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.&#160;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.&#160;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.&#160;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.

ecuaciones de curvas

Curve Equations

Las curvas son elementos geométricos esenciales que se caracterizan por la distancia tangente, la longitud de la cuerda, la ordenada media y la longitud total del arco. Estas medidas son cruciales para comprender las propiedades geométricas y espaciales de una curva y se definen por la relación entre su radio y su ángulo central.La distancia tangente (T) se refiere a la medida en línea recta desde el punto de intersección de dos tangentes hasta el inicio o el final de la curva. Esta distancia está influenciada por el radio de la curva (R) y el ángulo central (I), que juntos definen la disposición geométrica de la curva.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:12.48%;"><img style="aspect-ratio:571/276;" src="https://cloudfront.jove.com/files/ftp_upload/18120/18120_Equation_1_editor_18120.svg" alt="18120_Equation_1.svg"></figure>La longitud de la cuerda (L.C.) representa la distancia en línea recta más corta entre los puntos inicial y final de la curva. Se obtiene al considerar los puntos finales de la curva como la base de un triángulo isósceles, con los radios como lados iguales. La relación entre el ángulo central y esta configuración triangular determina la longitud de la cuerda, mostrando cómo varía con los cambios en el ángulo o el radio.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:12.42%;"><img style="aspect-ratio:571/276;" src="https://cloudfront.jove.com/files/ftp_upload/18120/18120_Equation_2_editor_18120.svg" alt="18120_Equation_2.svg"></figure>La ordenada media (M) es la distancia perpendicular máxima desde la cuerda hasta la curva, normalmente en el punto medio de la cuerda. Esta medida refleja la profundidad de la curva, influenciada por su radio y extensión angular. La relación entre el desplazamiento en el punto medio y el radio de la curva proporciona el valor de la ordenada media, enfatizando su dependencia de la geometría de la curva.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:12.79%;"><img style="aspect-ratio:673/276;" src="https://cloudfront.jove.com/files/ftp_upload/18120/18120_Equation_3_editor_18120.svg" alt="18120_Equation_3.svg"></figure>La longitud total de la curva (L), o longitud del arco, es directamente proporcional al radio y al ángulo central. Esta relación establece cómo la extensión angular de la curva determina la extensión física de su arco. Juntos, estos parámetros ofrecen un marco integral para analizar las dimensiones y las propiedades espaciales de una curva en aplicaciones geométricas y prácticas.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:12%;"><img style="aspect-ratio:454/276;" src="https://cloudfront.jove.com/files/ftp_upload/18120/18120_Equation_4_editor_18120.svg" alt="18120_Equation_4.svg"></figure>

Ecuaciones de curvas

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

Education

Educación

JoVE Core

Civil Engineering

Ingeniería Civil

Unit 1

Unidad 1

Horizontal and Vertical Curves

Curvas horizontales y verticales

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

Horizontal curves connect straight road or railroad segments and help vehicles move safely through a change in direction. They are used in highway and railroad design to create smooth transitions between tangents, which are the straight path sections before and after the curve. By reducing abrupt changes, these curves can support steadier travel speed and help lower the chance of accidents. A simple horizontal curve is defined by its geometry around the intersection point, or P.I., where two...

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 equations

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

Setting Out a Horizontal Curve

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,...

field procedure for staking out curves

Surveying Curve Layout with Deflection Angles

Surveying curve layout with deflection angles turns a designed curve into stakes placed in the field. In construction, this step helps keep structures aligned along a curved path. Surveyors use calculated positions so the physical markers match the plan. The process starts by finding the curve’s geometric values. These include the radius, central angle, and tangent distances. Surveyors also identify the Point of Curvature (PC), the Point of Tangency (PT), and the midpoint of the curve. These...

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.

elevation of intermediate points on vertical curves

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...

sight distance in a vertical curve

Road Safety on Crest and Sag Curves

Road safety on crest and sag curves depends on how far drivers can see ahead. In roadway design, vertical curves must give drivers enough sight distance to notice hazards and react in time. This affects safety, driver comfort, and how efficiently the transportation network works. Vertical curves are grouped into two main types: crest curves and sag curves. A crest curve is shaped like a hilltop, so the road can block the view of what is beyond it. For these curves, sight distance is based on...

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 provides a foundational exploration of essential construction materials in civil engineering, focusing on bricks, stones, steel, wood, and concrete through easy-to-understand animated videos. This collection covers the properties, manufacturing processes, and practical applications of these materials. Additionally, the accompanied science-in-action videos demonstrate the research experiments performed in laboratories.

civil engineering

Curve Measurements and Arc Length

Description

Transcript

Tags