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

ضع في اعتبارك منحنى محدد بنصف قطره وزاويته المركزية. مسافة المماس هي المسافة من نقطة التقاطع إلى نقطة بداية المنحنى أو نهايته.  يعتمد طول المماس هذا على نصف قطر المنحنى والزاوية المركزية ، مع تحديد علاقته من خلال التحليل الهندسي. طول الوتر هو الخط المستقيم الذي يربط بداية المنحنى ونقاط النهاية. يتم اشتقاق هذا الطول عن طريق تصور نقاط نهاية المنحنى على أنها تشكل مثلثا مع المركز ، حيث يعمل نصف قطر المنحنى كأضلاع له.  تعطي العلاقة بين زاوية هذا المثلث وقاعدته طول الوتر. يصف الإحداثي الأوسط الحد الأقصى للمسافة العمودية بين خط التوصيل المستقيم والمنحنى نفسه.  يتم تحديد هذه القيمة من خلال فحص عمق المنحنى بالنسبة إلى الوتر المتصل ، باستخدام العلاقة بين نصف قطر المنحنى وإزاحته عند نقطة المنتصف. يتناسب الطول الكلي للمنحنى طرديا مع الزاوية المركزية. يتم حسابه من خلال فهم كيفية ارتباط المدى الزاوي للمنحنى بنصف قطره ، مما يوفر مقياسا للامتداد المادي للقوس.

معادلات المنحنى

Curve Equations

المنحنيات هي عناصر هندسية أساسية تتميز بمسافة الظل وطول الوتر والإحداثي الأوسط وطول القوس الكلي. هذه القياسات ضرورية في فهم الخصائص الهندسية والمكانية للمنحنى ويتم تحديدها من خلال العلاقة بين نصف قطره وزاويته المركزية. تشير مسافة الظل ( T ) إلى قياس الخط المستقيم من نقطة تقاطع مماسين إلى بداية المنحنى أو نهايته. تتأثر هذه المسافة بنصف قطر المنحنى (R) والزاوية المركزية  (I) ، والتي تحدد معا التخطيط الهندسي للمنحنى. <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="18120_Equation_1.svg"></figure> يمثل طول الوتر أقصر مسافة خط مستقيم بين بداية المنحنى ونقاط النهاية. يتم اشتقاقه من خلال اعتبار نقاط نهاية المنحنى على أنها تشكل قاعدة مثلث متساوي الساقين ، مع نصف القطر كجوانب متساوية. تحدد العلاقة بين الزاوية المركزية وهذا التكوين الثلاثي طول الوتر ، مما يوضح كيف يختلف مع التغيرات في الزاوية أو نصف القطر. <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="18120_Equation_2.svg"></figure>الإحداثيات الوسطى (M ) هي المسافة القصوى العمودية من الوتر إلى المنحنى ، عادة عند نقطة منتصف الوتر. يعكس هذا المقياس عمق المنحنى ، ويتأثر بنصف قطره ومداه الزاوي. توفر العلاقة بين الإزاحة عند نقطة المنتصف ونصف قطر المنحنى قيمة الإحداثي الأوسط ، مع التركيز على اعتماده على هندسة المنحنى. <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="18120_Equation_3.svg"></figure>يتناسب الطول الكلي للمنحنى (L ) ، أو طول القوس ، طرديا مع نصف القطر والزاوية المركزية. تحدد هذه العلاقة كيف يحدد المدى الزاوي للمنحنى الامتداد المادي للقوس. توفر هذه المعلمات معا إطارا شاملا لتحليل أبعاد المنحنى وخصائصه المكانية في التطبيقات الهندسية والعملية. <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="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 ...

Education

JoVE Core

Civil Engineering

الهندسة المدنية

Unit 1

الوحدة الأولى

Horizontal and Vertical Curves

المنحنيات الأفقية والرأسية

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الرئيسية

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

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