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.

equazioni delle curve

Curve Equations

Le curve sono elementi geometrici essenziali, caratterizzati dalla distanza dalla tangente, lunghezza della corda, ordinata media e lunghezza totale dell'arco. Queste misurazioni sono cruciali per comprendere le propriet&#224; geometriche e spaziali di una curva e sono definite dalla relazione tra il suo raggio e il suo angolo al centro.La distanza dalla tangente (T) indica la misurazione in linea retta dal punto di intersezione di due tangenti all'inizio o alla fine della curva. Questa distanza &#232; influenzata dal raggio della curva (R) e dall'angolo al centro (I), che insieme definiscono la disposizione geometrica della curva.<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:13.29%;'><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 lunghezza della corda (L.C.) rappresenta la distanza pi&#249; breve in linea retta tra l'inizio e la fine della curva. Si ricava considerando i punti finali della curva come se formassero la base di un triangolo isoscele, con i raggi della curva come lati. La relazione tra l'angolo al centro e questa configurazione triangolare determina la lunghezza della corda, mostrando come questa vari in base ai cambiamenti nell'angolo o nel raggio.<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:13.7%;'><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>L'ordinata media (M) &#232; la distanza perpendicolare massima dalla corda alla curva, in genere nel punto medio della corda. Questa misura riflette la profondit&#224; della curva, influenzata dal suo raggio e dalla sua estensione angolare. La relazione tra l'offset nel punto medio e il raggio della curva fornisce il valore dell'ordinata media, sottolineandone la dipendenza dalla geometria della curva.<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:13.95%;'><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 lunghezza totale della curva (L), o lunghezza dell'arco, &#232; direttamente proporzionale al raggio e all'angolo centrale. Questa relazione stabilisce come l'estensione angolare della curva determina la campata fisica del suo arco. Insieme, questi parametri offrono un quadro completo per analizzare le dimensioni e le propriet&#224; spaziali di una curva, in applicazioni geometriche e pratiche.<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:11.9%;'><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>

Equazioni delle Curve

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

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Ingegneria Civile

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Horizontal and Vertical Curves

Curve orizzontali e verticali

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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

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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...

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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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Finding High and Low Points on Vertical Curves

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Plotting a Horizontal Curve from Field Data

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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

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