Understanding Simple Circular Curve Ratios

This solution explains how to match the given ratios related to a simple circular curve with its defining parameters: radius (R) and deflection angle (Δ°).

Matching Item I: Ratio of Long Chord to Tangent Length

First, let's recall the standard formulas for the long chord (LC) and the tangent length (T) of a simple circular curve:

• Long Chord (LC): The straight line connecting the beginning and end points of the curve. Its formula is given by: $$ LC = 2R \sin\left(\frac{\Delta\textdegree}{2}\right) $$
• Tangent Length (T): The distance from the point of intersection (PI) of the tangents to the tangent point (either back tangent or forward tangent). Its formula is: $$ T = R \tan\left(\frac{\Delta\textdegree}{2}\right) $$

Now, we calculate the ratio of the long chord to the tangent length:

We know that $ \tan(x) = \frac{\sin(x)}{\cos(x)} $. Substituting this into the ratio:

Simplifying the expression by canceling out common terms ($R$ and $ \sin\left(\frac{\Delta\textdegree}{2}\right) $):

This result matches option C.

Matching Item II: Ratio of Long Chord to Length of Curve

Next, we need the formula for the length of the simple circular curve (L).

• Length of Curve (L): The actual length along the arc of the curve. For a deflection angle Δ° and radius R, the length is calculated as: $$ L = R \times \left(\frac{\Delta\textdegree}{180\textdegree} \times \pi \right) $$ This formula converts the deflection angle from degrees to radians ($ \theta_{rad} = \Delta\textdegree \times \frac{\pi}{180\textdegree} $) and then uses the arc length formula $ L = R \theta_{rad} $.

Now, we calculate the ratio of the long chord to the length of the curve:

Simplifying the expression:

To get the numerator factor of 360, we multiply the numerator and denominator by 180:

This result matches option D.

Conclusion

Based on the calculations:

• Item I (Ratio of long chord to tangent length) matches C. $ 2 \cos\left(\frac{\Delta\textdegree}{2}\right) $.
• Item II (Ratio of long chord to the length of the simple circular curve) matches D. $ \frac{360\textdegree \sin\left(\frac{\Delta\textdegree}{2}\right)}{\Delta\textdegree \pi} $.

Therefore, the correct matching is I - C, II - D.