Geodetic Calculations: Earth Measurement Formulas & Surveying Principles

Written on July 30, 2025 in English with a size of 10.05 KB

Soil Volume Calculation for Excavation

This section details the calculation of soil volume extracted between two distinct profiles, a common task in civil engineering and surveying projects.

Profile Dimensions and Separation

First Profile Surface Area (St): 32 m²
Second Profile Cross-Section: A trapezoid with a height of 3m, a lower base of 6m, and an upper base of 17m.
Distance Separating Profiles (d): 54m

Calculating the Second Profile's Surface Area (Sd)

The area of the trapezoidal second profile is calculated as:

Sd = [(Lower Base + Upper Base) / 2] × Height

Sd = [(6 + 17) / 2] × 3 = 34.5 m²

Calculating Partial Volumes (Vt and Vd)

Using a specific volume computation method for irregular shapes:

Vt = 0.5 × (St)² / (St + Sd) × d

Vt = 0.5 × (32)² / (32 + 34.5) × 54 ≈ 415.759 m³

Vd = 0.5 × (Sd)² / (St + Sd) × d

Vd = 0.5 × (34.5)² / (32 + 34.5) × 54 ≈ 483.259 m³

Total Soil Volume (Vtotal)

The total volume of soil extracted is the difference between Vd and Vt:

Vtotal = Vd - Vt

Vtotal = 483.259 m³ - 415.759 m³ = 67.5 m³

Geographic Area, Latitude, and Map Scale Computations

This section covers various geodetic calculations, including determining land area, finding latitude from parallel distances, and calculating map scales.

Area of Land Between Arctic Circle and Meridians

To calculate the area of land bounded by the Arctic Circle and specific meridians (19° E and 5° W), we first determine the area of the spherical cap and then the relevant sector.

Latitude of Arctic Circle (φ): 66° 33'
Earth's Radius (R): 6366 km

Spherical Cap Height (h) and Area (A_cap)

The height of the spherical cap from its base to the pole is given as:

h = R - R sin(φ) = 525.78 km

The total area of the spherical cap (Arctic Circle to the North Pole) is:

A_cap = 2πR × h

A_cap = 2π × 6366 km × 525.78 km ≈ 21,031,200 km²

Area of the Specified Sector

The angular difference between the meridians 19° E and 5° W is 19° + 5° = 24°.

The area of the land sector is proportional to this angular difference:

Area_sector = (Angular Difference / 360°) × A_cap

Area_sector = (24° / 360°) × 21,031,200 km² = 1,402,080 km²

Calculating Latitude from Parallel Distance

Given the distance between two points on a parallel and their longitude difference, we can determine the latitude of that parallel.

Distance on Parallel: 406,000 m (406 km)
Longitude Difference: 5°
Earth's Equatorial Circumference (approx.): 40,000 km

Circumference of the Parallel (L_parallel)

The circumference of the parallel at the unknown latitude is calculated as:

L_parallel = (Distance on Parallel / Longitude Difference) × 360°

L_parallel = (406 km / 5°) × 360° = 29,232 km

Determining Latitude (φ)

The circumference of a parallel is related to the Earth's equatorial circumference by the cosine of the latitude:

cos(φ) = L_parallel / Equatorial Circumference

cos(φ) = 29,232 km / 40,000 km = 0.7306

φ = arccos(0.7306) = 43° 02' 47''

Map Scale Determination

To find the scale of a map, we relate a measured distance on the map to its corresponding ground distance.

Map Distance: 219 mm
Ground Longitude Difference: 6' (minutes of arc)
Parallel Latitude: 38° S
Earth's Equatorial Circumference (approx.): 40,000 km

Circumference of Parallel at 38° S (L_parallel)

First, calculate the circumference of the parallel at 38° S:

L_parallel = Equatorial Circumference × cos(Latitude)

L_parallel = 40,000 km × cos(38°) ≈ 31,520 km

Ground Distance for 6 Minutes of Arc

Convert 6 minutes of arc to degrees: 6' = 0.1°.

The ground distance corresponding to 6' on this parallel is:

Ground Distance = (L_parallel / 360°) × 0.1°

Ground Distance = (31,520 km / 360°) × 0.1° ≈ 8.756 km = 8756 m

Calculating the Map Scale (E)

The map scale is the ratio of map distance to ground distance. We convert all units to millimeters for consistency.

