Item

Explanation

Semi-major axis

a

Ellipsoid semi-major axis.

Semi-minor axis

b

Ellipsoid semi-minor axis. b= a(1-f)

Flattening

f

The relationship between the semi-major and semi-minor axes of the ellipsoid:
(a - b)/a

Inverse flattening

1/f

The reciprocal of the ellipsoid flattening. This is the value commonly used when specifying an ellipsoid (e.g. 1/f = 298.257).

Eccentricity squared

e²

(a² - b²)/a²

Second eccentricity squared

e'²

(a²- b²)/b²

Radius of curvature

r

Radius of curvature of the ellipsoid in the plane of the meridian.

n

Radius of curvature of the ellipsoid in the prime vertical.

R

Geometric mean radius of curvature: (r n )^1/2

R_a

Radius of curvature at a point, in a given azimuth. It may vary by thousands of metres, depending on the azimuth.

y

Ratio of the ellipsoidal radii of curvature (n /r )

r²

R² k₀² _{= r n} k₀²

r_m²

r n k₀² at f _m

Latitude

f

Geodetic latitude, negative south of the equator.

f ₁, f ₂

Geodetic latitude at points 1 and 2 respectively.

f _m

Mean latitude: (f ₁ + f ₂)/2

D f

Latitude difference: (f ₂ - f ₁)

Foot point latitude

f ’

Latitude for which the meridian distance (m) = N'/ k₀.
t’, y ’, r ’, n ’ are functions of the latitude f ’.

Longitude

l

Geodetic longitude measured from Greenwich, positive eastwards.

l ₁, l ₂

Geodetic longitude at points 1 and 2 respectively.

D l

Longitude difference: (l ₂-l ₁)

l ₀

Geodetic longitude of the central meridian

w

Geodetic longitude difference measured from the central meridian, positive eastwards:
l - l ₀

Azimuth

a

Horizontal angle measured from the ellipsoidal meridian, clockwise from north through 360°.

Ellipsoidal distance

s

Distance on the ellipsoid along either a normal section or a geodesic. The difference between the two is usually negligible, amounting to less than 20 millimetres in 3,000 kilometres. A line on the ellipsoid is projected on the grid as an arc.

Sea level or geoidal distance

s’

Distance reduced using heights above sea level or the geoid, which are often referred to as orthometric heights. Ellipsoidal distances should be used for GDA computations.

Easting

E’

Measured from a Central Meridian, positive eastwards

E

Measured from the false origin (E' + 500,000 metres for MGA94).

Northing

N’

Measured from the equator, negative southwards

N

Measured from the false origin (N' in the northern hemisphere; N' + 10,000,0000 metres in the southern hemisphere for MGA94).

Grid convergence

g

Angular quantity to be added algebraically to an azimuth to obtain a grid bearing:
Grid Bearing = Azimuth + Grid Convergence. In the southern hemisphere, grid convergence is positive for points east of the central meridian (grid north is west of true north) and negative for points west of the central meridian (grid north is east of true north).

Grid Bearing

b

Angle between grid north and the tangent to the arc at the point. It is measured from grid north clockwise through 360°.

Arc-to-chord Correction

d

Angular quantity to be added algebraically to a grid bearing to obtain a plane bearing:
q = b + d = a + g + d
The arc-to-chord corrections differ in amount and sign at either end of a line. Lines that do not cross the central meridian always bow away from the central meridian. In the rare case of a line that crosses the central meridian less than one-third of its length from one end, the bow is determined by the longer part. Note that D b = d ₁ - d ₂ and the sign is defined by the equations:
q = b + d = a + g + d .
The arc-to-chord correction is sometimes called the 't-T' correction.

Meridian convergence

D a

The change in the azimuth of a geodesic between two points on the spheroid:
Reverse Azimuth = Forward Azimuth + Meridian Convergence ± 180°:
a ₂₁ = a ₁₂ + D a ± 180°

Line curvature

D b

The change in grid bearing between two points on the arc.
Reverse grid bearing = Forward grid bearing + Line curvature ± 180°:
b ₂ = b ₁ + D b ± 180.

Plane bearing

q

The angle between grid north and the straight line on the grid between the ends of the arc formed by the projection of the ellipsoidal distance; measured clockwise through 360°.

Grid distance

S

The length measured on the grid, along the arc of the projected ellipsoid distance.

Plane distance

L

The length of the straight line on the grid between the ends of the arc of the projected ellipsoidal distance. The difference in length between the plane distance (L) and the grid distance (S) is nearly always negligible.
Using plane bearings and plane distances, the formulae of plane trigonometry hold rigorously: tan q = D E/ D N; D E = L sin q ; D N = L cos q .

Meridian distance

m

True distance from the equator, along the meridian, negative southwards.

G

Mean length of an arc of one degree of the meridian.

s

Meridian distance expressed as units G: s = m/G

Central scale factor

k₀

Scale factor on the central meridian ( 0.9996 for MGA94)

Point scale factor

k

Ratio of an infinitesimal distance at a point on the grid to the corresponding distance on the spheroid: k = dL/ds = dS/ds
It is the distinguishing feature of conformal projections, such as the Universal Transverse Mercator used for MGA94, that this ratio is independent of the azimuth of the infinitesimal distance.

Line scale factor

K

Ratio of a plane distance (L) to the corresponding ellipsoidal distance (s):
K = L/s » S/s.
The point scale factor will in general vary from point to point along a line on the grid,

Ellipsoidal height

h

Ellipsoidal Height (h) is the distance of a point above the ellipsoid, measured along the normal from that point to the surface of the ellipsoid used.

Dh

Change in ellipsoidal height (m)

Height above the geoid

H

Height of a point above the geoid measured along the normal from that point to the surface of the geoid. It is also referred to as the orthometric height.

Geoid-ellipsoid separation

N

Distance from the surface of the ellipsoid used, to the surface of the geoid measured along the normal to this ellipsoid. This separation is positive if the geoid is above the ellipsoid and negative if the geoid is below the ellipsoid.
h - H = geoid ellipsoid separation.

Earth-centred Cartesian coordinates.

X, Y, Z

A three dimensional coordinate system which has its origin at (or near) the centre of the earth. These coordinates are commonly used for satellite derived positions (e.g. GPS) and although they relate to a specific reference system they are independent of any ellipsoid.
The positive Z axis coincides with (or is parallel to) the earth’s mean axis of rotation and the X and Y axes are chosen to obtain a right-handed coordinate system; for convenience it can be assumed that the positive arm of the X axis passes through the Greenwich meridian.

Transformation parameters

Da

Change in ellipsoid semi-major axis (e.g. from ANS to GRS80) (m)

Df

change in ellipsoid flattening (e.g. from ANS to GRS80)

DX

origin shift along the X axis (m)

DY

origin shift along the Y axis (m)

DZ

origin shift along the Z axis (m)

Rx

Rotation of the X axis (radians); positive when anti-clockwise as viewed from the positive end of the axis looking towards the origin.

Ry

Rotation of the Y axis (radians); positive when anti-clockwise as viewed from the positive end of the axis looking towards the origin.

Rz

Rotation of the Z axis (radians); positive when anti-clockwise as viewed from the positive end of the axis looking towards the origin.

Sc

Change in scale (parts per million - ppm).