After the post about the limits of vertical acceleration, and even before, I was asked what was the original curve used to connect vertical gradients. What was first, the circle or the “egg”- the parabolic curve?
The expected answer was “the circular curve” because, presumably, a circle can be drawn easier than a parabola…

“How could our railway ancestors draw a parabola in the dark ages of pre-computerised era?
To draw a circle is easy … they used a compass!”

True! An arbitrary circle is easier to draw.
But this particular circle has one issue – for obvious reasons, from the early ages of railway track (and road) design, the vertical profile had a vertical distortion, the so called “anamorphic scale”.

On this distorted profile that circular curve becomes an ellipse. So, we are talking in fact about how to draw an ellipse not a circle.
Which of two can be drawn easier – the ellipse or the parabolic curve?

In the noble discipline of Descriptive Geometry there is the concept of ruled surfaces and ruled curves – those surfaces and curves that can be defined by moving straight lines in the 3D/2D space.
The parabolic curve is such a one – can be defined and drawn using straight lines.
How?
– Measure on each side of the point of vertical intersection half the length of the vertical curve
– Divide each gradient segment covered by this length in an equal number of segments
– And then connect them as shown below:

The curve drawn in this way fits exactly the definition of the vertical curve found in any alignment design standard.

It does not matter the vertical distortion nor the gradient of the two connected elements, this parabolic vertical curve could easily be drawn in this way with a ruler and a pencil.
Using this method the “ancient” designer was able to draw even a-symmetrical vertical parabolic curves, used on metro lines on sections with significant speed changes or over the hump of the marshalling yards.
To do this the designer needed only to compute for each of the connected gradients, G1 and G2, the lengths L1 and L2, related to the variable speeds at both ends of the vertical transition. Instead of measuring L/2 on the first and second gradients, the designer had to measure L1 for the first and L2 for the second and follow the steps shown in the animation above.
The result would be an asymmetrical vertical parabolic curve. Its length is L1 + L2.