Derivation of formula to set out the curves by the method of ordinates from a long chord

Derivation of formula to set out the curves by the Method of ordinates from a long chord.

When two roads intersect, a curve is established to connect them by measuring the offsets or ordinates as illustrated below. Derivation of formula to set out the curves by the method of ordinates from a long chord.

In the provided illustration, it is evident that AP and BP represent the two intersecting roads at point P. Through careful surveying, these 2 roads have been connected by establishing a curve T1CT2 to ensure the seamless transportation.

Now, we shall proceed to derive the formula that enables the calculation of the ordinates (or offsets) from a long chord in order to establish a curve.

Upon reviewing the illustration provided above, it is evident that..

Long chord = TIT2 = L

The radius of curve = OT1, OT2, or OE = R

The Mid-ordinate = CD =Oo

Ordinates at the distance x from the mid-ordinate = EF = Ox

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Now

The Mid-ordinate = Oo = CD = [OC – OD]

In the triangle T1DO,

T1O² = ( T1D² + OD² )

( By the Pythagoras theorem. )

or

OD² = ( T1O² – T1D² )

Here, T1D = Half of long chord = L/two, TIO = Radius = R

Therefore

OD² = {R² – (L/2)²}

OD = {√ R² – (L/2)²}

(1) Mid-ordinate 

= Oo = CD = [OC – OD]

Here, OC = Radius of the curve = R

Substituting the values of the OC & OD,

Oo = [ R – √ R² – (L/two)²]   ———– ①

From the above drawing,

EF=GD=Ox

GD = [OG – OD]

or

OG = [GD + OD]

Substituting the values of GD & OD

OG = [Ox + {√ R² – (L/2)²}]

or

Ox = [OG – {√ R² – (L/2)²}]

In the triangle OEG

OE² = EG² + OG²

( By the Pythagoras theorem. )

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 Here,

EG = Distance of the ordinates from mid-ordinate over long chord = x

OE = Radius of the curve = R

Therefore,

R² = x² + OG²

OG² = R² -x²

OG = √ (R² -x²)

Substituting the value of the OG in the above-derived equation, ©

Ox = [√ (R² -x²)  – {√ R² – (L/two)²}]  ———– ② 

The equations ① and ② are the formulas required to find out the mid-ordinate, ordinates, radius, etc. to set out the curve.

Let us rewrite the formulas for the further reference as follows

The Mid-ordinate = Oo = [ R – √ R² – (L/two)²] 

Ordinates = Ox = [√ (R² -x²)  – {√ R² – (L/two)²}]

Note:

Regardless of distance x along the long chord, the measurement of the ordinates Ox can be obtained from that point.

To gain a proper understanding of the concept, it is advisable to review the solved problems provided in the links below.

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