# 15.4: Perspective projection

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Consider two planes (Pi) and (Pi') in the Euclidean space. Let (O) be a point that does not belong neither to (Pi) nor (Pi').

A perspective projection from (Pi) to (Pi') with center at (O) maps a point (Pin Pi) to the intersection point (P'=Pi'cap (OP)).

In general, perspective projection is not a bijection between the planes. Indeed, if the line ((OP)) is parallel to (Pi') (that is, if ((OP)capPi'=emptyset)) then the perspective projection of (Pin Pi) is undefined. Also, if ((OP')parallel Pi) for (P'in Pi'), then the point (P') is not an image of the perspective projection.

For example, let (O) be the origin of ((x,y,z))-coordinate space and the planes (Pi) and (Pi') are given by the equations (z=1) and (x=1) respectively. Then the perspective projection from (Pi) to (Pi') can be written in the coordinates as

((x,y,1)mapsto (1,dfrac{y}{x},dfrac{1}{x}).)

Indeed the coordinates have to be proportional; points on (Pi) have unit (z)-coordinate, and points on (Pi') have unit (z)-coordinate.

The perspective projection, maps one plane to another. However, we can identify the two planes by fixing a coordinate system in each. In this case we get a partially defined map from the plane to itself. We will keep the name perspective transformation for such maps.

For the described perspective projection; we may get the map

[eta:(x,y)mapsto (dfrac{1}{x},dfrac{y}{x}).]

This map is undefined on the line (x=0). Also points on this line are not images of points under perspective projection.

Denote by (hat Pi) and (hat Pi') the projective completions of (Pi) and (Pi') respectively. Note that the perspective projection is a restriction of composition of the two bijections (hat PileftrightarrowPhi leftrightarrowhat Pi') constructed in the previous section. By Observation 15.3.1, the perspective projection can be extended to a bijection (hat Pileftrightarrowhat Pi') that sends lines to lines. (A similar story happened with inversion. An inversion is not defined at its center; moreover, the center is not an inverse of any point. To deal with this problem we passed to the inversive plane which is the Euclidean plane extended by one ideal point. The same strategy worked for perspective projection (Pi o Pi'), but this time we need to add an ideal line.)

For example, to define extension of the perspective projection (eta) in 15.4.1, we have to observe that

• The pencil of vertical lines (x=a) is mapped to itself.

• The ideal points defined by pencil of lines (y=m cdot x+ b) are mapped to the point ((0,m)) and the other way around — point ((0, m)) is mapped to the ideal point defined by the pencil of lines (y=m cdot x+ b).

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## 15.4: Perspective projection

In Perspective Projection the center of projection is at finite distance from projection plane. This projection produces realistic views but does not preserve relative proportions of an object dimensions. Projections of distant object are smaller than projections of objects of same size that are closer to projection plane. The perspective projection can be easily described by following figure :

1. Center of Projection –
It is a point where lines or projection that are not parallel to projection plane appear to meet.
2. View Plane or Projection Plane –
The view plane is determined by :

Types of Perspective Projection :
Classification of perspective projection is on basis of vanishing points (It is a point in image where a parallel line through center of projection intersects view plane.). We can say that a vanishing point is a point where projection line intersects view plane.

The classification is as follows :

One Point Perspective Projection –
One point perspective projection occurs when any of principal axes intersects with projection plane or we can say when projection plane is perpendicular to principal axis.

Application of Perspective Projection :
The perspective projection technique is used by artists in preparing drawings of three-dimensional objects and scenes.

## Perspective Projection, Drawing – Its Types, Objectives, Methods.

This is such type of pictorial drawing in which the shape of an object looks same as its construction.

The picture obtained in perspective drawing resembles the picture taken by a camera.

A common man cannot understand an orthographic drawing.

Since, in the perspective projection, the actual shape of the object looks like the true one, therefore, even a common man can understand it easily.

Perspective drawing is constructed for a building so that the concerned persons, like the owner of the house, officer-in-charge of the department could understand it easily, generally who are not expert technically.

The actual dimensions are not given in perspective projection.

The object is seen from one point for the construction of this drawing.

So, the horizontal lines do not look parallel but seem to meet at a point, and that point is called vanishing point.

### Objectives OF Perspective Projection.

Following are the Objectives of Perspective Drawing:

1. To represent the actual construction of buildings and other objects.

2. To prepare other Models.

3. To illustrate drawings in pamphlets.

4. To illustrate different parts of the assembly drawing.

5. To represent positions of moving things.

6. The artists construct such drawings to illustrate the actual view.

### Principles OF Perspective Projection.

Following are the principles of Perspective Projection:

1. The vertical axes of the drawing are shown perpendicular.

2. The horizontal lines of the drawing look to meet at a point called Vanishing Point.

3. The vanishing points are shown on the horizontal lines in the System of Horizontal Lines.

4. If a horizontal line lies in the horizontal plane, then its actual length will be visible.

5. If some lines is in the picture plane, then its length will be visible short to its actual length.

### Technical Terms OF Perspective Drawing.

Following are the technical terms of Perspective Drawing:

#### I. Station Paint.

It is the point from where the observer looks object.

#### Ii. Picture Plane.

It is the plane in which picture of the object forms.

#### Iii. Horizontal Plane.

This plane is parallel to the Earth and in line with the eye of the observer.

