How to Convert Latitude/Longitude Coordinates to Pixel Coordinates on a Mercator Projection?

Converting Latitude/Longitude Coordinates to Pixel Coordinates on a Mercator Projection

When working with geographic data, it often becomes necessary to translate latitude and longitude coordinates into pixel coordinates on a Mercator projection. This conversion is essential for overlaying geographic data onto images or maps.

The Mercator projection is a conformal projection that preserves angles, making it suitable for navigational charts. In this projection, meridians are represented as straight vertical lines, and parallels as straight horizontal lines. The spacing between meridians and parallels is uniform, except for the poles, which are distorted.

To derive projected pixel coordinates (x, y) from spherical latitude (φ) and longitude (λ), we utilize the following formulas:

x = (longitude + 180) * (mapWidth / 360)
y = (mapHeight / 2) - (mapWidth * ln(tan((PI/4) + (latitude * PI/180) / 2)) / (2 * PI))

Here, mapWidth and mapHeight represent the dimensions of the image in pixels, and the conversion from degrees to radians is achieved by multiplying by PI/180.

To demonstrate this conversion, consider a point located at latitude 41.145556 and longitude -73.995. For an image with a width of 200 pixels and a height of 100 pixels, the pixel coordinates of this point are:

x = (longitude + 180) * (mapWidth / 360)
= (-73.995 + 180) * (200 / 360)
= 79 pixels

y = (mapHeight / 2) - (mapWidth * ln(tan((PI/4) + (latitude * PI/180) / 2)) / (2 * PI))
= (100 / 2) - (200 * ln(tan((PI/4) + (41.145556 * PI/180) / 2)) / (2 * PI))
= 58 pixels

This conversion allows precise positioning of geographic data onto Mercator projection maps or images. Understanding the formulas and their application enables accurate data visualization and interpretation.

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