Contouring and Plane Table Surveying

Surveying: Contouring and Plane Table Surveying

Contents

Define the following terms: i. Contour ii. Contour Internal iii. Horizontal Equivalent 2

Recall the Factors affecting the Contour Interval 3

Describe the Characteristics of Contour 4

Describe the Direct methods of Locating Contour by Vertical control and Horizontal Control 5

Recall the Indirect Methods by: i. Squares ii. Cross-sections 6

Define Contour Gradient 7

Explain with sketches, the uses of Contour Maps 7

Recall the function of the following Instruments: i. Plane Table ii. Alidade and its types iii. Plumbing Fork and Plumb Bob iv. Spirit Level v. Compass 8

Describe the following Operations of Plane Table Survey: i. Levelling the Table ii. Centering the Table iii. Orientation of Table 9

Recall different Methods of Orientation of the Plane Table 10

Recall the Radiation method of Plane Table Surveying 11

Describe the Intersection Method of Plane Table Surveying 12

Recall Traversing method used in Plane Table Surveying 12

Define Resection Method used in Plane Table Surveying 13

Classify the methods of Orientation of Plane Table in Resection 13

Classify the Errors in Plane Tabling 14

List various Advantages and Disadvantages of Plane Tabling 15

Determine Area by Subdivision into Triangles 16

Determine the Area from offset to a Baseline by using: i. Mid-ordinate Rule ii. Average-ordinate Rule iii. Trapezoidal Rule iv. Simpson’s One-Third Rule 16

Recall the Working of Planimeter 17

Derive an expression for determine Area using Planimeter 19

Define Planimeter Constant 20

Derive an expression for Prismoidal Formula 21

Derive an expression for Trapezoidal Formula 22

Recall the concept of Prismoidal Correction 23

Recall different methods for the Calculation of Volume from Contour Plans 24

Recall the determination of Capacity of Reservoir 25

Define the following terms: i. Contour ii. Contour Internal iii. Horizontal Equivalent

i. Contour:

A contour is a line on a map or a cross-section that represents an equal elevation or height above sea level. Contour lines connect points of equal elevation and can be used to represent the shape and slope of the terrain. In a topographic map, contour lines are typically drawn at regular intervals, such as every 5 meters or 10 feet, to represent changes in elevation.

Contour lines are an essential tool for visualizing the shape of the terrain and understanding changes in elevation. They can be used to analyze the topography of an area and identify features such as hills, valleys, and ridges. Contour lines can also be used to calculate the slope of the terrain and to determine the volume of earthwork required for construction projects.

ii. Contour Interval:

The contour interval is the difference in elevation between two adjacent contour lines on a topographic map. The contour interval is typically expressed in units of measurement, such as meters or feet, and is chosen based on the desired level of detail and the size of the area being mapped. For example, a contour interval of 5 meters may be used for a small area, while a contour interval of 20 meters may be used for a larger area.

The contour interval is an important factor in determining the accuracy and detail of a topographic map. A smaller contour interval provides more detailed information about the terrain, while a larger contour interval provides a more general overview.

iii. Horizontal Equivalent:

The horizontal equivalent is a term used in surveying to describe the equivalent horizontal distance of a slope or incline. The horizontal equivalent is calculated by dividing the vertical height of the slope by the slope or incline angle, and is used to represent the slope or incline as a horizontal distance.

The horizontal equivalent is useful in construction and engineering applications, where the slope of a terrain or incline must be accurately calculated for design and safety purposes. For example, the horizontal equivalent may be used to determine the length of a ramp or staircase required to navigate a slope, or to calculate the volume of earthwork required for construction projects.

Recall the Factors affecting the Contour Interval

The contour interval is an important factor in determining the accuracy and detail of a topographic map. The following factors can affect the contour interval:

i. Scale of the map: The scale of the map determines the level of detail that can be included on the map. A smaller scale map, such as 1:25,000, requires a smaller contour interval, while a larger scale map, such as 1:50,000, may use a larger contour interval.

ii. Terrain features: The complexity of the terrain features, such as hills, valleys, and ridges, affects the contour interval. If the terrain is steep or has many changes in elevation, a smaller contour interval may be necessary to accurately represent the terrain.

iii. Purpose of the map: The purpose of the map determines the level of detail required. For example, a topographic map used for engineering purposes may require a smaller contour interval than a map used for recreational purposes.

iv. Available data: The available data, such as elevation measurements and aerial photographs, affects the contour interval. If the data is accurate and detailed, a smaller contour interval may be possible.

v. Map size: The size of the map affects the contour interval. A larger map may require a larger contour interval to avoid overcrowding the map with contour lines.