#### Iv. Ground Plane.

If some plane is taken on the surface of the Earth. This is called a Ground Plane.

#### V. Ground Line.

This is the line of intersection of the ground plane and the picture plane.

#### Vi. Horizontal Line.

This is such a line which starts from observer’s eye and meets the picture plane after passing through the horizontal plane.

#### Vii. Center of Vision.

This is the point on the picture plane which is the point of intersection of the horizontal line emerging from Station point.

#### Viii. Piercing Point.

This is the point on the picture plane which is formed by the intersection of the picture plane and the projection line.

#### Ix. Vanishing Point.

This is the point at infinity where the length, breadth, and height of an object vanish.

#### X. Axis of Vision.

This is the horizontal line which starts from the observer’s eye and makes an angle of 90° with the picture plane.

#### Location OF Station Point:

Station point is selected at such a position so, that the vision of the object is very clear.

It is selected nearly the middle of the object so, that the angle formed to the sides of the object be kept between 30° to 60°.

#### Position OF Object:

While drawing a perspective drawing, the horizontal plane is taken at a height equal to the observer’s eye.

If the object is small then drawing is drawn by placing the object somewhat above or below the horizontal plane.

If the object is placed above the horizontal plane, then its lower view will be seen while if it is placed somewhat below then, its upper view will be seen.

### Types OF Perspective Drawing:

Following are the two methods of Perspective Drawing:

1. Angular Perspective .

2. Parallel Perspective.

#### 1. Angular Perspective Drawing:

This type of perspective drawing is also called two-point perspective drawing.

This method involves the following steps:

i. The line of the picture plane is drawn at some suitable place in the drawing sheet. It is represented by P-P.

ii. The top view of the object is drawn making an angle of 30 to 60 degrees with the horizontal line on the picture plane.

iii. The station point is shown at some suitable place below the picture plane. All the points of the top view are joined with the station point.

iv. The lines joining the top view with the Station point will intersect the picture plane.

In this way, the points produced on the picture plane are called Piercing Points.

Perpendicular, parallel lines will be drawn downward from these points.

v. Lines are drawn parallel to the sides of the top view to intersect the picture plane.

Perpendicular lines will be drawn from these intersecting points in the downward direction.

vi. A horizontal line is drawn at some suitable position below the station point.

The points where this line intersects the previously drawn vertical lines will be the Vanishing Points.

vii. The front view of the object is drawn at some place below the horizontal line.

viii. Horizontal lines are drawn from the different points of the front view. These will intersect the vertical lines drawn from the top view.

In this way, the points produced by lines coming from a single point of top view and the front view will be the points of the Perspective drawing.

The perspective drawing will be completed by joining these points to the right and left vanishing points.

#### 2. Parallel Perspective Drawing:

Perspective projection is drawn mostly by this method. In this method, only one vanishing point is selected.

This type of perspective drawing is also called one point perspective drawing.

This method involves the following steps:

i. The top view is drawn at a suitable place and front view under it.

ii. The picture plane is shown under the top view and ground level under the front view.

iii. A station point is shown by one side of both the views at some suitable angle, and all the points of the top view are joined with it.

iv. The points arising from the intersection of the picture plane and the visual rays joining the point of top view to the station point will be piercing points.

v. The horizontal line is drawn between the picture plane and ground level. The vanishing point is put on the station point on the horizontal line.

All the points of the front view are joined with it.

vi. The vertical line under the top view and the line going from front view toward the vanishing point will intersect each other.

The point arising from the intersection of lines emerging from the same point of the two views will be the point of the perspective drawing.

By joining all such points, the perspective projection will be completed.

## 15.4: Perspective projection

Projection are defined as mapping of three-dimensional points to a two-dimensional plane. There are two type of projection parallel and perspective.

1. Parallel Projection :
Parallel projections are used by architects and engineers for creating working drawing of the object, for complete representations require two or more views of an object using different planes.

Parallel Projection use to display picture in its true shape and size. When projectors are perpendicular to view plane then is called orthographic projection. The parallel projection is formed by extending parallel lines from each vertex on the object until they intersect the plane of the screen. The point of intersection is the projection of vertex.

2. Perspective Projection :
Perspective projections are used by artist for drawing three-dimensional scenes.

In Perspective projection lines of projection do not remain parallel. The lines converge at a single point called a center of projection. The projected image on the screen is obtained by points of intersection of converging lines with the plane of the screen. The image on the screen is seen as of viewer’s eye were located at the centre of projection, lines of projection would correspond to path travel by light beam originating from object.

Two main characteristics of perspective are vanishing points and perspective foreshortening. Due to foreshortening object and lengths appear smaller from the center of projection. More we increase the distance from the center of projection, smaller will be the object appear.

