How To Do Change Of Variables For Quadratic Form

Linear vs. Quadratic

I of the almost common mistakes in nautical chart blueprint is to scale an surface area by 2 sides at the same time, producing a quadratic effect for a linear modify. That overstates the larger numbers and produces a badly skewed chart. A picayune care and some basic high-school math tin help avoid the problem.

The following detail from a information graphic produced past Princeton'due south International Networks Archive illustrates the problem (the numbers are presumably from 2002):

Fast Food Chains

Comparing Starbucks ($4.1bn) and KFC ($8.2bn), the problem becomes clear: the difference is a factor of 2, but the KFC logo has iv times the expanse of the Starbucks logo (even more because one is square and the other round). This tin be seen in a number of the graphics on that website, though they besides take some where they scale correctly.

The reason for the problem hither is the use of logos (or of images, more than generally) to make charts look better. Scaling a logo in merely one dimension (which would be done in a bar chart) does not work considering the image would look stretched and ugly. Then instead, the image is scaled in two dimensions, leading to a perceived deviation that is the square of the bodily difference.

In more than full general terms, a linear change (I will use a factor of 2 to illustrate this)

Linear Change

becomes a quadratic change:

Quadratic Change

Or, to utilise a bit of high-school math: The area of a square A = a², with a being the length of side of the square. If nosotros double the side length, we become A' = (2a)ⁱⁱ = 4a²: 4 times the surface area. This is the case for many other shapes every bit well, including circles and circle segments.

Quadratic Change in Circle

That final part was actually the subject of a give-and-take I had a while agone with a rather senior visualization person. He did not believe that changing a circumvolve segment's radius would pb to a quadratic increment in its area. It's like shooting fish in a barrel to show, though: a circle's expanse is r²π (r being the circumvolve's radius), the area of a circle segment that covers an angle θ is r²π·sin(θ). It is no more difficult to show that doubling the radius will quadruple the surface area than with the foursquare above.

Florence Nightingale already knew this in 1858 when she developed her coxcomb nautical chart (a predecessor to the pie chart): she represented the numbers of soldiers using the expanse, not the radius, of the circumvolve segments.

Nightingale's Coxcomb chart

A like event can be seen in the petal chart or star glyph, which connects points on a number of axes that radiate from a common betoken. Whether they are filled in or not, the impression is that of an enclosed expanse, and that changes in a quadratic way like to the circumvolve segments above.

star glyphs

The solution? Either use a better visualization (star glyphs in detail are very difficult to read) or scale your circles, squares, and other lengths by the square root of the value you want to represent. That makes the expanse calibration linearly with the value that is to be represented.