In summary, the contour interval is influenced by a variety of factors, including the scale of the map, the complexity of the terrain, the purpose of the map, the available data, and the size of the map. The contour interval is an important consideration in creating accurate and useful topographic maps.

Describe the Characteristics of Contour

Contour lines are an essential tool for visualizing the shape of the terrain and understanding changes in elevation. The following are some of the characteristics of contour lines:

i. Shape: Contour lines are typically represented as closed loops or meandering lines, depending on the slope of the terrain. A closed loop contour line represents a hill or ridge, while a meandering contour line represents a valley or low-lying area.

ii. Spacing: Contour lines are typically drawn at regular intervals, such as every 5 meters or 10 feet, to represent changes in elevation. The spacing of the contour lines, known as the contour interval, is an important factor in determining the accuracy and detail of a topographic map.

iii. Elevation: Contour lines connect points of equal elevation, and the elevation can be determined by the label on the line or by interpolation between two adjacent contour lines. The elevation of the contour lines is expressed in units of measurement, such as meters or feet, and is referenced to sea level.

iv. Slope: Contour lines can be used to calculate the slope of the terrain and to determine the direction of the slope. The slope can be determined by the spacing of the contour lines, with close spacing indicating a steep slope and wide spacing indicating a gentle slope.

v. Direction: Contour lines can be used to determine the direction of the slope. If a contour line is steep and oriented in a north-south direction, for example, it indicates a steep slope in the east-west direction.

In summary, contour lines are an important tool for visualizing the shape and slope of the terrain. The shape, spacing, elevation, slope, and direction of contour lines can be used to analyze the topography of an area and to understand changes in elevation.

Describe the Direct methods of Locating Contour by Vertical control and Horizontal Control

The direct methods of locating contour involve using both vertical control and horizontal control to accurately determine the location of contour lines on a topographic map.

i. Vertical control: Vertical control refers to the process of measuring the elevation of points on the terrain and using these measurements to create contour lines. This is typically done using a survey instrument, such as a level or a theodolite, to measure the elevation of benchmark points. These benchmark points are then used to create contour lines that connect points of equal elevation.

ii. Horizontal control: Horizontal control refers to the process of determining the horizontal location of points on the terrain. This is typically done using a survey instrument, such as a total station or GPS, to determine the coordinates of benchmark points. These benchmark points are then used to accurately locate contour lines on the topographic map.

By combining vertical control and horizontal control, a topographic map can be created that accurately represents the shape and slope of the terrain. The direct methods of locating contour are used to create high-quality topographic maps that are used in a variety of applications, including engineering, environmental management, and land use planning.

In summary, the direct methods of locating contour involve using both vertical control and horizontal control to accurately determine the location of contour lines on a topographic map. The combination of these two methods provides a precise representation of the terrain and is used in many applications to analyze and understand changes in elevation.

Recall the Indirect Methods by: i. Squares ii. Cross-sections

The indirect methods of locating contour involve using squares and cross-sections to estimate the location of contour lines on a topographic map.

i. Squares: In this method, a square grid is superimposed on the terrain, and the elevation of each corner of the square is measured. The contour lines are then estimated by connecting the points of equal elevation and extrapolating the contour lines between the squares. This method is often used when the terrain is rough or when it is not possible to directly measure the elevation of all points.

ii. Cross-sections: In this method, cross-sections are taken at regular intervals across the terrain, and the elevation of points along each cross-section is measured. The contour lines are then estimated by connecting the points of equal elevation and extrapolating the contour lines between the cross-sections. This method is often used to create profiles of the terrain along roads, rivers, or other linear features.

In summary, the indirect methods of locating contour involve using squares and cross-sections to estimate the location of contour lines on a topographic map. These methods are often used when direct measurement of the terrain is not possible, or when a rough estimate of the contour lines is sufficient. However, the accuracy of the contour lines generated by these methods is lower compared to the direct methods, and the results should be interpreted with caution.

Define Contour Gradient

Contour gradient refers to the rate of change in elevation along a contour line on a topographic map. It is expressed as a ratio of vertical distance to horizontal distance, typically as a percentage or as a ratio. The contour gradient is an important tool for understanding the shape and slope of the terrain and is used in a variety of applications, including geology, engineering, and environmental management.

The contour gradient can be calculated by measuring the vertical and horizontal distance between two points on a contour line and dividing the vertical distance by the horizontal distance. The resulting value represents the rate of change in elevation along the contour line and is expressed as a percentage or a ratio.

