Latest Geography NCERT Notes, Solutions and Extra Q & A (Class 8th to 12th)
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Class 11th Chapters
Fundamentals of Physical Geography
1. Geography As A Discipline	2. The Origin And Evolution Of The Earth	3. Interior Of The Earth
4. Distribution Of Oceans And Continents	5. Geomorphic Processes	6. Landforms And Their Evolution
7. Composition And Structure Of Atmosphere	8. Solar Radiation, Heat Balance And Temperature	9. Atmospheric Circulation And Weather Systems
10. Water In The Atmosphere	11. World Climate And Climate Change	12. Water (Oceans)
13. Movements Of Ocean Water	14. Biodiversity And Conservation
Indian Physical Environment
1. India — Location	2. Structure And Physiography	3. Drainage System
4. Climate	5. Natural Vegetation	6. Natural Hazards And Disasters: Causes, - Consequences And Management
Practical Work In Geography
1. Introduction To Maps	2. Map Scale	3. Latitude, Longitude And Time
4. Map Projections	5. Topographical Maps	6. Introduction To Remote Sensing

Chapter 4 Map Projections

Maps are essential tools used to represent the Earth or parts of it on a flat surface. However, since the Earth is a three-dimensional, curved object (specifically, a **geoid**), depicting it accurately on a two-dimensional plane presents a significant challenge. A globe is the best scaled model of the Earth's shape, showing areas, shapes, directions, and distances relatively accurately.

The globe is structured using a network of imaginary lines: horizontal lines representing **parallels of latitude** and vertical lines representing **meridians of longitude**. This grid is known as the **graticule**. For creating a map, this spherical graticule needs to be transferred onto a flat surface.

A **map projection** is the systematic method or system used to transfer this network of latitudes and longitudes from the curved surface of the Earth (or a globe representing it) onto a plane surface, typically at a chosen scale. It's the process of transforming the spherical graticule into a flat grid upon which map details can be drawn.

However, flattening a curved surface inevitably introduces **distortions**. When parallels (circles) and meridians (semi-circles) are transferred from a sphere to a flat plane, their relative shapes and spacing change. The need for map projections arises from the limitations of globes (cost, portability, inability to show fine detail or allow easy comparison) and the necessity of creating flat maps for detailed study and various uses.

Simply sticking a flat piece of paper onto a globe demonstrates the problem; it cannot cover a large surface without stretching or wrinkling away from the point or line where it touches the globe. Attempting to project the globe's features onto a flat surface using light from the center would result in increasing distortion with distance from the contact point. Since the Earth is not a "developable surface" (a surface that can be flattened without distortion), some distortion in properties like shape, size (area), distance, and direction is unavoidable in any map projection.

Map projection is therefore the study of the various methods developed over time to transfer the graticule onto a flat sheet while attempting to preserve certain desired properties and minimizing others. Different projections prioritize the accurate representation of different properties depending on the map's purpose.

Glossary Terms:

Term	Definition
Map projection	A systematic method for transforming the geographic coordinates (latitude and longitude) of points from the spherical Earth's surface onto a flat plane, preserving certain spatial properties at a chosen scale.
Loxodrome or Rhumb Line	A line on the Earth's surface that crosses all meridians at the same angle. On a Mercator projection, a loxodrome appears as a straight line, making it useful for navigation where maintaining a constant compass bearing is desired.
The Great Circle	Any circle on the Earth's surface whose plane passes through the center of the Earth. Great circles represent the shortest distance between two points on the sphere. Meridians of longitude and the Equator are examples of great circles.
Homolographic Projection	A map projection that preserves the relative area of geographical features; also known as an equal-area projection. While areas are correct, shapes are usually distorted, especially towards the edges of the map.
Orthomorphic Projection	A map projection that preserves the correct shape of small areas; also known as a true-shape projection. In an orthomorphic projection, the scale is the same in all directions around any given point, and meridians and parallels intersect at right angles on the map as they do on the globe. Preserving shape usually results in distorted areas, particularly over large regions.

