The Gnomonic Polar Zenithal Projection is a type of azimuthal projection that maps the surface of a sphere (like the Earth) onto a flat plane. In this projection, all great circles (shortest paths between two points on a sphere) are projected as straight lines.
The projection is created by placing the light source at the centre of the Earth (the sphere). The projection surface (a flat plane) touches the sphere at a specific point, which is the pole in the case of the polar version. Severe distortion occurs near the edges of the map. While the centre (pole) is accurate, distances and shapes become increasingly exaggerated as you move toward the equator.
Why the name Gnomonic?
The term “gnomonic” comes from the Greek word “gnomon” (γνώμων), which refers to the pointer of a sundial—the stick or rod that casts a shadow to indicate time. The concept behind the gnomonic projection is related to how a gnomon casts shadows. Just like a gnomon casts shadows from a single light source (the Sun), the gnomonic projection simulates a light source placed at the centre of the Earth. Every point on the sphere’s surface is projected along a straight line (like a shadow) from this central light source onto a flat plane.
What is Azimutal?
The term “azimuthal” comes from the Arabic word “as-sumut” (السموت), meaning directions or paths. In map projections, the term refers to how directions from a central point are preserved or represented accurately. Azimuthal projections are designed to show the surface of a sphere (like the Earth) from the perspective of one central point. In the case of the polar gnomonic projection, the central point is either the North Pole or the South Pole. In azimuthal projections, the angle (or azimuth) from the centre to any other point on the map matches the actual angle on the sphere. These projections are useful for navigation and plotting straight-line directions, especially over short distances.
Three Cases of Projection
The three cases—Polar, Equatorial, and Oblique—depend on the location where the projection plane touches the sphere. Each case serves different purposes, with the polar case focusing on poles, the equatorial case on areas near the equator, and the oblique case on regions at intermediate latitudes.
Polar
The projection plane touches the Earth at either the North Pole or the South Pole. This configuration is ideal for mapping polar regions, with meridians radiating outward from the centre and latitude circles appearing as concentric rings. It is commonly used for polar navigation and weather maps, as it offers a clear view of the Arctic or Antarctic.
Equitorial
The projection plane is tangent to the Earth at the equator. Here, the equator appears as a straight line across the map, and meridians extend radially from the centre. This type of projection is useful for showing areas around the equator with minimal distortion and is often applied in satellite coverage and communication maps.
Oblique
The projection plane touches the Earth at a point between the pole and the equator, typically somewhere in the mid-latitudes. This case allows for more specialized mapping of a specific region or location, such as airline routes or regional maps focused on cities or countries. It provides a balanced view, though some distortion increases toward the map’s edges.
Construction
Question: Construct a Gnomonic polar zenithal projection for the North Pole at a scale of 1:300,000,000 with meridian intervals of 30° and parallels at 45°, 60°, and 75°N.
Solution:
- Calculate the Radius of the Globe on the Given Scale:
- $$\text{Radius of the Globe} = \frac{6400 \times 1,00,000}{300,000,000} = 2.1 \, \text{cm}$$
- Thus, the radius of the globe on the scale of 1:300,000,000 is 2.1 cm.
- Draw the Globe Circle:
- Draw a circle with a radius of 2.1 cm. This circle represents the Earth on the given scale.
- Draw the Polar and Equatorial Diameters:
- Draw two perpendicular diameters passing through the centre O of the circle:
- NS for the polar diameter.
- WE for the equatorial diameter (perpendicular to NS).
- Set Up the Projection Plane:
- Place the projection plane (XY) at the North Pole (N), perpendicular to the polar diameter (NS).
- Draw Radii Corresponding to Parallels (45°, 60°, 75°):
- From the centre O, draw lines making 45°, 60°, and 75° angles with the equatorial diameter OE using a protractor.
- Extend these lines to meet the projection plane (XY) at points a, b, and c.
- Draw the Parallels:
- Use the North Pole (N) as the centre on the projection plane.
- Draw concentric circles with radii corresponding to the intersections of the projection plane:
- Na = 75°N
- Nb = 60°N
- Nc = 45°N
- Since the equator lies far from the North Pole, it won’t appear on the projection.
- Draw Meridians at 30° Intervals:
- From the centre N, draw straight lines at 30° intervals using a protractor:
- 0° meridian, 180° meridian opposite to it.
- 90°E and 90°W at right angles.
- Final Diagram:
- The resulting projection will show the North Pole at the centre.
- Meridians at 30° intervals radiate outward.
- Parallels at 45°, 60°, and 75°N are represented as concentric circles around the pole.