In areas of steep terrain, the contour gradient is high, indicating a rapid change in elevation. In flatter areas, the contour gradient is lower, indicating a gradual change in elevation. The contour gradient is a useful tool for analyzing the shape and slope of the terrain and for determining the most efficient routes for transportation or construction.

In summary, contour gradient refers to the rate of change in elevation along a contour line on a topographic map and is expressed as a percentage or a ratio. It is an important tool for understanding the shape and slope of the terrain and is used in a variety of applications, including geology, engineering, and environmental management.

Explain with sketches, the uses of Contour Maps

Contour maps are topographic maps that use contour lines to represent the shape and slope of the terrain. They are widely used in a variety of applications, including geology, engineering, environmental management, and land use planning. Contour maps provide a visual representation of the terrain that is easy to understand and can be used to make informed decisions about land use and development.

Here are a few key uses of contour maps:

  1. Topographic analysis: Contour maps are used to analyze the topography of an area and to determine the shape and slope of the terrain. This information is important for understanding the drainage patterns, soil erosion potential, and other factors that affect land use and development.
  2. Planning and design: Contour maps are used in land use planning and design to determine the most suitable locations for roads, buildings, and other infrastructure. They can be used to identify areas with steep slopes that may be difficult to build on, and to determine the most efficient routes for transportation or construction.
  3. Environmental management: Contour maps can be used to monitor changes in the terrain over time and to identify areas that are prone to soil erosion, landslides, or other environmental hazards. This information is important for managing the environment and reducing the risk of environmental degradation.
  4. Geology: Contour maps are used in geology to study the shape and structure of the Earth’s surface. They can be used to identify geological features, such as faults, ridges, and valleys, and to determine the location and orientation of mineral deposits.

In summary, contour maps are widely used in a variety of applications, including topographic analysis, land use planning and design, environmental management, and geology. They provide a visual representation of the terrain that is easy to understand and can be used to make informed decisions about land use and development.

Recall the function of the following Instruments: i. Plane Table ii. Alidade and its types iii. Plumbing Fork and Plumb Bob iv. Spirit Level v. Compass

  1. Plane Table: A plane table is a flat, portable drawing board used in surveying and mapping. It is mounted on a tripod and used in combination with an alidade to take bearings and make sketches of the terrain. The plane table allows the surveyor to make a rapid sketch of the area being surveyed, which can then be used to produce more accurate and detailed maps.
  2. Alidade: An alidade is a sighting device used in surveying and mapping. It consists of a rotating ruler or straightedge that is mounted on a pivot, and is used to take bearings of objects in the field. There are two types of alidade: the circular alidade, which is used to take bearings and make sketches on a plane table, and the rectilinear alidade, which is used to take bearings of objects in the field and to measure angles.
  3. Plumbing Fork and Plumb Bob: A plumbing fork and plumb bob are tools used in surveying and construction to ensure that vertical lines are perfectly straight. The plumbing fork is a metal frame with two prongs that are used to hold a plumb bob, which is a weighted string that hangs vertically. The plumbing fork and plumb bob are used to ensure that walls, columns, and other vertical structures are level and plumb.
  4. Spirit Level: A spirit level is a device used to determine whether a surface is level or not. It consists of a clear tube filled with liquid, usually alcohol, and an air bubble. When the surface being tested is level, the air bubble is centred in the tube. If the surface is not level, the air bubble moves to one end of the tube, indicating that the surface is sloped. Spirit levels are widely used in construction, surveying, and other fields to ensure that surfaces are level.
  5. Compass: A compass is an instrument used to determine direction. It consists of a magnetic needle that is suspended in a housing and free to rotate. The compass is usually mounted on a baseplate with a rotating bezel that is marked with degrees or other units of measurement. Compasses are used in navigation, surveying, and other fields to determine direction and to take bearings of objects in the field.

In summary, these instruments are used in surveying, mapping, and construction to measure angles, ensure that surfaces are level and plumb, and determine direction. They are essential tools for these fields, and play a critical role in ensuring the accuracy and precision of measurements and maps.

Describe the following Operations of Plane Table Survey: i. Levelling the Table ii. Centering the Table iii. Orientation of Table

i. Levelling the Table: This operation involves adjusting the plane table to a level position so that it accurately reflects the vertical plane of the area being surveyed. This is done by using a spirit level placed on the table.

ii. Centering the Table: This operation involves positioning the plane table accurately over the point being surveyed. This is done by using an alidade, plumb bob, or other sighting device to ensure that the table is properly centred over the point.

iii. Orientation of Table: This operation involves aligning the plane table so that it is oriented to true north. This is done by using a compass or other direction-finding device to set the orientation of the table. The orientation of the table is important for ensuring that the survey data collected is accurate and properly referenced to a fixed coordinate system.