Need For Map Projection

The fundamental need for map projections stems from the limitations inherent in using a globe as a representation of the Earth. While a globe is the only model that can truly represent Earth's surface properties without distortion, it is impractical for many applications.

Reasons for the necessity of map projections:

**Detailed Study:** Globes cannot display sufficient detail for thorough study of specific regions. Large-scale, detailed maps on flat paper are required for geographical analysis, planning, and other purposes.
**Comparison:** Comparing different regions or geographical phenomena is easier and more precise using maps on a flat surface than on a curved globe.
**Measurement:** Accurately measuring distances, areas, and angles for practical applications is more convenient on a flat map.
**Portability and Cost:** Globes are bulky, difficult to transport, and generally more expensive to produce and distribute than flat maps.
**Illustration of Specific Properties:** Maps can be designed using specific projections that emphasize or preserve particular properties (like area for distribution maps or direction for navigation charts), making them more useful for specific purposes than a general globe.

The challenge is how to transfer the spherical network of latitudes and longitudes (graticule) onto a flat sheet without introducing unacceptable levels of distortion for the map's intended use. Since a sphere is a non-developable surface, distortion is unavoidable. Map projections provide systematic ways to manage and distribute these distortions, allowing for the creation of functional flat maps.

Elements Of Map Projection

Understanding map projection involves considering several key conceptual elements:

Reduced Earth

Since it's impractical to work with the actual size of the Earth, map projections begin by scaling down the Earth to a manageable size represented on a flat sheet of paper. This scaled-down model, usually represented as a sphere or spheroid corresponding to the map's scale, is called the **"reduced earth."** The graticule is conceptually transferred from this reduced Earth model onto a developable surface or directly onto a plane.

Parallels Of Latitude

These are conceptual circles on the reduced earth, parallel to the equator and equidistant from the poles. Each parallel lies in a plane perpendicular to the Earth's axis. Their length decreases from the equator to the poles. On a projection, parallels appear as straight lines, curved lines, or arcs, depending on the type of projection.

Meridians Of Longitude

These are conceptual semi-circles on the reduced earth, running from the North Pole to the South Pole. Meridians lie in planes that intersect along the Earth's axis. All meridians are equal in length. By international agreement, the 0° meridian passing through Greenwich is the standard reference. On a projection, meridians appear as straight lines or curved lines, always converging at the poles.

Global Property

These are the spatial characteristics of features on the Earth's surface that cartographers attempt to preserve on a map. The four primary global properties are:

**Distance:** Maintaining correct relative distances between points.
**Shape:** Preserving the true shape of features (especially important for small areas).
**Size (Area):** Representing the relative sizes or areas of regions correctly.
**Direction (Bearing):** Showing the true direction of one point from another.

As mentioned, no single flat map projection can perfectly maintain all four of these properties simultaneously for large areas. Designing a projection involves prioritizing one or more of these properties while accepting distortion in others.

Classification Of Map Projections

Map projections can be classified based on several criteria, reflecting the different methods used in their construction and the properties they attempt to preserve.

Drawing Techniques

Based on how the graticule is conceptually transferred:

**Perspective Projections:** These projections can be conceptually created by imagining a light source placed at a specific point (e.g., the Earth's center, a pole, or infinity) and projecting the shadow of the globe's graticule onto a developable surface (like a cylinder, cone, or plane).
**Non-Perspective Projections:** These are developed using mathematical calculations without the aid of a light source and shadow projection.
**Conventional or Mathematical Projections:** Projections derived entirely through mathematical formulas, often designed to achieve specific properties, and may not have a simple conceptual geometric basis.

Developable Surface

A **developable surface** is a geometric shape (like a cylinder, cone, or flat plane) that can be unfolded or flattened into a plane without stretching, tearing, or distorting its area or shape. The Earth's sphere is a **non-developable surface**. Map projections are often derived by projecting the graticule onto a developable surface that is tangential to or intersects the globe, and then unfolding the surface into a plane (Figure 4.1 illustrates distortion from globe to flat surface). (Figure 4.1 shows distortion when flattening a globe's surface).

