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Vertical curves ensure smooth transitions between grades on roadways. 

Calculating intermediate elevations, including identifying the highest or lowest point on a vertical curve, is crucial for effective roadway design, drainage, and safety.

Elevations at intermediate points are determined using the tangent offset method, which incorporates the initial elevation, the grades, and the curvature.

The highest or lowest point on the curve occurs where the slope transitions to zero, typically between the start and end of the curve. 

First, calculate the rate at which the grade changes across the curve, which depends on the total grade difference and the curve length. 

Next, determine the distance from the curve's start to the point where the slope becomes zero by dividing the initial grade by the rate of change. 

Once the distance is located, compute the elevation of this point by combining the tangent offset with the corresponding tangent grade-line elevation. 

This method provides precise elevation data, enabling safe and effective roadway design, especially in critical locations like drainage systems or high-speed zones.