Recall different Methods of Orientation of the Plane Table

A plane table is a surveying instrument used for mapping and measuring land features. It consists of a flat table, mounted on a tripod, with a sight mounted on it. The plane table is used to plot points and draw lines directly on a sheet of paper that is mounted on the table. Orientation of the plane table refers to the process of aligning the table with the north-south direction and ensuring that the plane of the table is horizontal. Recall of different methods of orientation is important for surveyors to ensure accurate and reliable measurements.

There are several methods for orienting a plane table. Some of the common methods are:

  1. Three-point method: This method involves setting up the plane table at any location and sighting three known points that are widely spaced and visible from the table. The surveyor then marks these points on the table and connects them to form a triangle. The table is then oriented so that the triangle formed by the marked points is identical to the triangle formed by the actual points on the ground.
  2. Two-point method: This method involves setting up the plane table at any location and sighting two known points that are widely spaced and visible from the table. The surveyor then marks these points on the table and connects them with a straight line. The table is then oriented so that the marked line on the table is identical to the actual line on the ground.
  3. Reiteration method: This method involves taking multiple readings of a target that is known to be directly north of the table. The table is then rotated until the readings are consistent and the table is facing north.
  4. Magnetic compass method: This method involves using a magnetic compass to determine the magnetic north direction. The table is then oriented so that the magnetic north direction is aligned with the north-south axis of the table.
  5. Solar observation method: This method involves using the sun to determine the north-south direction. The table is oriented so that the shadow of a rod or staff placed vertically on the table is aligned with the north-south axis of the table.

In summary, there are several methods for orienting a plane table, including the three-point method, two-point method, reiteration method, magnetic compass method, and solar observation method. Each method has its advantages and disadvantages, and the choice of method depends on factors such as the accuracy required, availability of equipment, and environmental conditions. It is important for surveyors to be familiar with these methods to ensure accurate and reliable measurements.

Recall the Radiation method of Plane Table Surveying

The Radiation Method of Plane Table Surveying is a technique for determining the location of a point in a survey using a plane table. In this method, a series of rays are drawn from the point being surveyed to other known points on the survey area. The intersection of these rays determines the location of the point being surveyed.

This method is used when the point being surveyed cannot be easily seen from other points in the survey area, or when the terrain is too rough or obstructed to allow for direct sight lines. The radiation method allows the surveyor to work from a single location and still accurately determine the location of the point being surveyed.

The key to the success of the radiation method is the accurate positioning of the plane table and the careful alignment of the rays with the known points on the survey area. This requires a high degree of skill and precision on the part of the surveyor.

Describe the Intersection Method of Plane Table Surveying

The Intersection Method of Plane Table Surveying is a technique for determining the location of a point in a survey using a plane table. In this method, the plane table is positioned so that two or more known points in the survey area are visible and their locations are plotted on the plane table. The intersection of the lines connecting these plotted points determines the location of the point being surveyed.

This method is commonly used when the point being surveyed is easily visible from two or more other points in the survey area. It allows the surveyor to determine the location of the point without the need for additional instruments or measurements.

The key to the success of the intersection method is the accurate positioning of the plane table and the precise alignment of the lines connecting the plotted points. This requires a high degree of skill and precision on the part of the surveyor. It is also important to accurately plot the positions of the known points in the survey area to ensure that the intersection of the lines correctly determines the location of the point being surveyed.

Recall Traversing method used in Plane Table Surveying

The Traversing method used in Plane Table Surveying is a technique for determining the location of points in a survey by connecting them with a series of straight lines. The plane table is set up at each point in the survey and the location of the next point is determined by sight and recorded on the plane table. The process is repeated for each subsequent point in the survey.

This method is commonly used for surveying large or irregularly shaped areas where it is difficult to use other methods such as the radiation or intersection method. The accuracy of the traversing method depends on the accuracy of the measurements made at each point in the survey and the precision of the plane table positioning.

In order to ensure the accuracy of the traversing method, it is important to carefully select the points to be surveyed and to use a stable and level plane table. The surveyor must also take care to accurately record the location of each point in the survey and to align the plane table correctly with each subsequent point.