Illustration showing a curved surface (like part of a globe) being flattened into a plane, demonstrating stretching and tearing (distortion) needed for the transformation.

Based on the type of developable surface used, projections are classified as:

**Cylindrical Projections:** Conceptually obtained by projecting the graticule onto a cylinder wrapped around the globe. When the cylinder is cut open and flattened, it produces a cylindrical projection.
**Conical Projections:** Conceptually obtained by projecting the graticule onto a cone placed over the globe (Figure 4.2 shows a cone concept). When the cone is cut along a line and flattened, it produces a conical projection. (Figure 4.2 illustrates projecting from a Globe to a Cone).

Diagram showing a cone placed over a globe, illustrating how lines from the globe's surface might be projected onto the cone.

**Zenithal (or Planar/Azimuthal) Projections:** Obtained by projecting the graticule directly onto a flat plane that touches the globe at a single point (the point of tangency).

These projections can be further classified based on the position of the developable surface relative to the globe:

**Normal (or Equatorial):** Surface is tangential at the Equator (for cylindrical) or its axis is aligned with the Earth's axis and tangential at a parallel (for conical) or tangential at the Equator (for zenithal).
**Oblique:** Surface is tangential at a point or parallel between the pole and the Equator.
**Polar:** Surface is tangential at one of the poles (for conical and zenithal).

Global Properties

As discussed, no projection can maintain all four global properties simultaneously. Projections are often classified based on which property they preserve or prioritize:

**Equal Area (or Homolographic) Projections:** Preserve the relative areas of regions correctly. While areas are accurate, shapes are typically distorted, especially far from the center or standard lines.
**Orthomorphic (or Conformal/True-Shape) Projections:** Preserve the true shape of small areas and angles. Meridians and parallels intersect at right angles. While local shapes are correct, the area of regions is usually distorted, often increasingly towards the poles.
**Azimuthal (or True-Bearing) Projections:** Preserve the true direction or bearing from a central point (the point of tangency) to any other point on the map.
**Equi-distant (or True Scale) Projections:** Preserve correct distances or scale along specific lines or from a central point. No projection can have a true scale everywhere, but equi-distant projections aim to maintain accuracy along selected lines (like meridians or standard parallels).

Source Of Light

For perspective projections, the location of the imaginary light source used to project the graticule onto a developable surface provides another classification basis:

**Gnomonic Projection:** Light source is placed at the **center** of the globe.
**Stereographic Projection:** Light source is placed at the globe's **periphery**, at a point diametrically opposite to the point of tangency of the developable surface.
**Orthographic Projection:** Light source is placed at **infinity** from the globe, as if rays are parallel, opposite to the point of tangency.

Constructing Some Selected Projections

Here, we will look at the construction and characteristics of a few specific map projections.

Conical Projection With One Standard Parallel

This is a type of conical projection where a cone is conceptually wrapped around the globe so that it touches the globe along a single parallel of latitude. This parallel is called the **standard parallel**. Along this standard parallel, the scale is theoretically true. Parallels on either side of the standard parallel are represented as concentric arcs, and meridians are represented as straight lines converging at a point representing the pole (which is shown as an arc). Distortion increases as distance from the standard parallel increases. (Figure 4.3 illustrates the construction concept and resulting projection). (Figure 4.3 shows the construction steps and the resulting Conical Projection with one standard parallel).

Example. Construct a conical projection with one standard parallel for an area bounded by 10º N to 70º N latitude and 10º E to 130º E longitudes when the scale is 1:250,000,000 and latitudinal and longitudinal interval is 10º.

Answer:

Given:

Area extent: Latitudes $10^\circ$ N to $70^\circ$ N, Longitudes $10^\circ$ E to $130^\circ$ E.
Scale (R.F.): 1 : 250,000,000
Interval: $10^\circ$ for both latitudes and longitudes.

