Proportional symbols in three dimensions

Since I'm making my own Thematic Mapping Engine, I need to understand the math behind proportional symbol calculations. Originally, I thought I would need different equations for different geometric shapes, and my book in cartography gave me the same impression. But after reading this article, I realised that life was not that complicated.

This tutorial is a summary of a discussion on the CartoTalk forum. I especially want to thank Dominik Mikiewicz (mika) for his valuable comments and figures. The CartoTalk forum is highly recommended!

Equations for 1D, 2D and 3D proportional symbols:

1-dimensional symbols (height)
This is how the height of bars or prisms is calculated in TME.

Equation: symbolSize = (value / maxValue) * maxSize
PHP: $symbolSize = ($value / $maxValue) * $maxSize
JavaScript: symbolSize = (value / maxValue) * maxSize

Bars or prisms show “real” values scaled down to fit on a map, and you can easily see the relations and which is higher than the other. I’m not considering the problems caused by perspective and the curvature of the earth.

2-dimensional symbols (area)
This is how proportional images and regular polygons (e.g. circle, square) are scaled in TME.

Equation: symbolSize = power(value/maxValue; 1/2) * maxSize
PHP: $symbolSize = pow($value/$maxValue, 1/2) * $maxSize
JavaScript: symbolSize = Math.pow(value/maxValue, 1/2) * maxSize

2D symbols use areas as mean of expression and therefore you're dealing with a square root of a showed value. This makes it relatively difficult to assess a value.

3-dimensional symbols (volume)
This is how 3D Collada objects (e.g. cube, sphere) are scaled in TME.

Equation: symbolSize = power(value/maxValue; 1/3) * maxSize
PHP: $symbolSize = pow($value/$maxValue, 1/3) * $maxSize
JavaScript: symbolSize = Math.pow(value/maxValue, 1/3) * maxSize

3D objects use volumes as mean of expression so you’re showing a cube root of the value. This makes it difficult to assess a value.

It’s important to know that it’s one degree harder for the viewer to assess the relative size of 3-dimensional symbols compared to 3-dimensional, which again is harder to compare to 1-dimensional. This is clearly visualised on this figure (credit: Dominik Mikiewicz):

The figure compares the circle and sphere radius for the same values.

These three images shows GDP per capita (2006, UNdata) using bars (1D), circles (2D) and spheres (3D).

A colour scale and legend helps the user in assessing and comparing symbols.

The visual appearance of 2D and 3D symbols can also be improved by using a perceptual or logarithmic scale.

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