Combined Scale Factor

The Combined Scale Factor (CSF) converts between grid coordinates (on the map projection) and ground/plane coordinates (real-world distances). NullCad uses the following formula for MGA (Map Grid of Australia) / UTM projections.

Formula

CSF = PSF × HSF

Where:

• CSF = Combined Scale Factor
• PSF = Point Scale Factor (accounts for map projection distortion)
• HSF = Height Scale Factor (accounts for elevation above the ellipsoid)

Point Scale Factor (PSF)

PSF = 0.9996 + 1.23 × (E - 500000)² × 10⁻¹⁴

Where:

• E = Easting coordinate (metres)
• 500000 = False easting (central meridian offset for UTM/MGA zones)
• 0.9996 = Central meridian scale factor (k₀) for UTM

PSF equals exactly 0.9996 at the central meridian (E = 500,000m) and increases as you move east or west.

Height Scale Factor (HSF)

HSF = 1 - (H × 0.1571 × 10⁻⁶)

Where:

• H = Ellipsoidal height (metres)
• 0.1571 × 10⁻⁶ ≈ 1 / 6,371,000 (approximation of 1/Earth radius)

HSF reduces distances as elevation increases, since higher points are further from Earth’s centre.

Implementation (pseudocode)

def get_csf(easting, northing, height):
    # Point Scale Factor
    psf = 0.9996 + 1.23 * (easting - 500000)**2 * 1e-14

    # Height Scale Factor
    hsf = 1 - (height * 0.1571e-6)

    # Combined Scale Factor
    return psf * hsf

Usage

Worked example

For a point at E 456,789.000, N 6,123,456.000, H 150.000:

PSF = 0.9996 + 1.23 × (456789 - 500000)² × 10⁻¹⁴
    = 0.9996 + 1.23 × 1,867,422,121 × 10⁻¹⁴
    = 0.9996 + 0.00002297
    = 0.99982297

HSF = 1 - (150 × 0.1571 × 10⁻⁶)
    = 1 - 0.00002357
    = 0.99997643

CSF = 0.99982297 × 0.99997643
    = 0.99979941

A 100m ground distance would be 99.9799m on the grid.

Where to configure it

The active CSF is set in the survey settings and applied by the Traverse tool when converting between grid and ground distances.

Caveat

This formula is a simplified approximation suitable for most surveying work within a single UTM/MGA zone. For high-precision geodetic work spanning zone boundaries or requiring rigorous error bounds, use the full Transverse Mercator equations instead.