Calculations:

Calculate the radius of the reduced Earth (R) based on the given scale. The Earth's radius is approximately 640,000,000 cm. $$ R = \frac{\text{Earth's Radius}}{\text{Denominator of R.F.}} = \frac{640,000,000 \text{ cm}}{250,000,000} = 2.56 \text{ cm} $$ So, the radius of the circle representing the reduced Earth in the construction diagram will be 2.56 cm.
Determine the Standard Parallel. When using one standard parallel for an area spanning several latitudes, the standard parallel is typically chosen as the central parallel of the area. The latitudes are $10^\circ, 20^\circ, 30^\circ, 40^\circ, 50^\circ, 60^\circ, 70^\circ$. The middle latitude is $40^\circ$ N. So, the Standard Parallel is $40^\circ$ N.
Determine the Central Meridian. The longitudes are $10^\circ, 20^\circ, ..., 130^\circ$ E. The middle longitude is $70^\circ$ E. So, the Central Meridian is $70^\circ$ E.
Calculate the length of the standard parallel on the reduced Earth. Length of any parallel = $2 \pi R \cos(\text{latitude})$. For the standard parallel ($40^\circ$ N): Length = $2 \times \pi \times 2.56 \times \cos(40^\circ)$. (This length will be theoretically correct on the projection).
Calculate the distance between meridians along the standard parallel on the projection. This distance should correspond to the true length of the meridian interval ($10^\circ$) along the standard parallel on the reduced Earth. Circumference of the standard parallel on reduced earth = $2 \pi R \cos(40^\circ)$. Interval along the standard parallel = $\frac{\text{Circumference}}{360^\circ} \times \text{Longitudinal Interval}$ Interval = $\frac{2 \times \pi \times 2.56 \times \cos(40^\circ)}{360^\circ} \times 10^\circ$. (This calculation is conceptual for understanding; the text uses a geometric approach for construction).
Calculate the linear distance between parallels along a meridian on the reduced Earth. Distance between parallels is approximately constant. For a $10^\circ$ interval, this distance is approximately $10^\circ \times 111 \text{ km/degree} = 1110 \text{ km}$. On the reduced Earth scale: Distance = $1110 \text{ km} \times \frac{1 \text{ cm}}{250,000,000 \text{ cm}} = 1110 \times \frac{100000 \text{ cm}}{250000000} \text{ cm} = \frac{111000000}{250000000} \text{ cm} = 0.444 \text{ cm}$. So, the distance between parallels on the projection along the central meridian should be 0.444 cm. The text uses a geometric approach from the central angle. The arc distance CE in Figure 4.3 represents $10^\circ$ latitudinal interval on the reduced sphere.

Construction Steps (Following Figure 4.3 and text description):

