A **vertical curve calculator** is used in **road design** and **construction** to determine the **smooth transition** between two road grades. It helps engineers design **safe** and **comfortable roadways** by ensuring **gradual changes** in elevation.

**Examples of vertical curves include:**

**Crest curve**: Where two upward slopes meet, or an upward slope transitions to a downward slope.**Sag curve**: Where two downward slopes meet, or a downward slope transitions to an upward slope.

These calculators typically consider factors such as:

**Starting and ending grades****Length of the curve****Station and elevation**of the**Point of Vertical Intersection (PVI)**

## Vertical Curve Calculator

**PVC Station**: 10+00**PVC Elevation**: 100.00 feet**G₁**: 2%**G₂**: -3%**L**: 300 feet

Station | x (ft) | Tangent Elevation (ft) | Curve Elevation (ft) | E (ft) |
---|---|---|---|---|

10+00 | 0 | 100.00 | 100.00 | 0.00 |

10+50 | 50 | 101.00 | 100.96 | -0.04 |

11+00 | 100 | 102.00 | 101.67 | -0.33 |

11+50 | 150 | 103.00 | 102.13 | -0.87 |

12+00 | 200 | 104.00 | 102.33 | -1.67 |

12+50 | 250 | 105.00 | 102.29 | -2.71 |

13+00 | 300 | 106.00 | 102.00 | -4.00 |

## Vertical Curve Formula

The fundamental **parabolic equation** for a vertical curve is:

y = ax² + bx + c

Where:

**y**: Elevation at any point on the curve**x**: Horizontal distance from the beginning of the curve**a, b, c**: Constants determined by the curve’s characteristics

For practical applications, road designers often use a simplified form:

E = (G₁ – G₂) * x² / (200 * L)

Where:

**E**: Elevation difference between the curve and the tangent at any point**G₁**: Initial grade (%)**G₂**: Final grade (%)**x**: Distance from the beginning of the curve**L**: Total length of the curve

## How to Calculate PVI Vertical Curve

Calculating the **Point of Vertical Intersection (PVI)** is crucial for vertical curve design. Follow these steps:

**Determine the initial and final grades**(G₁ and G₂)**Identify the stations and elevations**of the**Point of Vertical Curvature (PVC)**and**Point of Vertical Tangency (PVT)****Calculate the horizontal distance**between PVC and PVT**Find the PVI station**by adding half the curve length to the PVC station**Calculate PVI elevation**using the tangent grades: PVI Elevation = PVC Elevation + (G₁ * L/2)

Where **L** is the total curve length.

## How do you calculate the K value for vertical curves?

The **K value** represents the **horizontal distance** required for a **1% change** in grade.

To calculate K:

K = L / A

Where:

**L**: Length of the vertical curve**A**: Algebraic difference between the initial and final grades (|G₁ – G₂|)

**Example**:

For a curve with L = 300 feet, G₁ = 2%, and G₂ = -3%:

A = |2 – (-3)| = 5%

K = 300 / 5 = 60

This K value of **60** means the curve provides **60 feet** of horizontal distance for each **1% change** in grade.

## How to find the elevation along a vertical curve?

To find the **elevation** at any point along the vertical curve:

**Calculate the station**of the point of interest**Determine the distance (x)**from the beginning of the curve**Use the parabolic equation**: Elevation = PVC Elevation + G₁ * x + E- Where E is calculated using the formula from the “Vertical Curve Formula” section.

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