =\; \backslash frac\{(z\_3\; +\; 2z\_6\; +\; z\_9)\; -\; (z\_1\; +\; 2z\_4\; +\; z\_7)\}\{8\backslash Delta\},\backslash \; \backslash frac\{\backslash partial\; z\}\{\backslash partial\; y\}\; =\; \backslash frac\{(z\_1\; +\; 2z\_2\; +\; z\_3)\; -\; (z\_7\; +\; 2z\_8\; +\; z\_9)\}\{8\backslash Delta\},\backslash \; S\; =\; \backslash arctan\backslash lVert\backslash nabla\; z\backslash rVert$$

Terrain curvature: total, profile and plan curvature (Zevenbergen-Thorne)

$$K=-2(D+E),K_{prof}=\frac{-2(D{G}^2+E{H}^2+FGH)}{G^2+H^2},K_{plan}=\frac{2(D{H}^2+E{G}^2-FGH)}{G^2+H^2}K\; =\; -2(D+E),\backslash quad\; K\_\{prof\}\; =\; \backslash frac\{-2(DG^2\; +\; EH^2\; +\; FGH)\}\{G^2\; +\; H^2\},\backslash quad\; K\_\{plan\}\; =\; \backslash frac\{2(DH^2\; +\; EG^2\; -\; FGH)\}\{G^2\; +\; H^2\}$$

Terrain ruggedness index (Riley), topographic position index and local relief

$$TRI=\sqrt{\sum _{i=1}^8(z_i-z_0{)}^2},TPI=z_0-\frac{1}{8}\sum _{i=1}^8{z}_{i}TRI\; =\; \backslash sqrt\{\backslash sum\_\{i=1\}^\{8\}(z\_i\; -\; z\_0)^2\},\backslash quad\; TPI\; =\; z\_0\; -\; \backslash frac18\backslash sum\_\{i=1\}^\{8\}z\_i$$

Watershed (catchment) of an outlet cell on a small DEM

$$W(o)=\{c:c\to \cdots \to o\}W(o)\; =\; \backslash \{c\; :\; c\; \backslash to\; \backslash cdots\; \backslash to\; o\backslash \}$$

Urban geography

Huff model: probabilities of shopping at each store from an origin (network distances)

$$P_{ij}=\frac{S_j^{\alpha }{d}_{ij}^{-\beta }}{\sum _k{S}_k^{\alpha }{d}_{ik}^{-\beta }}P\_\{ij\}\; =\; \backslash frac\{S\_j^\{\backslash alpha\}d\_\{ij\}^\{-\backslash beta\}\}\{\backslash sum\_k\; S\_k^\{\backslash alpha\}d\_\{ik\}^\{-\backslash beta\}\}$$

Rank-size rule (Zipf) fit of a settlement system

$$\ln P_r=\ln P_1-q\ln r\backslash ln\; P\_r\; =\; \backslash ln\; P\_1\; -\; q\backslash ln\; r$$

Residential segregation (evenness): dissimilarity, Gini and Theil entropy indices

$$D=\frac{\sum _i{t}_i\mathrm{\mid }{p}_i-P\mathrm{\mid }}{2TP(1-P)},G=\frac{\sum _i\sum _j{t}_i{t}_j\mathrm{\mid }{p}_i-p_j\mathrm{\mid }}{2T^{2}P(1-P)},H=\sum _i\frac{t_i(E-E_i)}{ET}D\; =\; \backslash frac\{\backslash sum\_i\; t\_i\backslash vert\; p\_i\; -\; P\backslash vert\; \}\{2TP(1-P)\},\backslash quad\; G\; =\; \backslash frac\{\backslash sum\_i\backslash sum\_j\; t\_it\_j\backslash vert\; p\_i\; -\; p\_j\backslash vert\; \}\{2T^2P(1-P)\},\backslash quad\; H\; =\; \backslash sum\_i\backslash frac\{t\_i(E\; -\; E\_i)\}\{ET\}$$

Residential segregation (exposure): isolation, interaction and correlation ratio

$${}_x{P}_x^{\ast }=\sum _i\frac{x_i}{X}\frac{x_i}{t_i},{}_x{P}_y^{\ast }=\sum _i\frac{x_i}{X}\frac{y_i}{t_i},{\eta }^2=\frac{{}_x{P}_x^{\ast }-P}{1-P}\{\}\_xP^*\_x\; =\; \backslash sum\_i\backslash frac\{x\_i\}\{X\}\backslash frac\{x\_i\}\{t\_i\},\backslash quad\; \{\}\_xP^*\_y\; =\; \backslash sum\_i\backslash frac\{x\_i\}\{X\}\backslash frac\{y\_i\}\{t\_i\},\backslash quad\; \backslash eta^2\; =\; \backslash frac\{\{\}\_xP^*\_x\; -\; P\}\{1-P\}$$

Spatial interaction (gravity) model fitted by Poisson regression

$$T_{ij}\sim \mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}({\mu }_{ij}),\ln {\mu }_{ij}=\ln k+\alpha \ln O_i+\beta \ln D_j-\gamma \ln d_{ij}T\_\{ij\}\; \backslash sim\; \backslash mathrm\{Poisson\}(\backslash mu\_\{ij\}),\backslash quad\; \backslash ln\backslash mu\_\{ij\}\; =\; \backslash ln\; k\; +\; \backslash alpha\backslash ln\; O\_i\; +\; \backslash beta\backslash ln\; D\_j\; -\; \backslash gamma\backslash ln\; d\_\{ij\}$$

Urban population density gradient (Clark's negative exponential model)

$$D(r)=D_0{e}^{-br},\ln D=\ln D_0-brD(r)\; =\; D\_0e^\{-br\},\backslash quad\; \backslash ln\; D\; =\; \backslash ln\; D\_0\; -\; br$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified geospatial analytics capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (ellipsoidal geodesy, coordinates and map projections, spatial statistics and geostatistics, remote sensing, terrain analysis, and urban and network analysis); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{geospatial_analytics_ai999x_1_0_0,
  title  = {geospatial-analytics-ai999x: an offline, verified geospatial analytics solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/geospatial-analytics-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/geospatial-analytics-ai999x/discussions or visit https://ai999x.com.

Support the project

geospatial-analytics-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com