Draw a quadrant (or half-circle) of radius R = 2.56 cm. Mark angles from the horizontal (Equator) at intervals of $10^\circ$ ($10^\circ, 20^\circ, ..., 70^\circ$). Label the points on the arc where these angles intersect. Label the horizontal line as the Equator base. Mark the $40^\circ$ N angle, say AOD, where A and B are points on the arc at $40^\circ$ from the Equator base E. The center is O.
Draw a line segment extending horizontally from point B (on the $40^\circ$ N arc) tangential to the arc at B. This line represents the $40^\circ$ N Standard Parallel on the projection. Similarly, conceptually, a tangent from A to P could also be drawn, meeting at apex P. BP represents the length from the pole P to the standard parallel on the projection.
Calculate the distance between meridians along the Standard Parallel ($40^\circ$ N) on the reduced Earth. This is the true distance we need to represent along the arc representing $40^\circ$ N on the projection. The text uses the distance XY in the diagram, which is the true distance of the longitudinal interval ($10^\circ$) along the standard parallel on the reduced Earth. This distance is calculated as the length of the $10^\circ$ arc along the $40^\circ$ parallel on the reduced Earth. Length of $1^\circ$ longitude at $40^\circ$ latitude = $111.3 \text{ km} \times \cos(40^\circ) \approx 111.3 \times 0.766 \approx 85.3 \text{ km}$. For a $10^\circ$ interval, this is $853 \text{ km}$. On the reduced Earth scale: Distance = $853 \text{ km} \times \frac{1 \text{ cm}}{250,000,000 \text{ cm}} = 853 \times \frac{100000}{250000000} \text{ cm} \approx 0.341 \text{ cm}$. (The text labels XY in Fig 4.3 and uses it in step vii; its exact derivation in the diagram might relate to distances on the tangential cone). Let's use the text's provided geometric method reference.
Draw a vertical line (N-S) to represent the Central Meridian ($70^\circ$ E). Mark the position of the Standard Parallel ($40^\circ$ N) on this line by taking the distance BP from the conceptual pole P downwards. (BP is the distance from Pole to $40^\circ$ N parallel on the tangential cone).
Mark the positions of other parallels ($10^\circ, 20^\circ, 30^\circ, 50^\circ, 60^\circ, 70^\circ$ N) on the Central Meridian (N-S line) by stepping off the distance between parallels (arc distance CE, which represents $10^\circ$ latitude interval on the reduced earth, approximately 0.444 cm as calculated earlier). Measure this distance upwards and downwards from the $40^\circ$ N parallel mark along the Central Meridian.
Draw arcs of circles through these marked points, centered at the apex P (or the point representing the pole N on the projection). These arcs represent the parallels of latitude. The $40^\circ$ N parallel is the Standard Parallel drawn as a straight line initially, but represented as an arc in the final projection. The pole N is represented by an arc (often the $90^\circ$ parallel).
Along the Standard Parallel arc ($40^\circ$ N), mark off the distances corresponding to the longitudinal interval ($10^\circ$ E and W from the Central Meridian). Use the distance XY from Figure 4.3. Mark these distances on both sides of the Central Meridian along the $40^\circ$ N arc for each $10^\circ$ longitude interval within the required range ($10^\circ$ E to $130^\circ$ E).
Draw straight lines from the pole point (apex P or N) through these marked points on the $40^\circ$ N parallel. These straight lines represent the meridians of longitude ($10^\circ, 20^\circ, ..., 130^\circ$ E). Label the Central Meridian as $70^\circ$ E.
Label the parallels with their respective latitudes ($10^\circ$ N to $70^\circ$ N) and the meridians with their longitudes ($10^\circ$ E to $130^\circ$ E).

Properties:

Parallels appear as concentric circular arcs, equally spaced along meridians.
Meridians are straight lines that converge at the pole point.
Meridians and parallels intersect at right angles.
The pole is represented as an arc of a circle (the $90^\circ$ parallel).
Scale is accurate along the Central Meridian and the Standard Parallel.
Scale is distorted away from the Standard Parallel; typically compressed poleward of the standard parallel and stretched equatorward.
Meridians are spaced correctly only along the standard parallel; spacing decreases too much towards the pole.
This projection is neither equal-area nor orthomorphic (shapes and areas are distorted).

Limitations:

Not suitable for a world map due to significant distortion in the hemisphere opposite to where the standard parallel is chosen.
Even within the chosen hemisphere, distortion increases significantly far from the standard parallel, limiting its use for very large areas.

Uses:

Best used for mapping areas with limited north-south extent but larger east-west extent, particularly in mid-latitudes where distortion is less severe.
Suitable for showing features like long, narrow railway lines or boundaries that run parallel to the standard parallel.
Examples include mapping areas like the Trans-Siberian Railway or parts of continents in the mid-latitudes.

Cylindrical Equal Area Projection

Also known as **Lambert's Cylindrical Equal Area Projection**. Conceptually derived by projecting the globe's graticule onto a cylinder that touches the globe at the Equator, using parallel rays from the Earth's center. When the cylinder is cut open, both parallels and meridians appear as straight lines intersecting at right angles. A key property is that the spacing of parallels is adjusted towards the poles to ensure that the **area** of any region on the map is proportional to its true area on the globe.

The Equator is the standard parallel and is shown at its true length. Parallels are straight lines parallel to the Equator, but their spacing decreases towards the poles to maintain area equality. Meridians are straight lines parallel to the central meridian, and they are equally spaced. The pole is represented as a line segment equal in length to the Equator, leading to extreme distortion in shape at high latitudes.