Difference Between Parallel Projection And Perspective Projection :

## Practical Work in Geography Class 11 Solutions Chapter 4 Map Projections

### Class 11 Practical Work in Geography Chapter 4 NCERT Textbook Questions Solved

1. Choose the right answer from the four alternatives given below:

Question 1(i).
A map projection least suitable for the world map:
(а) Mercator
(b) Simple Cylindrical
(c) Conical
(d) All the above
(c) Conical

Question 1(ii).
A map projection that is neither the equal area nor the correct shape and even the directions are also incorrect
(a) Simple Conical
(b) Polar zenithal
(c) Mercator
(d) Cylindrical
(a) Simple Conical

Question 1(iii).
A map projection having correct direction and correct shape but area greatly exaggerated.polewards is:
(a) Cylindrical Equal Area
(b) Mercator
(c) Conical
(d) All the above
(b) Mercator

Question 1(iv).
When the source of light is placed at the centre of the globe, the resultant projection is called:
(a) Orthographic
(b) Stereographic
(c) Gnomonic
(d) All the above
(c) Gnomonic

Question 2(i).
Describe the elements of map projection.

• Reduced Earth: A model of the earth is represented by the help of a reduced scale on a flat sheet of paper. This model is called the “reduced earth”.
• Parallels of Latitude: These are the circles running round the globe parallel to the equator and maintaining uniform distance from the poles.
• Meridians of Longitude: These are semi-circles drawn in north-south direction from one pole to the other, and the two opposite meridians make a complete circle, i.e. circumference of the globe.
• Global Property: In preparing a map projection the following basic properties of the global surface are to be preserved by using one or the other methods:
• Distance between any given points of a region
• Shape of the region
• Size or area of the region in accuracy
• Direction of any one point of the region bearing to another point.

Question 2(ii).
What do you mean by global property?
In preparing a map projection the following basic properties of the global surface are to be preserved by using one or the other methods:

1. Distance between any given points of a region
2. Shape of the region
3. Size or area of the region in accuracy
4. Direction of any one point of the region bearing to another point.

Question 2(iii).
Not a single map projection represents the globe truly. Why?
However, there is no such projection, which maintains the scale correctly throughout. It can be maintained correctly only along some selected parallels and meridians as per the requirement. Projection is a shadow of globe which has to be presented on a map. When shape of globe changes certainly inaccuracy comes in. Therefore, it is rightly said that not a single map projection represents the globe truly.

Question 2(iv).
How is the area kept equal in cylindrical equal area projection?
The area is kept equal in cylindrical equal area projection because latitudes and longitudes intersect each other at right angles in the straight line form.

Question 3(i).
Developable and non-developable surfaces

 Basis Developable Surface Non-developable Surface Meaning A developable surface is one, which can be flattened, and on which, a network of latitude and longitude can be projected. A non-developable surface is one, which cannot be flattened without shrinking, breaking or creasing. Example A cylinder, a cone and a plane have the property of developable surface. A globe or spherical surface has the property of non-developable surface

On the basis of nature of developable surface, the projections are classified as cylindrical, conical and zenithal projections.

Question 3(ii).
Homolographic and orthographic projections

 Basis Homolographic Projection Orthographic Projection Meaning A projection in which the network of latitudes and longitudes is developed in such a way that every graticule on the map is equal in area to the corresponding graticule on the globe. It is also known as the equal-area projection. A projection in which the correct shape of a given area of the earth’s surface is preserved.

Question 3(iii).
Normal and oblique projections

 Basis Normal Projection Oblique Projection Meaning If the developable surface touches the globe at the equator, it is called the equatorial or normal projection. If projection is tangential to a point between the pole and the equator, it is called the oblique projection.

Question 3(iv).
Parallels of latitude and meridians of longitude

 Basis Meridians of Longitude Parallels of Latitude Meaning The meridians of longitude refer to the angular distance, in degrees, minutes, and seconds, of a point east or west of the Prime (Greenwich) Meridian. The parallels of latitude refer to the angular distance, in degrees, minutes and seconds of a point north or south of the Equator. Name Lines of longitude are often referred to as meridians. Lines oflatitude are often referred to as parallels. Reference point 0° longitude is called prime meridian. 0° latitude is called equator. Division It divides the earth into eastern hemisphere and western hemisphere. It divides the earth into northern hemisphere and southern hemisphere. Number These are 360 in number: 180 in the eastern hemisphere and 180 in the western hemisphere. These are 180 in number: 90 in southern hemisphere and 90 in northern hemisphere. Importance It helps to determine time of a place. It helps to determine temperature of a place. Equality These are not equal. These are equal.

4. Answer the following questions in not more than 125 words:

Question 4(i).
Discuss the criteria used for classifying map projection and state the major characteristics of each type of projection.
Types of Map Projection:

1. On the basis of drawing techniques, map Projections maybe classified perspective, non-perspective and conventional or mathematical. Perspective projections can be drawn taking the help of a source of light by projecting the image of a network of parallels and meridians of a globe on developable surface. Non¬perspective projections are developed without the help of a source of light or casting shadow on surfaces, which can be flattened. Mathematical or conventional projections are those, which are derived by mathematical computation and formulae and have little relations with the projected image.

2. On the basis of developable surface, it can be developable surface and non developable surface. A developable surface is one, which can be flattened, and on which, a network of latitude and longitude can be projected. A globe or spherical surface has the property of non-developable surface whereas a cylinder, a cone and a plane have the property of developable surface. On the basis of nature of developable surface, the projections are classified as cylindrical, conical and zenithal projections.

3. On the basis of global properties, projections are classified into equal area, orthomorphic, azimuthal and equidistant projections.

4. On the basis of location of source of light, projections maybe classified as gnomonic, stereographic and orthographic.