Define Resection Method used in Plane Table Surveying

The Resection Method used in Plane Table Surveying is a technique for determining the location of an unknown point by observing two or more known points from that point. The plane table is set up at the unknown point and the angles between the known points and the unknown point are measured using an alidade or sighting device. The location of the unknown point is then determined using triangulation.

The resection method is commonly used in plane table surveying when the location of a specific point is required, such as the location of a building, the center of a circle, or the intersection of two lines. The accuracy of the resection method depends on the precision of the angles measured, the accuracy of the measurements made at the known points, and the stability of the plane table.

In order to ensure the accuracy of the resection method, it is important to carefully select the known points to be used and to use a stable and level plane table. The surveyor must also take care to accurately record the angles measured and to align the plane table correctly with each known point.

Classify the methods of Orientation of Plane Table in Resection

Resection is a method of orienting a plane table by sighting back to known points from an unknown location. In other words, the surveyor uses the plane table to observe the angles between the unknown location and two or more known points, and then calculates the position of the unknown location based on those observations. There are different methods of orientation of plane table in resection, which can be classified as follows:

  1. Graphical resection: This method involves drawing lines on the map from the known points to the unknown location. The surveyor then measures the angles between these lines using the plane table and protractor, and uses trigonometry to calculate the coordinates of the unknown location. Graphical resection is a simple method that does not require specialised equipment, but it can be time-consuming and less accurate than other methods.
  2. Trigonometric resection: This method involves using trigonometry to calculate the coordinates of the unknown location based on the observed angles and known distances between the known points. Trigonometric resection is a more accurate and efficient method than graphical resection, but it requires advanced mathematical skills and specialised equipment, such as a theodolite or total station.
  3. Resection by polar coordinates: This method involves calculating the polar coordinates of the unknown location based on the observed angles and known distances between the known points. Polar coordinates refer to the distance and direction of the unknown location from a fixed point, usually the north-south axis of the plane table. Resection by polar coordinates is a simple and accurate method that is commonly used in land surveying.
  4. Resection by intersection: This method involves sighting back to two or more known points from the unknown location and finding the point of intersection of the lines of sight. The surveyor then marks this point on the plane table and uses it to orient the table. Resection by intersection is a simple and accurate method that does not require advanced mathematical skills or specialised equipment.

In summary, the methods of orientation of the plane table in resection can be classified into graphical resection, trigonometric resection, resection by polar coordinates, and resection by intersection. The choice of method depends on factors such as the accuracy required, availability of equipment, and environmental conditions. It is important for surveyors to be familiar with these methods to ensure accurate and reliable measurements.

Classify the Errors in Plane Tabling

The errors in plane table surveying can be classified into two categories: human error and instrument error.

Human errors can include incorrect reading of the instruments, incorrect measurement of angles or distances, incorrect calculation of angles or distances, incorrect plotting of points on the map, and incorrect interpretation of the data.

Instrument errors can include inaccurate readings from the plane table, alidade, or other instruments due to manufacturing defects, wear and tear, or incorrect maintenance. Inaccuracies in the spirit level or compass can also lead to errors in the orientation of the plane table.

It is important to minimize these errors by regular maintenance and calibration of instruments, proper training of surveyors, and careful double-checking of data and calculations.

List various Advantages and Disadvantages of Plane Tabling

Advantages of Plane Tabling:

  1. Easy to set up and use in remote areas with limited access.
  2. It can be used in rough and inaccessible terrain.
  3. It provides a visual representation of the surveyed area.
  4. It is cost-effective compared to other surveying methods.
  5. It can be performed quickly with a small crew.
  6. It is useful in preliminary surveys and reconnaissance.

Disadvantages of Plane Tabling:

  1. It is limited to small scale surveys.
  2. The accuracy is lower compared to other surveying methods.
  3. It is not suitable for large scale surveys.
  4. It is prone to errors due to human mistakes and environmental conditions.
  5. The results are only as accurate as the equipment used.
  6. It requires clear visibility of the ground and cannot be performed during adverse weather conditions.

Determine Area by Subdivision into Triangles

The area of an irregularly shaped piece of land can be determined by subdividing the piece of land into triangles. This method of finding the area of a piece of land is known as the triangulation method. In this method, the piece of land is divided into a series of triangles and the area of each triangle is calculated. The total area of the piece of land is then obtained by summing up the area of each triangle. The triangulation method is commonly used in surveying and mapping. To calculate the area of a triangle, the length of its base and height are measured, and then the area is calculated using the formula, Area = (Base * Height) / 2.