Example. Construct a cylindrical equal area projection for the world when the R.F. of the map is 1:300,000,000 taking latitudinal and longitudinal interval as 15º.

Answer:

Given:

World map.
Scale (R.F.): 1 : 300,000,000.
Interval: $15^\circ$ for both latitudes and longitudes.

Calculations:

Calculate the radius of the reduced Earth (R). Earth's radius $\approx 640,000,000$ cm. $$ R = \frac{640,000,000 \text{ cm}}{300,000,000} = 2.133... \text{ cm} \approx 2.1 \text{ cm (as per text example)} $$ So, the radius of the circle representing the reduced Earth in the construction diagram will be 2.1 cm.
Calculate the length of the Equator on the projection. The Equator is the standard parallel and is shown at its true length. Length of Equator = $2 \pi R = 2 \times \frac{22}{7} \times 2.1 \text{ cm} = 2 \times 3.14159... \times 2.1 \text{ cm} \approx 13.19 \text{ cm} \approx 13.2 \text{ cm (as per text example)}$
Calculate the spacing between meridians along the Equator on the projection. This spacing should be equal for all longitudinal intervals. The total longitudinal extent is $360^\circ$. The interval is $15^\circ$. Number of intervals = $360^\circ / 15^\circ = 24$. Spacing = $\frac{\text{Length of Equator}}{\text{Number of Longitudinal Intervals}} = \frac{13.2 \text{ cm}}{24} = 0.55 \text{ cm}$. So, meridians are spaced 0.55 cm apart along the Equator.
Calculate the spacing between parallels of latitude to maintain equal area. In this projection, the distance of a parallel from the Equator is calculated by $y = R \sin(\text{latitude})$. The spacing between parallels is not constant. The text provides a geometric method involving calculating vertical distances based on sines of latitude, which results in decreasing spacing towards the poles. The vertical distance of the $15^\circ$ N/S parallel from the Equator is $2.1 \sin(15^\circ)$, $30^\circ$ is $2.1 \sin(30^\circ)$, etc.

Construction Steps (Following Figure 4.4 and text description):

Draw a circle of radius R = 2.1 cm. Mark angles from the horizontal (Equator) at intervals of $15^\circ$ ($15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ, 90^\circ$) in all four quadrants. These angles help determine the vertical position of parallels in the construction diagram.
Draw a horizontal line segment of length 13.2 cm to represent the Equator (0° latitude) on the projection. This line is the Central Parallel.
Divide this Equator line into 24 equal segments, each 0.55 cm long. Mark the points along the line representing longitudes at $15^\circ$ intervals (e.g., $0^\circ, 15^\circ$ E, $30^\circ$ E, ..., $180^\circ$ E/W). Label the central point as $0^\circ$.
Draw a vertical line through the center of the Equator line (the $0^\circ$ longitude point) perpendicular to the Equator. This represents the Central Meridian.
Determine the vertical position of each parallel ($15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ, 90^\circ$ N and S) from the Equator on the Central Meridian. The distance of a parallel $\phi$ from the Equator is $R \sin(\phi)$ in the construction diagram. Measure these distances along the Central Meridian upwards (North) and downwards (South) from the Equator.
Draw horizontal straight lines through the marked points on the Central Meridian. These lines represent the parallels of latitude ($15^\circ, 30^\circ, ..., 75^\circ$ N/S). The $90^\circ$ N and S parallels (the poles) are also drawn as lines equal in length to the Equator at the northernmost and southernmost calculated vertical positions.
Draw vertical straight lines through the marked points on the Equator line. These lines represent the meridians of longitude ($15^\circ$ E/W, $30^\circ$ E/W, etc.). Label the Central Meridian as $0^\circ$.
Label the parallels with their latitudes and the meridians with their longitudes. The resulting grid is the cylindrical equal area projection.

