The Mollweide projection is an equal-area projection that represents the world in an elliptical shape. This means that areas on the map are proportional to their corresponding areas on the globe. This makes it particularly useful for displaying global data, such as population density, climate zones, or land use, where the preservation of area is essential.
Mollweide’s projection is often used in world maps to balance the distortion of shapes and sizes, providing an accurate visual comparison of different regions.
Key Features
- Equal Area: The Mollweide projection preserves areas across the entire map, ensuring that the relative sizes of countries and continents are displayed accurately.
- Elliptical Shape: The projection forms an ellipse, with the equator being the longest horizontal line.
- Curved Meridians: Unlike cylindrical projections, the meridians (lines of longitude) curve and converge toward the poles, reflecting the Earth’s curvature.
The Mathematics Behind Mollweide’s Projection
Using specific equations, the Mollweide projection transforms geographic coordinates (latitude and longitude) into planar coordinates (x and y). These equations ensure the projection remains equal-area while providing a reasonable visual representation of the globe.
Key Equations
- Auxiliary Angle Equation:
$$2\theta + \sin(2\theta) = \pi \sin(\phi)$$
Here, $$(\theta)$$ is an auxiliary angle, $$(\phi)$$ is the latitude, and $$(\pi)$$ is the mathematical constant. This equation must be solved numerically for each latitude. - X-Coordinate (Longitude):
$$x = \frac{2 \sqrt{2}}{\pi} R (\lambda – \lambda_0) \cos(\theta)$$
Where:
- (R) is the Earth’s radius (or scale factor),
- $$(\lambda)$$ is the longitude,
- $$(\lambda_0)$$ is the central meridian,
- $$(\cos(\theta))$$ is the cosine of the auxiliary angle $$(\theta)$$
- Y-Coordinate (Latitude):
$$y = \sqrt{2} R \sin(\theta)$$
Where:
- (R) is the Earth’s radius,
- $$(\sin(\theta))$$ is the sine of the auxiliary angle $$(\theta)$$
Step-by-Step Example
Let’s assume we want to project a point at latitude
$$( \phi = 30^\circ )$$
and longitude $$( \lambda = 60^\circ )$$ using the Mollweide projection. For simplicity, we use a map scale where ( R = 1 ) (for relative distance).
Step 1: Solve for the Auxiliary Angle $$(\theta)$$
For $$( \phi = 30^\circ )$$, solve the equation $$( 2\theta + \sin(2\theta) = \pi \sin(30^\circ) )$$
This gives $$( 2\theta + \sin(2\theta) = \pi/2 )$$ which we can solve numerically to find $$( \theta \approx 0.6435 )$$ radians.
Step 2: Calculate X and Y Coordinates
- Using $$( \theta = 0.6435 )$$ radians, $$( \lambda = 60^\circ )$$, and $$( \lambda_0 = 0^\circ )$$:
$$x = \frac{2 \sqrt{2}}{\pi} (60^\circ – 0^\circ) \cos(0.6435) \approx 1.029$$ - For the y-coordinate:
$$y = \sqrt{2} \sin(0.6435) \approx 0.905$$
So, the point at $$((30^\circ \text{N}, 60^\circ \text{E}) $$ in geographic coordinates becomes $$((x \approx 1.03, y \approx 0.91) $$ on the Mollweide projection.
Example
Question: Draw a graticule for the Mollweide’s Projection showing the entire world on a scale of 1:100,000,000. The graticules should have 30° intervals for both latitudes and longitudes.
Solution:
To draw the Mollweide projection as shown in the image with a scale of 1:100,000,000, follow these step-by-step instructions:
Step 1: Set Up the Parameters
- Radius of the Earth (R): 6,371,000 meters or 6,371 km.
