E4004 Surveying Computations A Circular Curves

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

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

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

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

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

Circular Curves - Area Circle A C O X B

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

A C O X B R R

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

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

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

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

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

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

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

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

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

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