Simple Curve External Distance

In highway or railway alignment, a simple curve is a circular arc connecting two tangent lines. Key points on a simple curve include:

• Point of Curve (PC): Where the curve begins and the tangent ends.
• Point of Tangency (PT): Where the curve ends and the tangent begins.
• Vertex (V) or Point of Intersection (PI): The point where the two tangents intersect.
• Middle point of the curve (M): The point on the curve midway between PC and PT along the arc.
• Center of the curve (O): The center of the circle of which the curve is an arc.

Several distances are used to define and lay out a simple curve. These include:

• Tangent Length (T): The distance from the Vertex (V) to the Point of Curve (PC) or Point of Tangency (PT). It is given by $T = R \tan(\frac{\Delta}{2})$, where R is the radius and $\Delta$ is the deflection angle (angle between the tangents).
• Long Chord (L): The straight line distance between the PC and the PT. It is given by $L = 2R \sin(\frac{\Delta}{2})$.
• External Distance (E): This is the distance from the Vertex (V) to the middle point of the curve (M). It is measured along the line from the Vertex towards the center of the curve. This distance is also sometimes called the Apex Distance or Rise.
• Middle Ordinate (M.O.): The distance from the middle point of the long chord to the middle point of the curve (M). It is measured perpendicular to the long chord.

Defining External Distance

Based on the definitions, the external distance is specifically the distance between the Vertex (V) and the middle point of the curve (M). This distance represents how far the curve "bows out" from the point where the tangents would have met.

The formula for External Distance (E) is:

\( E = R \left( \sec\left(\frac{\Delta}{2}\right) - 1 \right) \)

Where:

• \( R \) is the radius of the curve
• \( \Delta \) is the deflection angle (the angle at the Vertex between the two tangents)

Analyzing the Options

Let's look at the given options in light of this definition:

1. Vertex and middle point of the curve: This matches the definition of External Distance (E).
2. Vertex and point of the curve: A simple curve has infinitely many points. This is not a specific defining distance like the external distance.
3. Point of the curve and point of tangency: This describes a segment of the curve itself, not the external distance from the vertex.
4. Vertex and the center of the curve: This distance is the distance from V to O. Using the geometry, this distance is $R / \cos(\Delta/2)$, which is $R \sec(\Delta/2)$. This is different from the external distance E.

Therefore, the external distance is defined as the distance between the Vertex and the middle point of the curve.