- Map Radius (R_map): The map scale is 1:100,000,000. So, convert the Earth’s radius to the map scale:
$$R_{map} = \frac{6,371,000 \, \text{cm}}{100,000,000} = 6.371 \, \text{cm}$$ - Central meridian $$( \lambda_0 = 0^\circ ) (Greenwich Meridian)$$
Step 2: Draw the Elliptical Boundary
- Draw a horizontal axis that is 4 times the radius $$(2R_{map})$$ (use 12.742 cm as the total width of the ellipse, i.e., the major axis).
- Draw the vertical axis with a height of $$(2R_{map} \sqrt{2})$$ (use 9.01 cm as the total height, i.e., the minor axis).
- Use a compass to draw the elliptical boundary using these measurements.
Step 3: Mark the Parallels (Latitudes)
- The parallels should be spaced at 30° intervals (from 90°N to 90°S).
- Use the following table to calculate the distance of each latitude from the centre (equator) on the map. The formula to use is $$(y = \sqrt{2} \cdot R_{map} \cdot \sin(\theta))$$ where $$( \theta )$$ is the auxiliary angle derived from $$(2\theta + \sin(2\theta) = \pi \sin(\phi))$$
- The approximate values for each latitude (in cm) from the centre are:
| Latitude (°) | $$( \theta )$$ (radians) | Distance from Equator (y) (cm) |
|---|---|---|
| 90°N | 1.571 | 9.01 |
| 60°N | 1.047 | 5.53 |
| 30°N | 0.523 | 2.87 |
| 0° (Equator) | 0 | 0 |
| 30°S | -0.523 | -2.87 |
| 60°S | -1.047 | -5.53 |
| 90°S | -1.571 | -9.01 |
Mark the distances for each parallel on the vertical axis.
Step 4: Mark the Meridians (Longitudes)
- The meridians should be spaced at 30° intervals (from 180°W to 180°E).
- Use the following formula to calculate the x-coordinates of each meridian:
$$x = \frac{2\sqrt{2}}{\pi} R_{map} (\lambda – \lambda_0) \cos(\theta)$$
where $$( \lambda )$$ is the longitude and $$( \theta )$$ is the auxiliary angle corresponding to the latitude.
For example, for $$( \lambda = 30° )$$, use the values of $$( \theta )$$ calculated earlier. The x-coordinate values for 30° intervals will be:
| Longitude (°) | X-Coordinate (cm) |
|---|---|
| 0° | 0 |
| 30°E/W | ±2.54 |
| 60°E/W | ±5.08 |
| 90°E/W | ±7.62 |
| 120°E/W | ±10.16 |
| 150°E/W | ±12.71 |
| 180°E/W | ±15.25 |
Mark the points for each meridian along the horizontal axis.
Step 5: Draw the Graticules
- Draw the parallels (latitude lines) as horizontal lines spaced according to the distances calculated in Step 3.
- Draw the meridians (longitude lines) as curved lines that pass through the x-coordinates calculated in Step 4, ensuring they meet at the poles.
How to draw an ellipse?
- Set up the Axes:
- Start by drawing two perpendicular lines, XY (the major axis) and CD (the minor axis), intersecting at point O.
- Identify Point G:
- Using X as the centre and the length of the minor axis CD as the radius, mark point G along the major axis XY.
- Divide Segment GB:
- Split the line segment GB into three equal sections (1,2,3).
- Locate Points M and L:
- From point O, measure a distance equal to twice one of the divisions of GB, and use it to mark points M and L on the major axis XY.
- Create Intersection Points P and Q:
- Using the length of ML as the radius, draw arcs from points M and L, which will intersect at points P and Q along the minor axis.
- Draw Arcs from Points P and Q:
- With P as the centre and PD as the radius, draw an arc that passes through point D.
- Similarly, use Q as the centre and QC as the radius to create another arc passing through point C.
- Drawing the Ellipse:
- Using L as the centre and LY as the radius, draw a curve through point Y.
- With M as the centre and MX as the radius, draw another curve passing through point X.
- To complete the ellipse, use French curves to smoothly connect all the arcs passing through points C and D.
