Name: curve measurements and arc length
Uploaded: 2025-03-26T15:36:35.000Z
Duration: 1 min 17 s
Channel: JoVE

Consider a curve defined by its radius and central angle. The tangent distance is the distance from the point of intersection to the curve's start or end point. This tangent length depends on the curve's radius and the central angle, with its relationship defined through geometric analysis.The length of the chord is the straight line connecting the curve's start and endpoints. This length is derived by visualizing the curve's endpoints as forming a triangle with the center, where the curve's radius serves as its sides. The relationship between this triangle's angle and base gives the length of the chord.The middle ordinate describes the maximum perpendicular distance between the straight connecting line and the curve itself. This value is determined by examining the curve's depth relative to the connecting chord, using the relationship between the curve's radius and its offset at the midpoint.The curve's total length is directly proportional to the central angle. It is calculated by understanding how the curve's angular extent relates to its radius, offering a measure of the arc's physical span.

curve equations

Curve Equations

Curves are essential geometric elements characterized by tangent distance, chord length, middle ordinate, and total arc length. These measurements are crucial in understanding a curve's geometric and spatial properties and are defined by the relationship between its radius and its central angle.The tangent distance (T) refers to the straight-line measurement from the intersection point of two tangents to either the start or end of the curve. This distance is influenced by the curve's radius (R) and the central angle (I), which together define the geometric layout of the curve. <figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:12.04%;"><img style="aspect-ratio:571/276;" src="https://cloudfront.jove.com/files/ftp_upload/18120/18120_Equation_1_editor_18120.svg" alt="Static equilibrium equation T=Rtan(l/2), mathematical formula, educational concept." data-altupdatedat="1762173717000" data-alt="18120_Equation_1.svg" /></figure>The length of the chord (L.C.) represents the shortest straight-line distance between the curve's start and endpoints. It is derived by considering the curve's endpoints as forming the base of an isosceles triangle, with the radius as equal sides. The relationship between the central angle and this triangular configuration determines the chord length, showing how it varies with changes in the angle or radius.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:11.94%;"><img style="aspect-ratio:571/276;" src="https://cloudfront.jove.com/files/ftp_upload/18120/18120_Equation_2_editor_18120.svg" alt="Formula for arc length; L.C. = 2R sin(l/2); trigonometric calculation; educational math diagram." data-altupdatedat="1762173717000" data-alt="18120_Equation_2.svg" /></figure>The middle ordinate (M) is the maximum perpendicular distance from the chord to the curve, typically at the midpoint of the chord. This measure reflects the curve's depth, influenced by its radius and angular extent. The relationship between the offset at the midpoint and the curve's radius provides the value of the middle ordinate, emphasizing its dependence on the curve's geometry.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:12.77%;"><img style="aspect-ratio:673/276;" src="https://cloudfront.jove.com/files/ftp_upload/18120/18120_Equation_3_editor_18120.svg" alt="Mechanical advantage formula M=R(1-cos(&#952;/2)); equation related to static equilibrium studies." data-altupdatedat="1762173717000" data-alt="18120_Equation_3.svg" /></figure>The total length of the curve (L), or arc length, is directly proportional to the radius and the central angle. This relationship establishes how the curve's angular extent determines its arc's physical span. Together, these parameters offer a comprehensive framework for analyzing a curve's dimensions and spatial properties in geometric and practical applications.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:9.07%;"><img style="aspect-ratio:454/276;" src="https://cloudfront.jove.com/files/ftp_upload/18120/18120_Equation_4_editor_18120.svg" alt="L=100I/D formula, geometric optics diagram, lens calculation, light intensity analysis." data-altupdatedat="1762173717000" data-alt="18120_Equation_4.svg" /></figure>

Curves are essential geometric elements characterized by tangent distance, chord length, middle ordinate, and total arc length. These measurements are ...

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JoVE Core

Civil Engineering

Unit 1

Horizontal and Vertical Curves

KEY TERMS AND CONCEPTS

curve measurements and arc length

Curve measurements describe the key parts of a curved line. The main values are tangent distance, chord length, middle ordinate, and total arc length. These measurements help explain a curve’s shape and size using its radius and central angle.

Tangent distance, or T, is the straight-line distance from the point where two tangents meet to the start or end of the curve. It depends on the curve radius, R, and the central angle, I. These two values work together to set the curve’s layout.

Chord length, or L.C., is the shortest straight-line distance between the start and end points of the curve. The endpoints and the radius can be viewed as forming an isosceles triangle, where the two sides are equal. The central angle controls how that triangle changes, so the chord length changes with the angle and the radius.