The correctness of area, shape, direction and distances are the four major global properties to be preserved in a map. But none of the projections can maintain all these properties simultaneously. Therefore, according to specific need, a projection can be drawn so that the desired quality may be retained.

Question 4(ii).
Which map projection is very useful for navigational purposes? Explain the properties and limitations of this projection.
Mercator’s Projection is very useful for navigational purposes. A Dutch cartographer Mercator Gerardus Karmer developed this projection in 1569. The projection is based on mathematical formulae.

• It is an orthomorphic projection in which the correct shape is maintained.
• The distance between parallels increases towards the pole.
• Like cylindrical projection, the parallels and meridians intersect each other at right angle. It has the characteristics of showing correct directions.
• A straight line joining any two points on this projection gives a constant bearing, which is called a Laxodrome or Rhumb line.
• All parallels and meridians are straight lines and they intersect each other at right angles.
• All parallels have the same length which is equal to the length of equator.
• All meridians have the same length and equal spacing. But they are longer than the corresponding meridian on the globe.
• Spacing between parallels increases towards the pole.
• Scale along the equator is correct as it is equal to the length of the equator on the globe but other parallels are longer than the corresponding parallel on the globe hence the scale is not correct along them.
• Shape of the area is maintained, but at the higher latitudes distortion takes place.
• The shape of small countries near the equator is truly preserved while it increases towards poles.
• It is an azimuthal projection.
• This is an orthomorphic projection as scale along the meridian is equal to the scale along the parallel.
• There is greater exaggeration of scale along the parallels and meridians in high latitudes. As a result, size of the countries near the pole is highly exaggerated.
• Poles in this projection cannot be shown as 90° parallel and meridian touching them are infinite.

Question 4(iii).
Discuss the main properties of conical projection with one standard parallel and describe its major limitations.
A conical projection is one, which is drawn by projecting the image of the graticule of a globe on a developable cone, which touches the globe along a parallel of latitude called the standard parallel. As the cone touches the globe located along AB, the position of this parallel on the globe coinciding with that on the cone is taken as the standard parallel.

• All the parallels are arcs of concentric circle and are equally spaced.
• All meridians are straight lines merging at the pole. The meridians intersect the parallels at right angles.
• The scale along all meridians is true.
• An arc of a circle represents the pole.
• The scale is true along the standard parallel but exaggerated away from the standard parallel.
• Meridians become closer to each other towards the pole.
• This projection is neither equal area nor orthomorphic.
• It is not suitable for a world map due to extreme distortions in the hemisphere opposite the one in which the standard parallel is selected.
• Even within the hemisphere, it is not suitable for representing larger areas as the distortion along the pole and near the equator is larger.
• This projection is commonly used for showing areas of mid-latitudes with limited latitudinal and larger longitudinal extent.
• A long narrow strip of land running parallel to the standard parallel and having east-west stretch is correctly shown on this projection.
• Direction along standard parallel is used to show railways, roads, narrow river valleys and international boundaries.
• This projection is suitable for showing the Canadian Pacific Railways, Trans- Siberian Railways, international boundaries between USA and Canada and the Narmada Valley.

1. Construct graticule for an area stretching between 30° N to 70° N and 40° E to 30° W on a simple conical projection with one standard parallel with a scale of 1 : 200,000,000 and interval at an 10° apart.
Attempt yourself.

2. Prepare graticule for a Cylindrical Equal Area Projection for the world when R.F. is 1: 150,000,000 and the interval is 15° apart.
Construction
(frac<6,40,000,000><150,000,000>)
=4.26cm
(round off to 4.3 cm)
Draw a circle of 4.3 cm radius
Mark the angles of 15°, 30°, 45°, 60°, 75° and 90° for both, northern and southern hemispheres
Length of the equator = 2π r
=2 × (frac<22><7>) × 4.3 = 27.03 cm,
Draw a line of 27.03 cm.
Divide it into 24 equal parts at a distance of 1.1262 cm apart.
This line represents the equator
Draw a line perpendicular to the equator at the point where 0° is meeting the circumference of the circle
Extend all the parallels equal to the length of the equator from the perpendicular line and complete the projection as shown in figure given below:

3. Draw a Mercator Projection for the world map when the R.F. is 1:400,000,000 and the interval between the latitude and longitude is 20°.
Calculation
(frac<250,000,000><400,000,000>)=0.625
Radius of the reduced earth R is 0.625″ is 1: 400,000,000
Length of the equator 2πR or
2 × (frac<22><7>) × 0.625
= 3.93″ inches
Interval along equator
(frac < 3.93 imes 20 > < 360 >) =0.218

• Draw a line of 3.93″ inches representing the equator as Equation.
• Divide it into 24 equal parts. Determine the length of each division using the following formula: Length of the equator multiplied by interval divided by 360°.
• Calculate the distance for latitude with the help of the table given below:

Complete the projection as shown in Figure given below:

### Class 11 Practical Work in Geography Chapter 4 NCERT Extra Questions

#### Class 11 Practical Work in Geography Chapter 4 Multiple Choice Questions

Question 1.
(a) Mercator Gerardus Karmer
(b) Lambert
(c) Plato
(d) Hambolt
(a) Mercator Gerardus Karmer

Question 2.
Which of the following geographical feature is not there in a map?
(a) Area
(b) Direction
(c) Shape
(d) Topography
(d) Topography

