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):
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggMxRPXZMelHzp7-3TyIJNSROek_gxRDQZrptbWtKBGCeqU_RoXeySYDB9FEfVNsHK-vQJlAof5HTm2UyXLruY-4XjOCoJp3VotbaOvWVz3-mqbjfEezFAOHp2iDKjZtFQ1IwCiDb0bQg/s400/propsymbolsfigure.png)
These three images shows GDP per capita (2006, UNdata) using bars (1D), circles (2D) and spheres (3D).
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4C_n6hF2gewTQBTAY-JCIl-MKlMWl4Jl9zz6_7vgiTfpYc_Ln7ehKUOZ931Bb13vVwcOZOYhT_g8TedMsbxDAGS0TxtePnFFgYEZkEU_OSfABAq9nLglADtvwxZoh4yDPFUheDQCjZmg/s400/propcompare1.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6XJEQjBsVhf6cg_VvX2fAvWE8MAifW-VPFbD6TJdGWMPrHBIfkfk_zgLTULZijc0gZabdLjQWgPF4Dy3vvOuzZuOSXEB9XPN_FnfolXmABaRgtDs_IsUgtmfRk1-b1S-c861h_RsLCcM/s400/propcompare2.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUD5THGUPbZdZnpE5QujlmL7XvNcD7wlR-zRbZ7hqJpQXoV_W-5WIzhATFcoMWztE_UkNzmaSLh8VgprDznsxiEmfbbAcPSx4p2XX8TRaFrnqVP8mXw6h0BQr3gjLOm-K7pXCCXthLF_8/s400/propcompare3.png)
The visual appearance of 2D and 3D symbols can also be improved by using a perceptual or logarithmic scale.
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