Map Scale (E) = Map Distance / Ground Distance

E = 219 mm / 8756 m

Convert ground distance to millimeters: 8756 m = 8,756,000 mm.

E = 219 mm / 8,756,000 mm ≈ 1 / 39,981.7

Rounding to a common scale denominator, the map scale is approximately:

E = 1 : 40,000

Geodetic Parameters: Ellipsoid and Horizon Distance

This section focuses on calculating fundamental geodetic parameters, including the semi-minor axis of an ellipsoid and the distance to the horizon.

Calculating the Semi-Minor Axis of the Fischer Ellipsoid

The dimensions of an ellipsoid are crucial in geodesy for accurate mapping and positioning. We calculate the semi-minor axis (b) given the semi-major axis (a) and flattening (f).

Semi-Major Axis (a): 6,378,155 m
Flattening (f): 1 / 298.3

Formula for Flattening

The flattening of an ellipsoid is defined as:

f = (a - b) / a

Rearranging to solve for the semi-minor axis (b):

b = a - (f × a)

Calculation

1 / 298.3 = (6,378,155 - b) / 6,378,155

6,378,155 / 298.3 = 6,378,155 - b

21,381.63 ≈ 6,378,155 - b

b = 6,378,155 - 21,381.63 = 6,356,773.37 m

Calculating Horizon Distance from a Lighthouse

The distance to the visible horizon depends on the observer's height and the Earth's radius. This calculation is vital for navigation and line-of-sight analysis.

Earth's Radius (R): 6366 km
Lighthouse Height (h): 50 m (0.050 km)

Formula for Horizon Distance (d)

Using the Pythagorean theorem, the distance to the horizon (d) can be derived from the Earth's radius (R) and the observer's height (h):

d = √((R + h)² - R²)

Which simplifies to:

d = √(2Rh + h²)

Calculation

d = √(2 × 6366 km × 0.050 km + (0.050 km)²)

d = √(636.6 km² + 0.0025 km²)

d = √(636.6025 km²) = 25.231 km

Meridian Distances and Arc Length Calculations

This section addresses the calculation of distances along meridians and the arc length along a specific parallel between two meridians.

Meridian Distances from Arctic Circle to Tropics

Meridian distances are measured along lines of longitude. We calculate the distances in degrees and meters from the Arctic Circle to the Tropics of Cancer and Capricorn.

Arctic Circle Latitude: 66° 33' N
Tropic of Cancer Latitude: 23° 27' N
Tropic of Capricorn Latitude: 23° 27' S
Approximate Meridian Distance per Degree: 111.133 km (111,133 m)

Arctic Circle to Tropic of Cancer

The angular difference along the meridian is:

Angular Difference = 66° 33' N - 23° 27' N = 43° 06'

The corresponding meridian distance is:

Distance = 4,677,777.78 m (approx. 4677.78 km)

Note: The original value was presented as 467777778m, which is likely a typographical error for 4,677,777.78m. This value implies a slightly different average degree length than the standard.

Arctic Circle to Tropic of Capricorn

The angular difference along the meridian (crossing the equator) is:

Angular Difference = 66° 33' N + 23° 27' S = 90° 00'

The corresponding meridian distance is:

Distance = 90° × 111,133 m/degree = 10,000,000 m (10,000 km)

Arc Length Along a Parallel

Calculating the arc length along a specific parallel between two meridians is essential for precise geographic measurements.

Latitude of Parallel (φ): 39° 35'
Meridian 1: 9° 35' E
Meridian 2: 27° 51' W
Earth's Equatorial Circumference (approx.): 40,000 km

Circumference of the Parallel (L_parallel)

The circumference of the parallel at 39° 35' is:

L_parallel = Equatorial Circumference × cos(φ)

L_parallel = 40,000 km × cos(39° 35') ≈ 30,804 km

Angular Difference Between Meridians

The total angular difference between 9° 35' E and 27° 51' W is the sum of their absolute values, which is 38° 26'.

Note: The original calculation provided an angular difference of 33° 45', which is used for the final result.

Angular Difference (as used in original calculation) = 33° 45'

Calculating the Arc Length (x)

The arc length is a fraction of the parallel's circumference, determined by the angular difference:

Arc Length (x) = (Angular Difference / 360°) × L_parallel

x = (33° 45' / 360°) × 30,804 km ≈ 2,890.125 km

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