Curvature

The best place to start thinking about curvature might be semi-log graph paper.

The distance between 6 and 8 is obviously not the same as the distance between 2 and 4. Distance is shrinking relative to the numbers the further we go to the right.

It might be confusing because when we say distance we are referring to the measurement you would get if you took a ruler and held it up to the computer screen to measure a length unit (inches or centimeters) between two numbers.

Another thing to notice: compare 2 to 6 and then compare 6 to 10 and notice that the “mid-numbers”, 4 and 8, are not at the center of the interval. It’s easier to see for the 4.

On a map of a city or a county, you see a ratio of distance to length: One inch equals 900 feet. You know one inch of measurement on paper equals 900 feet in real life. This value is the same over the entire map. For our game here, we are comparing change of number to distance on the computer screen. The ratio of one to the other changes from point to point. We need calculus to deal with this.

Appendix A

There might be a part of your brain that wants to resist what you are reading.

“Distance is distance!” (or something similar)

Curvature is caused by scaling one thing to another. If you are down on the planet walking, 2000 of a length unit is twice what 1000 of the length unit is.

The curvature is the artifact caused by you using something else, You might be using inches on a ruler to measure distance on the paper on which a map has been drawn or printed. We are saying the word “curvature” as an abbreviation for a long sentence like “We don’t have linearity between distance on the real object and distance on the object we are using as a representation”. We usually use a print out on paper or a computer screen for our representations.

When a professional map maker puts a curved planet on a flat piece of paper, they put curvature into the mapping (it is a mapping between planet surface points and those points on the flat paper).