Question 3.
Which of the following is called equal area projection?
(a) Orthomorphic Projection
(b) Azimuthal Projection
(c) Equidistant projections
(d) Homolographic projection
(d) Homolographic projection

Question 4.
Which projection is obtained by putting the light at the centre of the globe?
(a) Gnomonic Projection
(b) Azimuthal Projection
(c) Equidistant projections
(d) Homolographic projection
(a) Gnomonic Projection

Question 5.
In which projection different parts of the . earth are shown accurately?
(а) Orthomorphic Projection
(b) Azimuthal Projection
(c) Equidistant projections
(d) Homolographic projection
(d) Homolographic projection

Question 6.
Which of the following projection is not classified on the basis of source of light?
(a) Gnomonic Projection
(b) Stereographic Projection
(c) Equal area projection
(d) Orthographic Poijection
(c) Equal area projection

Question 7.
Which of the following is not a quality of globe?
(a) Accurate shape of a region
(b) Accurate area of a place
(c) Showing direction of one pace from another place
(d) Showing light
(d) Showing light

Question 8.
Which of the following is not a developable surface?
(a) Angle
(b) Cylindrical
(c) Plane
(d) Map
(d) Map

Question 9.
Which of the following does not have qualities of developable surface?
(a) Conical
(b) Cylindrical
(c) Plane
(d) Globe
(d) Globe

#### Class 11 Practical Work in Geography Chapter 4 Very Short Answer Type Questions

Question 1.
What is the shape of meridians and parallels in Mercator projection?
All parallels and meridians are straight lines and they intersect each other at right angles. All parallels have the same length which is equal to the length of equator. All meridians have the same length and equal spacing. But they are longer than the corresponding meridian on the globe. Spacing between parallels increases towards the pole.

Question 2.
Classify projections on the basis of method of construction.
On the basis of method of construction, projections are generally classified into perspective, non-perspective and conventional or mathematical.

Question 3.
What do you mean by non-developable surface?
A non-developable surface is one, which cannot be flattened without shrinking, breaking or creasing. A globe or spherical surface has the property of non-developable surface.

Question 4.
What is Lexodrome or Rhumb line?
Lexodrome or Rhumb Line is a straight line drawn on Mercator’s projection joining any two points having a constant bearing. It is very useful in determining the directions during navigation.

Question 5.
What is mathematical or conventional projection?
Mathematical or conventional projec-tions are those, which are derived by mathematical computation and formulae and have little relations with the projected image.

Question 6.
How can perspective and non-perspective projections be drawn?
Perspective projections can be drawn taking the help of a source of light by projecting the image of a network of parallels and meridians of a globe on developable surface. Non-perspective projections are developed without the help of a source of light or casting shadow on surfaces, which can be flattened.

Question 7.
What is developable surface?
A developable surface is one, which can be flattened, and on which, a network of latitude and longitude can be projected. A cylinder, a cone and a plane have the property of developable surface.

Question 8.
How can we obtain projection on a plane surface?
When the cylinder is cut open, it provides a cylindrical projection on the plane sheet. A Conical projection is drawn by wrapping a cone round the globe and the shadow of graticule network is projected on it. When the cone is cut open, a 1 projection is obtained on a flat sheet.

Question 9.
What are limitations of Mercator
Projection?

• There is greater exaggeration of scale along the parallels and meridians in high
latitudes. As a result, size of the countries near the pole is highly exaggerated.
• Poles in this projection cannot be shown as 90° parallel and meridian touching them are infinite.

#### Class 11 Practical Work in Geography Chapter 4 Short Answer Type Questions

Question 1.
How are conical projections drawn?
A Conical projection is drawn by wrapping a cone round the globe and the shadow of graticule network is projected
on it. When the cone is cut open, a projection is obtained on a flat sheet. A conical projection is one, which is drawn by projecting the image of the ‘ graticule of a globe on a developable cone, which touches the globe along a parallel of latitude called the standard parallel. As the cone touches the globe located along AB, the position of this parallel on the globe coinciding with that on the cone is taken as the standard parallel. The length of other parallels on either side of this parallel are distorted.

Question 2.
What is map projection?
It is the system of transformation of the spherical surface onto a plane | surface. It is carried out by an orderly
and systematic representation of the parallels of latitude and the meridians of longitude of the spherical earth or part of it on a plane surface on a conveniently chosen scale. In map projection we try to represent a good model of any part of the earth in its true shape and dimension. But distortion in some form or the other is inevitable.

To avoid this distortion, various methods have been devised and many types of projections are drawn. Due to this reason, map projection is also defined as the study of different methods which have been tried for transferring the lines of graticule from the globe to a flat sheet of paper.

Question 3.
What are the qualities and limitations of a globe?
Qualities of globe can be expressed as follows:

• A globe is the best model of the earth. Due to this property of the globe, the shape and sizes of the continents and oceans are accurately shown on it.
• It shows the directions and distances very accurately.
• The globe is divided into various segments by the lines of latitude and longitude.
• It is expensive.
• It can neither be carried everywhere easily nor can a minor detail be shown on it.
• Besides, on the globe the meridians are semi-circles and the parallels are circles. When they are transferred on a plane surface, they become intersecting straight lines or curved lines.

