Vertical curves are essential in roadway design because they provide smooth transitions between varying roadway grades. Designing vertical curves involves calculating intermediate elevations and identifying the curve's highest or lowest point, which is essential for optimal roadway performance.

Intermediate elevations on a vertical curve are determined using the tangent offset method. This method considers the initial elevation at the start of the curve, the grades, and the curve's geometry. The highest or lowest point of the curve occurs where the slope of the roadway changes to zero. This point typically lies between the curve's start and end, and identifying it requires analyzing the grade change across the curve.

1. Calculate the rate of grade change, also known as the vertical curve's curvature. This is obtained by dividing the total grade difference by the length of the curve.
2. Next, determine the distance from the curve's start to the critical point by dividing the initial grade by the rate of grade change.
3. Compute the elevation at this point by adding the tangent grade-line elevation to the tangent offset. 

Accurate calculation of these elevations is essential for ensuring effective drainage and safety. In drainage systems or high-speed zones, precise elevation data prevents water accumulation and provides adequate sight distance. Additionally, these calculations support the creation of safe, smooth roadway profiles, enhancing overall driving conditions.