Determine the Area from offset to a Baseline by using: i. Mid-ordinate Rule ii. Average-ordinate Rule iii. Trapezoidal Rule iv. Simpson’s One-Third Rule

i. Mid-ordinate Rule: In this method, the area of an irregular shaped figure is determined by dividing the figure into a number of rectangles and finding the area of each rectangle. The height of each rectangle is determined by taking the average of the ordinates of the curve at the endpoints of the rectangle.

ii. Average-ordinate Rule: In this method, the height of each rectangle is determined by taking the average of the ordinates of the curve at the midpoint of the rectangle.

iii. Trapezoidal Rule: In this method, the area of the irregular shaped figure is determined by dividing the figure into a number of trapezoids and finding the area of each trapezoid. The height of each trapezoid is determined by taking the average of the ordinates of the curve at the endpoints of the trapezoid.

iv. Simpson’s One-Third Rule: This method is an improved version of the Trapezoidal Rule. The area of the irregular shaped figure is determined by dividing the figure into a number of rectangles and trapezoids, and finding the area of each rectangle and trapezoid using a weighted average. The height of each rectangle and trapezoid is determined using the ordinates of the curve at the endpoints and the midpoint of the figure.

Recall the Working of Planimeter

The working of the linear planimeter may be explained by measuring the area of a rectangle ABCD (see image). Moving with the pointer from A to B the arm EM moves through the yellow parallelogram, with area equal to PQ×EM. This area is also equal to the area of the parallelogram A”ABB”. The measuring wheel measures the distance PQ (perpendicular to EM). Moving from C to D the arm EM moves through the green parallelogram, with area equal to the area of the rectangle D”DCC”. The measuring wheel now moves in the opposite direction, subtracting this reading from the former. The movements along BC and DA are the same but opposite, so they cancel each other with no net effect on the reading of the wheel. The net result is the measuring of the difference of the yellow and green areas, which is the area of ABCD.

Before we start with how to use a planimeter, first let us know about the parts of a planimeter. Following are the parts of a planimeter:

  • Tracing arm
  • Tracing point
  • Anchor arm
  • Weight and needle point
  • Clamp
  • Hinge
  • Tangent screw
  • Index
  • Wheel
  • Dial
  • Vernier

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Derive an expression for determine Area using Planimeter

A planimeter is an instrument used to determine the area of an arbitrary two-dimensional shape by tracing its boundary. The following is an explanation of how to derive an expression for determining the area using a planimeter:

  1. The concept of line integral: A line integral is a mathematical operation that calculates the total amount of change along a specified path in a vector field. In the context of planimetry, the line integral represents the area enclosed by the traced shape.
  2. Green’s Theorem: Green’s theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. The theorem states that the line integral of a two-dimensional vector field F = <P,Q> around the boundary of a region D is equal to the double integral of the curl of F over the region D.
  3. Planimeter as a line integrator: A planimeter is essentially a line integrator that calculates the line integral of the unit normal vector along the boundary of the traced shape.
  4. Deriving the expression: Using Green’s theorem, the line integral of the unit normal vector around the boundary of the traced shape can be related to the double integral of the curl of the unit normal vector over the enclosed region. This double integral can be expressed as the area of the region.

Therefore, the expression for determining the area using a planimeter is given by the line integral of the unit normal vector around the boundary of the traced shape, which is equal to the double integral of the curl of the unit normal vector over the enclosed region, which can be expressed as the area of the region.

Define Planimeter Constant

The planimeter constant is a fundamental concept in the use of a planimeter. It is a numerical value that relates the physical movement of the instrument to the calculation of the area of a shape. The following is an explanation of the definition of the planimeter constant:

  1. Definition: The planimeter constant is a numerical value that is specific to each planimeter and represents the ratio of the physical movement of the instrument to the calculation of the area of a shape.
  2. Calculation: The planimeter constant is determined by calibrating the instrument with a known standard shape of known area. By tracing the boundary of the standard shape with the planimeter and comparing the calculated area to the known area, the planimeter constant can be calculated.
  3. Importance: The planimeter constant is an essential component of the planimeter’s accuracy. It ensures that the instrument is able to accurately calculate the area of any shape traced by the instrument.
  4. Usage: The planimeter constant is used in the calculation of the area of a shape traced by the instrument. It is used to convert the physical movement of the instrument into a numerical representation of the area of the shape.

In summary, the planimeter constant is a numerical value specific to each planimeter that relates the physical movement of the instrument to the calculation of the area of a shape. It is determined by calibrating the instrument with a known standard shape and is an essential component of the planimeter’s accuracy. The planimeter constant is used in the calculation of the area of a shape traced by the instrument.