Question 4.
Classify the projections on the basis of method of construction.
On the basis of method of construction, projections are generally classified into perspective, non-perspective and conventional or mathematical.

• Perspective projections: These can be drawn taking the help of a source of light by projecting the image of a network of parallels and meridians of a globe on developable surface.
• Non-perspective projections: These are developed without the help of a source of light or casting shadow on surfaces, which can be flattened.
• Mathematical or conventional projections: These are those, which are derived by mathematical computation and formulae and have little relations with the projected image.

Question 5.
Classify projections on the basis of global properties.
On the basis of global properties, projections are classified into:

• Equal Area Projection
• Orthomorphic Projection,
• Azimuthal Projection and
• Equidistant Projections.
• Equal Area Projection: It is also called homolographic projection. It is that projection in which areas of various parts of the earth are represented correctly.
• Orthomorphic or True-Shape projection: It is one in which shapes of various areas are portrayed correctly. The shape is generally maintained at the cost of the correctness of area.
• Azimuthal or True-Bearing projection: It is one on which the direction of all points from the centre is correctly represented.
• Equidistant or True Scale projection: It is that where the distance or scale is correctly maintained.

Question 6.
Write a short note on developable surface and zenithal projections.
A developable surface is one, which can be flattened, and on which, a network of latitude and longitude can be projected. A cylinder, a cone and a plane have the property of developable surface. On the basis of nature of developable surface, the projections are classified as cylindrical, conical and zenithal projections.

1. Cylindrical Projections: These are made through the use of cylindrical developable surface. A paper-made cylinder covers the globe, and the parallels and meridians are projected on it.

2. Zenithal projection: It is directly obtained on a plane surface when plane touches the globe at a point and the graticule is projected on it. Generally, the plane is so placed on the globe that it touches the globe at one of the poles. These projections are further subdivided into normal, oblique or polar as per the position of the plane touching the globe.

• Normal Projection: If the developable surface touches the globe at the equator, it is called equatorial or normal projection.
• Oblique Projection: If it is tangential to a point between the pole and the equator, it is called the oblique projection
• Polar Projection: If it is tangential to the pole, it is called the polar projection.

Question 7.
What is the need of map projection?
The need for a map projection mainly arises to have a detailed study of a region, which is not possible to do from a globe. Similarly, it is not easy to compare two natural regions on a globe. Therefore, drawing accurate large-scale maps on a flat paper is required. It gives birth to a problem. The problem is how to transfer these lines of latitude and longitude on a flat sheet. If we stick a flat paper over the globe, it will not coincide with it over a large surface without being distorted. If we throw light from the centre of the globe, we get a distorted picture of the globe in those parts of paper away from the line or point over which it touches the globe.

The distortion increases with increase in distance from the tangential point. So, tracing all the properties like shape, size and directions, etc. from a globe is nearly impossible because the globe is not a developable surface.

Map projection helps to solve this problem. In map projection we try to represent a good model of any part of the earth in its true shape and dimension. But distortion in some form or the other is inevitable. To avoid this distortion, various methods have been devised and many types of projections are drawn. Due to this reason, map projection is also defined as the study of different methods which have been tried for transferring the lines of graticule from the globe to a flat sheet of paper.

#### Class 11 Practical Work in Geography Chapter 4 Long Answer Type Questions

Question 1.
Explain the qualities of Mercator projection.
Mercator’s Projection is very useful for navigational purposes. A Dutch cartographer Mercator Gerardus Karmer developed this projection in 1569. The projection is based on mathematical formulae.
Properties:

• It is an orthomorphic projection in which the correct shape is maintained.
• The distance between parallels increases towards the pole.
• Like cylindrical projection, the parallels and meridians intersect each other at right angle. It has the characteristics of showing correct directions.
• A straight line joining any two points on this projection gives a constant bearing, which is called a Laxodrome or Rhumb line.
• All parallels and meridians are straight lines and they intersect each other at right angles.
• All parallels have the same length which is equal to the length of equator.
• All meridians have the same length and equal spacing. But they are longer than the corresponding meridian on the globe.
• Spacing between parallels increases towards the pole.
• Scale along the equator is correct as it is equal to the length of the equator on the globe but other parallels are longer than the corresponding parallel on the globe hence the scale is not correct along them.
• Shape of the area is maintained, but at the higher latitudes distortion takes place.
• The shape of small countries near the equator is truly preserved while it increases towards poles.
• It is an azimuthal projection.
• This is an orthomorphic projection as scale along the meridian is equal to the scale along the parallel.

Question 2.
Explain properties, limitations and uses of cylindrical equal area projection.
The cylindrical equal area projection is also known as the Lambert’s projection. It has been derived by projecting the surface of the globe with parallel rays on a cylinder touching it at the equator. Both the parallels and meridians are projected as straight lines intersecting one another at right angles. The pole is shown with a parallel equal to the equator hence, the shape of the area gets highly distorted at the higher latitude.

• All parallels and meridians are straight lines intersecting each other at right angle.
• Polar parallel is also equal to the equator.
• Scale is true only along the equator.
• Distortion increases as we move towards the pole.
• The projection is non-orthomorphic.
• Equality of area is maintained at the cost of distortion in shape.
• The projection is most suitable for the area lying between 45° N and S latitudes.
• It is suitable to show the distribution of tropical crops like rice, tea, coffee, rubber and sugarcane.

