3.2 Plane and Spherical Trigonometry for Surveying and Astronomical Applications

3.2 Plane and Spherical Trigonometry for Surveying and Astronomical Applications

Trigonometric analysis is the cornerstone of geodetic positioning and field astronomy. While plane trigonometry assumes a flat surface (valid for local boundary and construction surveys covering less than $250 \text{ km}^2$), spherical trigonometry accounts for Earth's curvature on large geodetic control networks and celestial observations.

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Plane Trigonometry Laws in Surveying Computations

Plane triangles formed by survey control stations are solved using standard trigonometric relations.

Law of Sines and Law of Cosines

For any plane triangle with side lengths $a, b, c$ opposite to interior angles $A, B, C$:

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Law of Cosines for Sides: $a^2 = b^2 + c^2 - 2 b c \cos A$ $b^2 = a^2 + c^2 - 2 a c \cos B$ $c^2 = a^2 + b^2 - 2 a b \cos C$

Area of Plane Triangles

• Side-Angle-Side Area Formula: $\text{Area} = \frac{1}{2} a b \sin C$
• Heron's Formula (Three Sides Known): $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \frac{a+b+c}{2}$

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Right Spherical Triangles and Napier's Rules

A spherical triangle is formed on the surface of a sphere by three intersecting great circle arcs. The sides $a, b, c$ are measured as central angles in angular units (degrees, minutes, seconds, or radians). The interior angles at vertices are denoted $A, B, C$.

Napier's Circle of Circular Parts

For a right spherical triangle where $C = 90^\circ$, Napier's Circle arranges five circular parts in sequence, excluding the right angle $C$:

1. Side $a$
2. Side $b$
3. Complement of Angle $B$: $\bar{B} = 90^\circ - B$
4. Complement of Hypotenuse $c$: $\bar{c} = 90^\circ - c$
5. Complement of Angle $A$: $\bar{A} = 90^\circ - A$

      Napier's Circle (5 Parts):
               [a]
         /             \
     [b]                 [90° - A]
       \                 /
   [90° - B] ------- [90° - c]

The Two Fundamental Rules of Napier

• SIN-CO-OP Rule: The sine of any middle part equals the product of the cosines of the opposite parts: $\sin(\text{middle part}) = \cos(\text{opposite}_1) \times \cos(\text{opposite}_2)$
• SIN-CO-AD Rule: The sine of any middle part equals the product of the cotangents of the adjacent parts: $\sin(\text{middle part}) = \cot(\text{adjacent}_1) \times \cot(\text{adjacent}_2)$

Summary Table of Napier's Formulas

Worked Example: Solving a Right Spherical Triangle

Problem: In a right spherical triangle ($C = 90^\circ$), side $a = 30^\circ 00'$ and side $b = 45^\circ 00'$. Compute the length of hypotenuse side $c$.

Solution:

1. Select the Napier formula relating hypotenuse $c$ to legs $a$ and $b$: $\cos c = \cos a \cos b$
2. Substitute values $a = 30^\circ$ and $b = 45^\circ$: $\cos c = \cos(30^\circ) \cos(45^\circ) = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{6}}{4} \approx 0.612372$
3. Take inverse cosine: $c = \arccos(0.612372) \approx 52.24^\circ$

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Oblique Spherical Triangles

When none of the interior angles of a spherical triangle equals $90^\circ$, oblique spherical formulas are required.

Spherical Law of Sines

$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B} = \frac{\sin c}{\sin C}$

Spherical Law of Cosines for Sides

$\cos a = \cos b \cos c + \sin b \sin c \cos A$ $\cos b = \cos a \cos c + \sin a \sin c \cos B$ $\cos c = \cos a \cos b + \sin a \sin b \cos C$

Spherical Law of Cosines for Angles

$\cos A = -\cos B \cos C + \sin B \sin C \cos a$ $\cos B = -\cos A \cos C + \sin A \sin C \cos b$ $\cos C = -\cos A \cos B + \sin A \sin B \cos c$

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Spherical Excess, Girard's Theorem, and Legendre's Theorem

The sum of interior angles of a spherical triangle always exceeds $180^\circ$ ($180^\circ < A + B + C < 540^\circ$).

