E4004 Surveying Computations A

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