A civil engineer is designing a new high-speed rail line and encounters three different alignment challenges:

Recommend the specific type of horizontal curve required for each challenge.

A highway designer is routing a mountain road. They need to connect two tangent sections that meet at an awkward angle. They have two options:

Explain the safety and superelevation (banking) challenges associated with a Reverse Curve compared to a Compound Curve, and why a tangent section is usually required between them.

The intersection angle ($I$) of two tangents is $40^\circ 00'$. The radius ($R$) of the proposed simple circular curve is $300.00 \text{ m}$.

Calculate the Tangent Distance ($T$), the Length of Curve ($L_c$), the Long Chord ($LC$), and the Middle Ordinate ($M$).

Using the data from Problem 1 ($I = 40^\circ 00'$, $R = 300.00 \text{ m}$, $T = 109.19 \text{ m}$, $L_c = 209.44 \text{ m}$): The stationing of the Point of Intersection (PI) is $10 + 250.00$ ($10250.00 \text{ m}$ from the project start).

Determine the stationing of the Point of Curvature (PC) and the Point of Tangency (PT). Use the metric standard where 1 Station = $20 \text{ m}$ (represented as $1+000$ for $1000 \text{ m}$ or $0+020$ for $20 \text{ m}$).

A compound curve consists of two simple curves ($C_1$ and $C_2$). The first curve has a radius $R_1 = 400.00 \text{ m}$ and an intersection angle $I_1 = 30^\circ$. The second curve has a radius $R_2 = 250.00 \text{ m}$ and an intersection angle $I_2 = 45^\circ$.

Calculate the tangent distances of each individual curve ($t_1$ and $t_2$) and the total length of the compound curve ($L_{total}$).

A symmetrical parabolic crest vertical curve has a length ($L$) of $200 \text{ m}$. The incoming grade ($g_1$) is $+3.0\%$. The outgoing grade ($g_2$) is $-2.0\%$. The Point of Vertical Intersection (PVI) is at Station $5 + 000$ with an elevation of $150.00 \text{ m}$.

Calculate the elevations of the Point of Vertical Curve (PVC), the Point of Vertical Tangency (PVT), and the highest point on the curve.