Question 3.
Explain properties of Conical Projection with one Standard Parallel.
A conical projection is one, which is drawn by projecting the image of the graticule of a globe on a developable cone, which touches the globe along a parallel of latitude called the standard parallel. As the cone touches the globe located along AB, the position of this parallel on the globe coinciding with that on the cone is taken as the standard parallel.

• All the parallels are arcs of concentric circle and are equally spaced.
• All meridians are straight lines merging at the pole. The meridians intersect the parallels at right angles.
• The scale along all meridians is true.
• An arc of a circle represents the pole.
• The scale is true along the standard parallel but exaggerated away from the standard parallel.
• Meridians become closer to each other towards the pole.
• This projection is neither equal area nor orthomorphic.

Question 4.
Explain the limitations and uses of Conical Projection with one Standard Parallel.
Limitations

• It is not suitable for a world map due to extreme distortions in the hemisphere opposite the one in which the standard parallel is selected.
• Even within the hemisphere, it is not suitable for representing larger areas as the distortion along the pole and near the equator is larger.
• This projection is commonly used for showing areas of mid-latitudes with limited latitudinal and larger longitudinal extent.
• A long narrow strip of land running parallel to the standard parallel and having east-west stretch is correctly shown on this projection.
• Direction along standard parallel is used to show railways, roads, narrow river valleys and international boundaries.
• This projection is suitable for showing the Canadian Pacific Railways, Trans- Siberian Railways, international boundaries between USA and Canada and the Narmada Valley.

Question 5.
Prepare graticule for a Cylindrical Equal Area Projection for the world when R.F. is 1: 300,000,000 and the interval is 15° apart.
Construction

• Draw a circle of 2.1 cm radius
• Mark the angles of 15°, 30°, 45°, 60°, 75° and 90° for both, northern and southern hemispheres
• Draw a line of 13.2 cm and divide it into 24 equal parts at a distance of 0.55cm apart.
• This line represents the equator
• Draw a line perpendicular to the equator at the point where 0° is meeting the circumference of the circle
• Extend all the parallels equal to the length of the equator from the perpendicular line and Complete the projection as shown in figure given below:

Question 6.
Draw a Mercator Projection for the world map when the R.F. is 1:250,000,000 and the interval between the latitude and longitude is 15°.
Calculation: Radius of the reduced earth R is “1 is 1: 250,000,000 Length of the equator 2πR or

• Draw a line of 6.28″ inches representing the equator as Equation.
• Divide it into 24 equal parts. Determine the length of each division using the following formula: Length of the equator multiplied by interval divided by 360°.
• Calculate the distance for latitude with the help of the table given below:
Latitude Distance 15° 0.25 x 1 = 0.25″ inch 30° and so on, Complete the projection as shown in Figure given below:

#### Class 11 Practical Work in Geography Chapter 4 Viva Questions

Question 1.
What is other name for cylindrical equal-area projection?
The cylindrical equal-area projection is also known as Lambert’s projection.

Question 2.
What is Lexodrome or Rhumb Line?
It is a straight line drawn on Mercator’s projection joining any two points having a constant bearing. It is very useful in determining the directions during navigation.

Question 3.
Which map projection is very useful for navigational purposes? Who developed it and on what is it based?
Mercator’s Projection is very useful for navigational purposes. A Dutch cartographer Mercator Gerardus Karmer developed this projection in 1569. The projection is based on mathematical formulae.

Question 4.
Name different types of projections on the basis of method of construction.
On the basis of method of construction, projections are generally classified into perspective, non-perspective and conventional or mathematical.

Question 5.
Name some developable surface.
A cylinder, a cone and a plane have the property of developable surface.

Question 6.
What is mathematical or conventional projection?
Mathematical or conventional projections are those, which are derived by mathematical computation and formulae and have little relations with the projected image.

Question 7.
what is the need of map projection?
The need for a map projection mainly arises to have a detailed study of a region, which is not possible to do from a globe. Similarly, it is not easy to compare two natural regions on a globe. Therefore, drawing accurate large-scale maps on a flat paper is required.

## First: method -distort perspective

The distortion method perspective will make sure that straight lines in the source image will remain straight lines in the destination image. Other methods, like barrel or bilinearforward do not: they will distort straight lines into curves.

The -distort perspective requires a set of at least 4 pre-calculated pairs of pixel coordinates (where the last one may be zero). More than 4 pairs of pixel coordinates provide for more accurate distortions. So if you used for example:

(for readability reasons using more blanks between the mapping pairs than required) would mean:

1. From the source image take pixel at coordinate (1,2) and paint it at coordinate (3,4) in the destination image.
2. From the source image take pixel at coordinate (5,6) and paint it at coordinate (7,8) in the destination image.
3. From the source image take pixel at coordinate (9,10) and paint it at coordinate (11,12) in the destination image.
4. From the source image take pixel at coordinate (13,14) and paint it at coordinate (15,16) in the destination image.

You may have seen photo images where the vertical lines (like the corners of building walls) do not look vertical at all (due to some tilting of the camera when taking the snap). The method -distort perspective can rectify this.