Spherical Excess ($E$)

$E = A + B + C - 180^\circ$

Girard's Theorem for Surface Area

The surface area of a spherical triangle on a sphere of radius $R$ is directly proportional to its spherical excess in radians ($E_{\text{rad}}$):

$\text{Area} = R^2 E_{\text{rad}} = \frac{\pi R^2 E^\circ}{180^\circ}$

Legendre's Theorem for Small Geodetic Triangles

Legendre's Theorem states that if a spherical triangle whose sides are small compared to the sphere radius $R$ is replaced by a plane triangle having the exact same side lengths $a, b, c$, the interior angles of the plane triangle ($A', B', C'$) equal the spherical angles reduced by one-third of the spherical excess:

$A' = A - \frac{E}{3}, \quad B' = B - \frac{E}{3}, \quad C' = C - \frac{E}{3}$

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The Astronomical P-Z-S Celestial Triangle

Field astronomy uses the celestial sphere to establish true North azimuths, latitude $\phi$, and longitude $\lambda$ by observing stars or the Sun. The fundamental celestial triangle $P-Z-S$ connects three points:

1. $P$ (North Celestial Pole): Point where Earth's rotation axis meets the celestial sphere.
2. $Z$ (Observer's Zenith): Point directly above the observer's instrument station.
3. $S$ (Celestial Body / Star / Sun): Observed star or celestial object.

       Celestial Triangle P-Z-S:
                 P (Pole)
                / \
               /   \
   (90° - φ)  /     \  (90° - δ)
             /       \
            /    t    \
           Z-----------S
            (Zenith)  (Star)
              (90° - h)

Side Lengths of Triangle P-Z-S

• Side $PZ = 90^\circ - \phi$ (Co-latitude)
• Side $PS = 90^\circ - \delta$ (Co-declination or Polar Distance, where $\delta$ is star declination)
• Side $ZS = 90^\circ - h = z$ (Co-altitude or Zenith Distance $z$, where $h$ is corrected altitude)

Interior Angles of Triangle P-Z-S

• Angle at Pole $P = t$ (Local Hour Angle of the star)
• Angle at Zenith $Z = A$ (Azimuth angle of the star measured from North)
• Angle at Star $S = q$ (Parallactic angle)

Determination of True Azimuth ($A$)

Applying the Spherical Law of Cosines for side $PS$ ($90^\circ - \delta$):

$\cos(90^\circ - \delta) = \cos(90^\circ - \phi) \cos z + \sin(90^\circ - \phi) \sin z \cos A$ $\sin \delta = \sin \phi \cos z + \cos \phi \sin z \cos A$

Rearranging to solve for the True Star Azimuth Angle $A$:

$\cos A = \frac{\sin \delta - \sin \phi \cos z}{\cos \phi \sin z}$

Worked Example: True Star Azimuth Calculation

Problem: An observer at latitude $\phi = 20^\circ 00' \text{ N}$ measures a star's corrected zenith distance $z = 30^\circ 00'$. The star's tabulated declination is $\delta = +40^\circ 00'$. Calculate the star's true azimuth angle $A$.

Solution:

1. Identify given angles: $\phi = 20^\circ$, $\delta = 40^\circ$, $z = 30^\circ$.
2. Calculate trigonometric values: $\sin(40^\circ) = 0.642788, \quad \sin(20^\circ) = 0.342020, \quad \cos(20^\circ) = 0.939693$ $\sin(30^\circ) = 0.500000, \quad \cos(30^\circ) = 0.866025$
3. Substitute into the azimuth cosine formula: $\cos A = \frac{0.642788 - (0.342020 \times 0.866025)}{0.939693 \times 0.500000}$
4. Evaluate numerator: $\text{Numerator} = 0.642788 - 0.296198 = 0.346590$
5. Evaluate denominator: $\text{Denominator} = 0.939693 \times 0.5 = 0.469846$
6. Compute $\cos A$ and inverse cosine: $\cos A = \frac{0.346590}{0.469846} \approx 0.737667$ $A = \arccos(0.737667) \approx 42.46^\circ$