Properties:

All parallels and meridians are straight lines.
Parallels and meridians intersect at right angles.
Parallels are parallel to the Equator, but their spacing decreases towards the poles.
All parallels are equal in length, the same as the Equator.
All meridians are equal in length and are equally spaced.
**Areas** of regions are correctly represented (it is equal area).
Scale is true only along the Equator. Along all other parallels, the scale along the parallel is exaggerated, while the scale along the meridian is compressed to compensate and maintain equal area.

Limitations:

**Shape distortion** is significant, increasing towards the poles. Features at high latitudes appear stretched horizontally.
It is a non-orthomorphic projection because shapes are not preserved.

Uses:

Most suitable for mapping areas that span a wide range of longitudes but have limited latitudinal extent, particularly in the middle latitudes (e.g., between $45^\circ$ N and S).
Useful for thematic maps showing the distribution of phenomena where maintaining correct area relationships is important (e.g., distribution of crops like rice, tea, coffee, rubber, sugarcane, or population distribution).

Mercator’s Projection

Developed by Gerardus Mercator in 1569, this is a famous cylindrical projection based on mathematical formulas. It is an **orthomorphic** or true-shape projection, meaning it preserves the shapes of small areas and angles. Meridians and parallels are straight lines intersecting at right angles, as in the cylindrical equal area projection. However, to maintain correct shapes, the spacing between parallels increases towards the poles.

A unique property of the Mercator projection is that any straight line drawn on it represents a line of **constant compass bearing** or a **Loxodrome (Rhumb Line)**. This makes it exceptionally useful for navigation, as sailors can maintain a constant direction by following a straight line on the chart. However, Great Circle routes (the shortest distance between two points on a sphere) appear as curved lines on a Mercator projection (Figure 4.6). (Figure 4.6 illustrates Rhumb lines and Great Circles on Mercator Projection).

Diagram on a Mercator projection showing a straight line connecting two points labeled as a Rhumb Line or Loxodrome, and a curved line connecting the same two points labeled as a Great Circle.

To maintain shape (orthomorphism), the scale at any point on the Mercator projection is the same in all directions. This requires the exaggeration of distances along parallels to be equal to the exaggeration of distances along meridians at that latitude. Since the parallels are shown as straight lines equal to the length of the Equator (while their true length decreases towards poles), they are increasingly exaggerated away from the Equator. To maintain shape, the spacing between parallels is increased by the same factor, leading to the spacing between parallels increasing rapidly towards the poles.

Example. Draw a Mercator’s projection for the world map on the scale of 1:250,000,000 at 15º interval.

Answer:

Given:

World map.
Scale (R.F.): 1 : 250,000,000.
Interval: $15^\circ$ for both latitudes and longitudes.

Calculations:

Calculate the radius of the reduced Earth (R). Earth's radius $\approx 250,000,000$ inches (approx. 6350 km, using inches convention for English system context from text). $$ R = \frac{250,000,000 \text{ inches}}{250,000,000} = 1 \text{ inch} $$ So, the radius of the circle representing the reduced Earth in the construction diagram will be 1 inch.
Calculate the length of the Equator on the projection. The Equator is the standard parallel and is shown at its true length. Length of Equator = $2 \pi R = 2 \times \pi \times 1 \text{ inch} \approx 6.283 \text{ inches}$. So, the Equator will be a line of approximately 6.28 inches on the projection.
Calculate the spacing between meridians along the Equator on the projection. This spacing should be equal for all longitudinal intervals. The total longitudinal extent is $360^\circ$. The interval is $15^\circ$. Number of intervals = $360^\circ / 15^\circ = 24$. Spacing = $\frac{\text{Length of Equator}}{\text{Number of Longitudinal Intervals}} = \frac{6.283 \text{ inches}}{24} \approx 0.262 \text{ inches}$. So, meridians are spaced about 0.26 inches apart along the Equator. (Text shows 0.26 inches).
Calculate the vertical distance of each parallel from the Equator on the projection. This is the key step for Mercator, derived using specific formulas that increase spacing towards the poles. The text provides a table of values for distances from the Equator for different latitudes (using 1 inch as R). Distance of parallel $\phi$ from Equator $\approx R \times \text{meridional part for } \phi$. Meridional parts are calculated using logarithms. The table provided in the text gives sample distances from the Equator (assuming R=1 inch):
- $15^\circ$ N/S: Distance $\approx 0.265 \times R = 0.265 \times 1" = 0.265"$
- $30^\circ$ N/S: Distance $\approx 0.549 \times R = 0.549 \times 1" = 0.549"$
- $45^\circ$ N/S: Distance $\approx 0.881 \times R = 0.881 \times 1" = 0.881"$
- $60^\circ$ N/S: Distance $\approx 1.317 \times R = 1.317 \times 1.317"$
- $75^\circ$ N/S: Distance $\approx 2.027 \times R = 2.027 \times 2.027"$
- $90^\circ$ N/S: Distance approaches infinity.
(Note: The values in the text table "Distance" seem to be directly the distances from the Equator in inches when R=1 inch, rather than a multiplier, and the calculation in step (iii) "Distance for latitude" seems to multiply a given value by 1, which is consistent if the given values are the distances for R=1).