The middle ordinate, or M, is the greatest perpendicular distance from the chord to the curve. It is usually measured at the midpoint of the chord. This value shows the depth of the curve and depends on the curve’s radius and angular extent.

The total length of the curve, or L, is the arc length. Arc length is directly related to both the radius and the central angle. Together, tangent distance, chord length, middle ordinate, and arc length give a full picture of a curve’s geometry in practical applications.

Curve Measurements and Arc Length

introduction to horizontal curves

Horizontal Curve Parts and Road Alignment

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degree of curvature and radius of curvature

degree-of-curvature-and-radius-of-curvature

Curvature Measures in Civil Engineering Curves

Degree of curvature and radius of curvature help describe how sharply a curve bends in civil engineering. The degree of curvature measures the steepness of the bend. The radius of curvature shows how tight or smooth the curve is. The degree of curvature can be found by the chord basis or the arc basis. In the chord basis method, the degree is the central angle subtended by a chord of 30.48 meters. In the arc basis method, it is the central angle subtended by an arc of length 30.48 meters. <br...

Curve measurements describe the key parts of a curved line. The main values are tangent distance, chord length, middle ordinate, and total arc length. These measurements help explain a curve’s shape and size using its radius and central angle. Tangent distance, or T, is the straight-line distance from the point where two tangents meet to the start or end of the curve. It depends on the curve radius, R, and the central angle, I. These two values work together to set the curve’s layout. Chord...

horizontal curve: problem solving

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A horizontal curve is set out by using its radius, intersection angle, and stationing of key points. In this example, the radius is 400 meters, the intersection angle is 30 degrees, and the station of the point of curvature, or P.C., is 0 + 150 meters. The goal is to find the station values for the point of intersection, or P.I., the point of tangency, or P.T., and the midpoint of the curve. The length of the long chord is also needed. The first step is to calculate the tangent distance, T,...

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introduction to vertical curves

Designing Smooth Road Grade Transitions

Vertical curves are parabolic transitions that connect different grades on highways and railroads. They help create a smooth alignment between the back tangent and the forward tangent. The back tangent is the first grade, and the forward tangent is the next grade. These curves may be symmetrical or nonsymmetrical. In a symmetrical vertical curve, the tangent lengths are equal. In a nonsymmetrical curve, the tangent lengths are different. The main points used to describe the curve are the Point...

vertical curve: problem solving

Calculating Elevations on a Vertical Curve

Vertical curves connect two roadway grades and help make the road safer, smoother, and more functional. To find the elevation at specific stations on the curve, engineers follow a set of steps based on the curve length, the grades, the Point of Vertical Intersection, or P.V.I., and the P.V.I. elevation. The first step is to locate the Point of Vertical Curvature, or P.V.C., and the Point of Vertical Tangency, or P.V.T. The P.V.C. marks the start of the curve, and the P.V.T. marks the end.

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elevation-of-intermediate-points-on-vertical-curves

Finding High and Low Points on Vertical Curves

Vertical curves help roadways move smoothly from one grade to another. Engineers use them to connect changing slopes and keep the road profile safe and even. To design these curves, they calculate intermediate elevations and find the highest or lowest point on the curve. Intermediate elevations on a vertical curve are found with the tangent offset method. This method uses the starting elevation, the grades at each end, and the curve shape. It gives the elevation of any point along the curve by...

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design example: setting a curve using design data

design-example-setting-a-curve-using-design-data

Plotting a Horizontal Curve from Field Data

Plotting a horizontal curve from field data starts with careful geometric design. The curve in this example has a radius of 200 meters and an intersection angle of 20 degrees. Engineers use the method of perpendicular offsets from the long chord to place the curve on the ground. The long chord is the straight line between the two ends of the curve. Here, its length is calculated as 69.46 meters. To improve accuracy, 3-meter intervals are marked along the chord. Offset distances at each point...

key terms and concepts

Learn the main measurements of a curve, including tangent distance, chord length, middle ordinate, and arc length, using radius and central angle.

Curve Measurements and Arc Length | JoVE K12

JoVE Core Civil Engineering offers a foundational exploration of essential construction materials such as bricks, stones, steel, wood, and concrete, along with fluid mechanics topics like flow over immersed bodies, open channel flow, and dimensional modeling. It also covers key surveying concepts, including leveling, traverse adjustments, curve design, topographic mapping, and geospatial technologies (GPS & GIS). Science-in-action videos showcase laboratory research on materials, fluids, and surveying practices in real-world engineering systems.

civil engineering

Curve Measurements and Arc Length

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