GELE cheat sheet

GRS80 ellipsoid: a = 6,378,137 m, 1/f = 298.257222101; WGS84 uses 1/f = 298.257223563.

Geoid undulation N = h − H, where h is ellipsoidal height and H is orthometric height.

Meridian radius M = a(1−e²)/(1−e² sin²φ)^(3/2); prime-vertical radius N = a/(1−e² sin²φ)^(1/2).

Geodetic minus geocentric latitude reaches a maximum of about 11.5 arcminutes near latitude 45°.

Deflection of vertical components: ξ = Φ − φ (meridian) and η = (Λ − λ)cos φ (prime vertical).

Seven-parameter Helmert transformation: X′ = T + (1 + s)·R·X, with three translations, three rotations, and one scale factor.

UTM: 60 zones each 6° wide; central meridian scale factor 0.9996, false easting 500,000 m, false northing 0 (north) or 10,000,000 m (south).

Meridian arc: 1° of latitude is about 110.6 km at the equator and 111.7 km at the poles; 1 arcminute ≈ 1,843 m.

Clairaut gravity formula: g = ge(1 + β sin²φ), with β ≈ 0.0053024.

Order of boundary evidence: natural monuments control over artificial monuments, which control over courses and distances, which control over area.

Torrens registration confers indefeasible title; the state registers and guarantees the registered proprietor's title.

Adverse possession requires open, notorious, hostile, exclusive, and continuous possession for the statutory period.

Accretion (alluvion) is gradual and imperceptible and belongs to the riparian owner; avulsion is sudden and does not move the boundary.

Metes and bounds descriptions give bearings and distances from a point of beginning tied to a monument or meridian.

Lot area is the weakest element of a description and yields to monuments, bearings, and distances.

An easement grants rights of use, not ownership; a prescriptive easement requires adverse use for the statutory period.

A subdivision plan shows technical descriptions, tie lines, and curve data: radius R, central angle Δ, arc length L, tangent T, long chord C, and chord bearing.

Observation equation L + v = Ax; least squares minimizes the weighted sum of squared residuals vᵀPv.

Normal equations: (AᵀPA)x = AᵀPL; solution x = (AᵀPA)⁻¹AᵀPL.

Reference variance σ0² = vᵀPv/(n − u); parameter covariance Qx = (AᵀPA)⁻¹; σxi = σ0√qii.

Latitude = d·cos(bearing); departure = d·sin(bearing).

Closed loop: Σlat = 0 and Σdep = 0; closed polygon interior angles sum to (n − 2)·180°.

Compass (Bowditch) rule correction: c_i = −misclosure × (length_i / Σlength).

Transit rule apportions misclosure in proportion to the magnitude of each latitude or departure, not to side length.

Error propagation: Σy = J Σx Jᵀ, where J is the Jacobian of partial derivatives.

Traverse precision is expressed as linear misclosure divided by perimeter, in the form 1:N.

Scale 1:N means one map unit equals N ground units; 1:24,000 gives 1 cm = 240 m.

Larger scale means a smaller denominator and more detail; smaller scale means a larger denominator and less detail.

Conformal projections preserve angles and local shape; equal-area projections preserve area; equidistant projections preserve distance along standard lines.

Contour interval is constant on a given map; index contours are heavier lines, usually every fifth contour.

Vector model stores points, lines, and polygons with topology; raster model stores grid cells with one attribute value per cell.

NMAS: 90% of well-defined points within 1/30 inch at map scale horizontally, and 90% within one-half contour interval vertically.

Overlay operations include union, intersect, clip, and erase; proximity operations include buffer and near.

Generalization operators include simplification, smoothing, aggregation, displacement, exaggeration, and selection.

Geographic coordinates are angular latitude/longitude; projected coordinates are linear metres on a chosen datum.

Combined curvature and refraction correction C = 0.0675·K² metres, with K in kilometres; curvature alone is 0.0785·K².

Tape corrections: temperature Ct = αL(T − T0); tension Cp = (P − P0)L/(AE); sag Cs = −w²L³/(24P²).

1 hectare = 10,000 m²; 1 acre = 4,046.86 m² = 0.404686 ha.

Law of sines: a/sin A = b/sin B = c/sin C; law of cosines: c² = a² + b² − 2ab·cos C.

EDM accuracy is stated as ±(a mm + b ppm); atmospheric correction depends on temperature, pressure, and humidity.

Differential leveling: ΔElev = BS − FS; arithmetic check ΣBS − ΣFS = last RL − first RL.

Systematic instrument errors include collimation error, vertical index error, and plate bubble error; double centering removes collimation error.

Azimuth from bearing: NE = bearing; SE = 180° − bearing; SW = 180° + bearing; NW = 360° − bearing.

Photo scale S = f/H = photo distance / ground distance.

Relief displacement d = r·h/H, measured radially outward from the nadir point.

Parallax relation: H = B·f/p; elevation difference Δh = Δp·H/(p + Δp).

Standard aerial overlap: endlap about 60% and sidelap about 20–30%.

Airbase B = H·w·(1 − overlap)/f for photo width w.

Ground sample distance GSD = H·pixel size / f.

NDVI = (NIR − Red)/(NIR + Red), ranging from −1 to +1.

Resolution types are spatial, spectral, radiometric, and temporal.

An orthophoto is rectified for tilt and relief displacement, giving uniform scale over the image.