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Understanding Circular Curves in Surveying: Computations of Chords, Arcs, and Angles

This study guide delves into essential concepts of circular curves within surveying, focusing on the computations of arcs, chords, and angles. Key topics include the geometric properties of circular arcs, area calculations for sectors and segments, and chord relationships. The text discusses the intersection of lines, tangent lengths, and various area formulas for triangles and segments related to circular curves to aid in precise surveying work. The importance of understanding bearings, deflection angles, and subtended angles is also highlighted for comprehensive surveying analysis.

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Understanding Circular Curves in Surveying: Computations of Chords, Arcs, and Angles

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  1. E4004 Surveying Computations A Circular Curves

  2. Circular Curves - Chord • Let O be the centre of circular arc AC • Arc AC subtends an angle of D at O • Line AC is a chord A chord C O

  3. Circular Curves - Sector • Area OACO is a sector A C O

  4. Circular Curves - Segment • Area ACA is a segment A C O

  5. Circular Curves • Let X bisect the chord AC such that • AX = XC • Angle AXO is 90° • AX cuts the curve AC at B B A X C O

  6. Circular Curves - Arc Length B A X C R R O

  7. Circular Curves - Area Circle B A X C O

  8. Circular Curves - Sector Area B A X C R R O

  9. Circular Curves - Sector Area B A X C R R O

  10. Circular Curves - Triangle Area B A X C R R O

  11. Circular Curves - Segment Area B A X C R R O

  12. Circular Curves - Chord Length • Consider triangle AXO B A X C R R O

  13. Circular Curves - Intersection Pt • Let AI have bearing B1 and IC have bearing B2 - note the bearing directions • Extend the tangents at A and C to intersect at I I B1 B2 B A X C • X and B lie on the line OI R R O

  14. Circular Curves - Tangent Length • Consider triangle AIO I B1 B2 B A X C R R O

  15. Circular Curves - Area OAICO • Again consider triangle AIO I B1 B2 B A X C R R O

  16. Circular Curves - Area Outer Segment • Consider Area AICBA - “Outer Segment” I B1 B2 B A X C R R O

  17. Circular Curves - Secant Distance BI I B1 B2 B A X C R R O

  18. Circular Curves - Deflection Angle & Subtended Angle Extend AI to D Consider the quadrilateral OAIC D Quadrilateral internal angles = 360° I B1 B2 B A Line AID is straight X C R R O

  19. Circular Curves - Deflection Angle & Subtended Angle Provided bearings are expressed in one direction D I B1 B2 B A X C R R O

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