It can even achieve things like this, 'straightening' or 'rectifying' one face of a building that appears in the 'correct' perspective of the original photo:

==>

The control points used for this distortion are indicated by the corners of the red (source controls) and blue rectangles (destination controls) drawn over the original image:

==>

This particular distortion used

Complete command for your copy'n'paste pleasure:

## 4D to 3D perspective projection

Im trying to calculate the position of 4D point in 3D world. I started with 2D and tried to extend it to the 3D and then to 4D. Firstly, I found out that its easy to calculate the projected position of 2D point on the line.

Now I figured out that the same will apply in the 3D world if I split the P(X,Y,Z) to the P1(X,Z) and P2(Y,Z), calcualte their Q and then build a point of P'(Q1,Q2) (Assuming Im looking Z axis positive infinity from C(0,-a) point and rendering to the XY plane).

Then I thought its just as simple as adding next point P3, and came up with

I felt it was weird, becouse W (new axis) actually affects only Z of the last point, and referring to the tesseract it should affect all dimensions.

This isn't working, so I'd like to ask if you can possibly provide some details of what Im doing wrong. Im pretty sure that its the "point splitting" problem, and the equation should be more complex. Please, don't attack me with matrixes and quaternions. I just want to have a simple static camera at (0,-1) looking at (0,0).

## Analytic Treatment of the Perspective View of a Circle

One is taught in drawing class, that circular objects in three-dimensional Euclidean Space are drawn in perspective as ellipses. The usual construction is to draw a square around the circle, and then project the perspective view of the square by finding its edges using the vanishing points and measuring points, the center by drawing the diagonals, and then sketching the projected circle by drawing it tangent to the projected square. A beginner will sometimes make the mistake of trying to make the tangency points the same as the endpoints of the axes of the ellipse, but they are not the same as seen in the p. 17 figure. But why is the image exactly the ellipse and not some other closed curve?

We shall answer this question by figuring out the equation of the image of the circle on the perspective drawing. We'll be using the methods of analytic geometry, where curves are represented by equations. Thus we shall describe a circle in three space by describing it as the locus of points satisfying certain equations. We then compute the corresponding perspective locus in terms of the Cartesian coordinates of the drawing plane. Finally, after some simplification, we will be able to recognize the curve as an ellipse.

The conic sections in the plane are given as the locus, that is the set of all points (u,v) in E 2 which satisfy a quadratic equation of the form

au 2 + 2buv + cv 2 + eu + fv + g = 0,

where a , b , c , d , e , f , g are constants. This can be deduced from the geometric description of the conic section as the intersection in three space of a plane with a right circular cone. All possible conic sections arise this way including degenerate ones such as lines and points and the empty set. For example if a=b=c=0 then

is the equation of a line and if a=c=1 , b=0 , d=-2u0 , e=-2v0 , g=-u0 2 -v0 2 then

a u 2 + c v 2 + e u + f v + g = (u-u0) 2 + (v-v0) 2 = 0

is satisfied only by one point (u,v)=(u0,v0) whereas

has no real solution at all. On the other hand if the discriminant

is negative, then the conic is a hyperbola, if D=0 the conic is a parabola and if D is positive the conic is an ellipse. The easiest to see are the canonical conic curves given by the formulae

Of course if a=b the ellipse is a circle.

Now let's see what a projective transformation looks like analytically. For simplicity, we assume that the set is located in front of the observer (all points of the circle satisfy y0 .) Then the horizontal and vertical coordinates of the drawing plane (points which satisfy y=1 ) are

where (x, y, z) runs through all points of the original set. Now suppose that we consider a circle in space with center (x0, y0, z0) and radius r and which lies on a plane not parallel to the drawing plane. By a rotation around the y -axis, we may arrange that the intersection line of the circle plane and the drawing plane is horizontal. In other words, the equation of the plane through the center of the circle sloping away from the drawing plane with slope m is given by

To be able to see the circle, we require that the eyepoint (0, 0, 0) is not on the plane of the circle, which means z0 does not equal m y0 . The circle also lies on the sphere of radius r centered at (x0, y0, z0) , which has the equation

The circle is the collection of points satisfying both (3.) and (4.) These are projected using (2.) to the drawing plane. By substituting (3.) into (4.),

(x - x0) 2 + (1 + m 2 )(y - y0) 2 = r 2 .

We are trying to see how these equations relate u to v . Using (2.), we substitute in the equations (3.) and (5.)

Substituting into equation (5.) and multiplying by (v - m) 2 yields

Multiplying out and collecting factors of u 2 , uv , . yields

Thus (u,v) satisfy a quadratic equation in the plane. The discriminant is

Since the eyepoint is not on the plane of the circle z0 - m y0 > 0. Since the circle is in front of the y=0 plane, the point (x0, 0, z0 + m y0) which is both in the y=0 plane and on the circle plane is can't be on the circle, in fact it is farther from the center than any point of the circle, hence

Thus D > 0 and the locus is an ellipse.

Perspective view of the circle

Here is a diagram from Alberti's treatise. The square that surrounds the circle projects to a trapezoid. The circle itself projects to an ellipse which is tangent to all four sides of the trapezoid. Observe that the left and right endpoints of the axes of the ellipse where the ellipse is widest occur below the tangency points. But be careful when drawing the ellipse which is not centered on the eyepoint to centerpoint line!

## 15.4: Perspective projection

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