Vertical Curve Calculator

Created by Mihir Ketkar
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Oct 26, 2022

The vertical curve calculator helps you find the elevation of points on a curve, which may be useful when planning a transition between two sloped roadways. While such curves may be circular or parabolic, our tool can only use the vertical curve formula for the latter case, which is more common.

Read the following article to understand how our calculator works, learn what vertical curve equations are, and how to use the length of the vertical curve formula in practice under different scenarios.

What is a vertical curve?

A vertical curve is a transition between two sloped roadways. Due to its wide range of applications, it is one of the most important calculations in the field of civil engineering when it comes to road construction.

But before we can go ahead with the equations, it would be a good idea to familiarize ourselves with a few basic concepts related to vertical curves.

Basic terminology

These are the definitions we need to understand the length of vertical curve formula:

  1. Gradient - A point's gradient is basically how steep the curve is at that point. We also refer to it as the point's inclination. The gradient at point X is denoted by gxg_x in the vertical curve formula. The value of the gradient is a unitless number (a rational number, to be precise) and is usually reported as a percentage.

You can learn more about gradient and slope in our elevation grade calculator or our angle of depression calculator. Note that elevation grade and elevation are two different things.

  1. Elevation - The height at which the given point stands, as measured from a reference point (usually sea-level). In the vertical curve equation, it is denoted by ExE_x.

  2. BVC - stands for the Beginning of Vertical Curve. It denotes the point at which the vertical curve starts. There are two values associated with the BVC:

    • EBVCE_\text{BVC} - The elevation of the BVC; and

    • g1g_1 - The gradient at the BVC.

  3. EVC - stands for End of Vertical Curve. It denotes the point at which the vertical curve ends. There are two values that are associated with the EVC:

    • EEVCE_\text{EVC} - The elevation of the EVC.

    • g2g_2 - The gradient at the EVC.

  4. Length of curve - We define the horizontal distance between the BVC and the EVC as the length of the vertical curve. To represent it, we use the letter LL.

  5. PVI - Stands for Point of Vertical Intersection. It is defined as the point where the gradient lines extending from BVC and EVC, g1g_1 and g2g_2, respectively, intersect. For symmetric vertical curves, the PVI is equidistant from the EVC and BVC.

What is the vertical curve formula?

For symmetric curves, the formula is as follows:

Ex=EBVC+g1×x+(g2g1)×x22×L\footnotesize E_\text{x}\! =\! E_\text{BVC}\! +\!g_1\!\times\! x\! +\! (g_2\! -\! g_1)\!\times\! \frac{x^2}{2\!\times\! L}

Here, xx is the horizontal distance of the point in question, and the other variables have their usual meanings, as explained above.

If you require the elevation of PVI, we can create a simpler formula by substituting the appropriate values into the vertical curve equation:

EPVI=EBVC+g1×L2\footnotesize E_\text{PVI} = E_\text{BVC} + g_1\times \frac{L}{2}

Similarly, we can find out the elevation of EVC through the same process. When we substitute the length of the curve LL in, we get:

EEVC=EBVC+g1×L+(g2g1)×L2\footnotesize E_\text{EVC}\! =\! E_\text{BVC}\! +\! g_1\! \times\! L\! +\! (g_2\! -\! g_1)\! \times\! \frac{L}{2}

How to use the vertical curve calculator?

Our vertical curve calculator is a straightforward tool. All you need to do is start filling in the values, and our elevation calculator will estimate the rest of them automatically. If you don't feel confident using it, check out the following instructions:

  1. To find the elevation of PVI using the vertical curve calculator, simply fill in the first four fields in the tool.

  2. Remember to check the units. Check the dropdown menu on the right-hand side of the box to see which units you can use.

  3. If you want to perform a series of calculations on similar values, lock certain values by tapping the gray region on the right of each field.

  4. Adjust the position of the EVC, BVC, and PVI using the "Horizontal distance" section of the calculator.

  5. To enter completely new values, click the refresh button. You can also use this elevation calculator in multiple directions, e.g., to find out what the length of the vertical curve formula is.

Loved using the vertical curve elevation calculator? Check out some of the other calculators we offer:

Mihir Ketkar
Elevation of BVC
Initial gradient (g₁)
Final gradient (g₂)
Length of curve
Elevation of PVI
Elevation of EVC
Horizontal distances
Horizontal distance of BVC
Horizontal distance of PVI
Horizontal distance of EVC
Find the elevation at any point on the vertical curve
Horizontal distance of point x
Elevation of point x
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