##### Horizontal Curve and its Components

A horizontal curve provides a transition between two tangent strips of roadway, allowing a vehicle to negotiate a turn at a gradual rate rather than a sharp cut. The design of the curve is dependent on the intended design speed for the roadway, as well as other factors including drainage and friction. These curves are semicircles as to provide the driver with a constant turning rate with radius.

#### Geometric Properties

Horizontal curves occur at locations where two roadways intersect, providing a gradual transition between the two. The intersection point of the two roads is defined as the **Point of Tangent Intersection**. The location of the curve’s start point is defined as the **Point of Curve** while the location of the curve’s end point is defined as the **Point of Tangent**.

Central angle of the curve is ɸ . This angle is equal to the supplement of the interior angle or angle of intersection between the two road tangents.

Here

- ADC is chord length
- ABC is curve length
- BD is mid ordinate/ versed sine
- BE is apex distance
- AE is tangent length
- OA/OB/OC is radius(R)

#### Chord Length (ADC)

We know, P/H = sinѲ

In triangle ADO,

AD/AO = sinɸ/2

AD = R sinɸ/2

where AD =AC

therefore ADC = AD + AC = 2AD = 2(R sinɸ/2)

**Chord Length = ADC = 2(R sinɸ/2)**

#### Curve length (ABC)

We know, Ѳ = Arc / Radius

ɸ = ABC / R

ABC = Rɸ = Rɸ (pi/180)

**Curve length = ABC = Rɸ (pi/180) **

#### Mid ordinate/ Versed sine (BD)

BD = OB – OD

Here in triangle ADO,

We know B/H = cosѲ

OD / R = cosɸ/2

OD = R cosɸ/2

So, BD = R – (R cosɸ/2) = R (1 – cosɸ/2)

**Mid ordinate/ Versed sine = BD = R (1 – cosɸ/2)**

#### Apex distance (BE) OR (EB)

EB = OE – OB

Here in triangle AEO,

We know B/H = cosѲ

Or H/B = secѲ

OE/R = secɸ/2

OE = R (secɸ/2)

So, EB = R (secɸ/2) – R = R (secɸ/2 – 1)

**Apex distance = BE or EB = R (secɸ/2 – 1)**

#### Tangent length (AE)

Here in triangle AEO,

We know P/B = tanѲ

AE/R = tanɸ/2

AE = R (tanɸ/2)

**Tangent length = AE = R (tanɸ/2) **

#### Radius of curve (R) when AD (chord length = a) and BD (mid ordinate = h) is known

R = (a^2 + h^2) / 2h

**Radius of curve = R = (a^2 + h^2) / 2h**