Construction Steps (Following Figure 4.5 and text description):

Draw a horizontal line segment of length 6.28 inches to represent the Equator (0° latitude). Label it EQ.
Divide this Equator line into 24 equal segments, each 0.26 inches long, representing the $15^\circ$ longitudinal intervals. Mark the points and label the longitudes (e.g., $0^\circ, 15^\circ$ E/W, $30^\circ$ E/W, ..., $180^\circ$ E/W). Label the center as $0^\circ$.
Draw a vertical line through the center of the Equator line (the $0^\circ$ longitude point) perpendicular to the Equator. This represents the Central Meridian.
Determine the vertical positions of the parallels of latitude ($15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ$ N and S) using the calculated distances from the Equator (or the values provided in the table, assuming R=1 inch = 2.54 cm, convert table values from inches to cm if drawing in cm, or use inches). Measure these distances along the Central Meridian upwards (North) and downwards (South) from the Equator line.
Draw horizontal straight lines through the marked points on the Central Meridian. These lines represent the parallels of latitude. Their length is equal to the Equator line (6.28 inches). Label the parallels with their respective latitudes ($15^\circ, 30^\circ, ..., 75^\circ$ N/S).
Draw vertical straight lines through the marked points on the Equator line. These lines represent the meridians of longitude ($15^\circ$ E/W, $30^\circ$ E/W, etc.). Label the Central Meridian as $0^\circ$.
Label the longitudes. The resulting grid is the Mercator projection. Note that the poles ($90^\circ$ N/S) cannot be shown as their vertical distance from the Equator is theoretically infinite. The projection is typically truncated before $90^\circ$.

Properties:

All parallels and meridians are straight lines.
Meridians and parallels intersect at right angles.
All parallels are equal in length, which is the same as the Equator (exaggerated lengths except for the Equator).
All meridians are equal in length (but longer than the corresponding meridian on the globe) and are equally spaced.
Spacing between parallels **increases** towards the poles.
**Shapes** of small areas are preserved (it is orthomorphic).
**Directions** (bearings) from any point are correct, and rhumb lines are straight lines.
It is also an azimuthal projection from the perspective of direction from the center parallel (Equator).

Limitations:

**Area is greatly exaggerated** towards the poles. Regions at high latitudes appear disproportionately large compared to areas near the Equator. For example, Greenland appears larger than South America, although it is much smaller in reality.
The **poles ($90^\circ$ N/S) cannot be shown** on this projection because the spacing between parallels becomes infinite as latitude approaches $90^\circ$.
It is not suitable for showing the entire world in a single uninterrupted map unless significant area distortion at high latitudes is accepted.

Uses:

Widely used for **navigation charts**, especially for sea and air routes, because rhumb lines (lines of constant bearing) appear as straight lines.
Useful for world maps showing geographical features or phenomena where maintaining correct shapes and directions is important and distortion of area at high latitudes is acceptable (e.g., some world atlas maps).
Appropriate for showing distribution of ocean currents, wind directions, or other elements where direction and shape are key.

Exercise

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Activity

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