Derivation of formula to set out curves by the method of ordinates from a long chord./ Deriving long chord method for ordinates to draw a curve.

 When two roads intersect, we set out a curve to connect them by measuring the offsets or ordinates, as shown below.

As you can observe in the above drawing, AP & BP are the two roads intersecting at point P. By surveying, the two roads are joined by setting out a curve T1CT2 for smooth transportation.

 Now, let us derive a formula to calculate the ordinates (or offsets) from a long chord to set a curve.

 As you can observe in the above drawing,

Long chord = TIT2 = L

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

Mid-ordinate = CD =Oo

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

Now,

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

In triangle T1DO,

T1O² = [T1D² + OD²]

( By Pythagoras theorem. )

 or

OD² = [T1O² - T1D²]

Here, T1D = Half of long chord = L/2, 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 OC & OD,

Oo = [ R - √ R² - (L/2)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 triangle OEG

OE² = EG² + OG²

( By Pythagoras theorem. )

 Here,

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

OE = Radius of curve = R

Therefore,

R² = x² + OG²

OG² = R² -x²

OG = √ (R² -x²)

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

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

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

Let us rewrite the formulas for further reference as follows

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

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

Note: Whatever may be the distance of x over the long chord, you will get the measurement of ordinates Ox from that point.

To understand the concept correctly, go through the solved problems as linked below.

👀.  How to calculate the offsets for a curve from a long chord?

👀.  How to calculate the ordinates for a curve by the long chord method?

Thank you for going through these calculation steps❤. Have a good day 😄.

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