Lecture 2

Trigonometric Leveling

• Trigonometric leveling determines elevation differences by measuring slope distance (or horizontal distance) and zenith angle (or altitude angle).
• The zenith angle is the vertical angle to the target, measured from a horizontal plane at the instrument's position.
• As the scope of the instrument rotates to sight on a prism, the zenith angle increases in size.
• The zenith angle to a point directly above the instrument is 0°
• Zenith angle less than 90° indicates an uphill sight.
• Zenith angle equal to 90° indicates a level sight (horizontal).
• Zenith angle greater than 90° indicates a downhill sight

Calculating Difference in Elevation (ΔH)

• ΔH = S * cos(Z), where S is the slope distance and Z is the zenith angle (converted to decimal degrees)
  • Use the 8th decimal place for the conversion of zenith angles from degrees, minutes, and seconds to decimal to avoid errors.
• ΔH = H * tan(α), where H is the horizontal distance and α is the altitude angle (converted to decimal degrees).
• For slope distances over 1000 feet, use the formula ΔH = (SD) (cos Z) + (0.0206)[(SD/1000) (sin Z)²] to correct for curvature and refraction.
  • where SD is the measured slope distance and Z is the zenith angle in decimal degrees.

Leveling Instrument Height and Prism Height

• To eliminate calculation errors, the instrument height (hi) and prism height (r) should remain consistent (equal).

Level Vial Types

• Tube Level: A glass tube precisely shaped to a specific radius, containing a small air bubble. The bubble moves towards the higher elevation when the tube is tilted. A larger radius leads to increased sensitivity.
• Bull's-Eye Level: Spherical-shaped level vial with a concentric circle. Centering the air bubble within the circle indicates horizontal alignment, but with lower sensitivity than the tube level.

Tilting Levels

• Used for precise leveling. Levelling screws are used for approximate leveling followed by precise leveling using prisms to view the bubble.

Automatic Levels

• Feature a three-screw head to quickly center the bull's-eye bubble.
• An automatic compensator maintains the level line of sight (horizontal).

Digital Levels

• Electronic and most modern automatic levels.
• Read directly using electronic displays (bar code).
• Store readings in memory.

Tripods

• Essential component for stable leveling instrument setup.

This quiz covers the fundamentals of trigonometric leveling, including the measurement of elevation differences using slope distances and zenith angles. Understand how to calculate differences in elevation and the significance of zenith angles in the leveling process.