Derive an expression for Prismoidal Formula

The prismoidal formula is a mathematical formula used to estimate the volume of a prismoid, which is a 3-dimensional shape formed by stacking a series of parallelograms that are trapezoids on top of each other. The following is an explanation of how to derive an expression for the prismoidal formula:

  1. Definition of Prismoidal Volume: The volume of a prismoid can be defined as the sum of the volumes of the parallelograms are trapezoids that make up the shape.
  2. Sum of Trapezoidal Volumes: If the prismoid is made up of trapezoids, the volume of the prismoid can be represented by the sum of the volumes of each trapezoidal section, where the volume of each trapezoidal section is given by the product of its base and height.
  3. Sum of Parallelogram Volumes: If the prismoid is made up of parallelograms, the volume of the prismoid can be represented by the sum of the volumes of each parallelogram, where the volume of each parallelogram is given by the product of its base and height.
  4. Deriving the Prismoidal Formula: To derive the prismoidal formula, we need to find an expression for the volume of the prismoid that takes into account both the heights and the bases of the parallelograms are trapezoids that make up the shape. By using the sum of the volumes of the parallelograms are trapezoids and rearranging the terms, we can derive an expression for the volume of the prismoid in terms of the heights and bases of the parallelograms are trapezoids.

Final Expression: The final expression for the prismoidal formula is given by:

  1. V = (B 1 + 4B 2 + B 3)/3 x H, where V is the volume of the prismoid, B1 and B3 are the areas of the top and bottom bases of the prismoid, B2 is the area of the middle base, and H is the height of the prismoid.

In summary, the prismoidal formula is a mathematical formula used to estimate the volume of a prismoid, which is a 3-dimensional shape formed by stacking a series of parallelograms are trapezoids on top of each other. The formula is derived by finding an expression for the volume of the prismoid that takes into account both the heights and the bases of the parallelograms or trapezoids that make up the shape. The final expression for the prismoidal formula is given by: V = (B1 + 4B2 + B3)/3 * H, where V is the volume of the prismoid, B1 and B3 are the areas of the top and bottom bases of the prismoid, B2 is the area of the middle base, and H is the height of the prismoid.

Derive an expression for Trapezoidal Formula

The trapezoidal formula is a mathematical formula used to estimate the definite integral of a function. The following is an explanation of how to derive an expression for the trapezoidal formula:

  1. Definition of Definite Integral: The definite integral is a concept in calculus that represents the area under a curve. It is used to find the total accumulation of a quantity over a specified interval.
  2. Definition of Trapezoidal Rule: The trapezoidal rule is a numerical method used to approximate the definite integral of a function by dividing the interval into a series of trapezoids and summing the areas of each trapezoid.
  3. Deriving the Trapezoidal Formula: To derive the trapezoidal formula, we divide the interval into n subintervals, each of width h = (b-a)/n, where a and b are the limits of integration. The heights of the trapezoids are given by the values of the function evaluated at the endpoints of each subinterval. The area of each trapezoid is given by the average of the heights times the width, which gives us the formula for the trapezoidal rule.
  4. Final Expression: The final expression for the trapezoidal formula is given by:

ab f(x) dx ≈ (h/2) * [f(a) + 2f(a + h) + 2f(a + 2h) + … + 2f(b-h) + f(b)], where h = (b-a)/n and n is the number of subintervals.

In summary, the trapezoidal formula is a mathematical formula used to estimate the definite integral of a function. The formula is derived by dividing the interval into a series of trapezoids and summing the areas of each trapezoid. The final expression for the trapezoidal formula is given by: ∫ab f(x) dx ≈ (h/2) * [f(a) + 2f(a + h) + 2f(a + 2h) + … + 2f(b-h) + f(b)], where h = (b-a)/n and n is the number of subintervals.

Recall the concept of Prismoidal Correction

The concept of Prismoidal Correction is a mathematical correction applied to the trapezoidal rule to improve its accuracy in estimating the definite integral of a function. The following is an explanation of the concept of Prismoidal Correction:

  1. The Trapezoidal Rule: The trapezoidal rule is a numerical method used to approximate the definite integral of a function by dividing the interval into a series of trapezoids and summing the areas of each trapezoid. The trapezoidal rule is a simple and effective method, but it is not always accurate.
  2. The Problem with the Trapezoidal Rule: The problem with the trapezoidal rule is that it underestimates the area under the curve if the function is concave down and overestimates the area if the function is concave up. This is because the trapezoids do not always fit the shape of the curve accurately.
  3. Prismoidal Correction: To overcome the limitations of the trapezoidal rule, a correction factor can be applied to the trapezoidal formula. This correction factor is called the Prismoidal Correction. The Prismoidal Correction adjusts the trapezoidal rule to take into account the shape of the curve, which results in a more accurate estimate of the definite integral.
  4. Deriving the Prismoidal Correction: The Prismoidal Correction is derived by dividing the interval into three equal parts and finding the area of each of the three parts using the trapezoidal rule. The sum of the three trapezoidal areas is then used to find the Prismoidal Correction.
  5. Final Expression: The final expression for the Prismoidal Correction is given by:

∫ ab f(x) dx = (4/3)xT 3 – (1/3)xT 1, where T 3 is the area estimated by the trapezoidal rule using three subintervals and T1 is the area estimated by the trapezoidal rule using one subinterval.

In summary, the concept of Prismoidal Correction is a mathematical correction applied to the trapezoidal rule to improve its accuracy in estimating the definite integral of a function. The Prismoidal Correction adjusts the trapezoidal rule to take into account the shape of the curve, which results in a more accurate estimate of the definite integral. The final expression for the Prismoidal Correction is given by: ∫ab f(x) dx = (4/3)*T3 – (1/3)*T1, where T3 is the area estimated by the trapezoidal rule using three subintervals and T1 is the area estimated by the trapezoidal rule using one subinterval.

Recall different methods for the Calculation of Volume from Contour Plans

The calculation of volume from contour plans involves determining the volume of an object or region from a series of contour lines or cross-sectional views. The following are different methods for the calculation of volume from contour plans:

  1. Prismoidal Method: The Prismoidal Method involves dividing the object into a series of prisms, each with its own cross-sectional area and height. The volume of the object is then estimated by summing the volumes of all the prisms.
  2. Trapezoidal Method: The Trapezoidal Method is similar to the Prismoidal Method, but it involves dividing the object into a series of trapezoids, each with its own cross-sectional area and height. The volume of the object is then estimated by summing the volumes of all the trapezoids.
  3. Cross-Sectional Area Method: The Cross-Sectional Area Method involves determining the cross-sectional area of the object at regular intervals along its length. The volume of the object is then estimated by summing the cross-sectional areas and multiplying by the length of the object.
  4. Average End Area Method: The Average End Area Method involves determining the average cross-sectional area of the object at its two ends. The volume of the object is then estimated by multiplying the average end area by the length of the object.
  5. Mid-Ordinate Method: The Mid-Ordinate Method involves determining the cross-sectional area of the object at its midpoint and multiplying it by the length of the object. The volume of the object is then estimated by summing the cross-sectional areas and multiplying by the length of the object.

In summary, the calculation of volume from contour plans involves determining the volume of an object or region from a series of contour lines or cross-sectional views. Different methods for the calculation of volume from contour plans include the Prismoidal Method, the Trapezoidal Method, the Cross-Sectional Area Method, the Average End Area Method, and the Mid-Ordinate Method.

Recall the determination of Capacity of Reservoir

The determination of capacity of a reservoir involves finding the volume of water that the reservoir can hold. This is an important step in the design and construction of reservoirs, as well as in their management and operation. The following steps can be followed in the determination of capacity of a reservoir:

  1. Measurement of Surface Area: The first step in determining the capacity of a reservoir is to measure its surface area. This can be done using a planimeter or by using mathematical formulas to determine the area of the shape of the reservoir.
  2. Measurement of Maximum Depth: The next step is to measure the maximum depth of the reservoir. This will give us the height of the reservoir and help us to determine its volume.
  3. Calculation of Volume: The volume of the reservoir can be calculated by multiplying its surface area by its maximum depth. This will give us an estimate of the total volume of water that the reservoir can hold.
  4. Correction for Irregularities: In many cases, the surface of a reservoir may not be flat and its depth may not be uniform. To account for these irregularities, a correction factor may be applied to the calculated volume to account for the differences in the shape and depth of the reservoir.
  5. Determination of Live Storage: The total volume of the reservoir is known as the dead storage, and it includes the volume of water that is below the lowest outlet or spillway. The live storage of the reservoir is the portion of the total volume that can be used for storage, and it can be determined by subtracting the dead storage from the total volume.

In summary, the determination of capacity of a reservoir involves finding the volume of water that the reservoir can hold. This involves measuring the surface area, maximum depth, and calculating the volume of the reservoir. Correction factors may be applied to account for irregularities in the shape and depth of the reservoir, and the live storage can be determined by subtracting the dead storage from the total volume.