Blog Archives

A book I’d like (someone) to write

Here’s an idea I had for a book. Maybe someone has already written it. If you know of such a book, please let me know. Differential geometry has a huge ratio of definitions to theorems. It seems like you do

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Spotting sensitivity in an equation

The new book Heavenly Mathematics describes in the first chapter how the medieval scholar Abū Rayḥān al-Bīrūnī calculated the earth’s radius. The derivation itself is interesting, but here I want to expand on a parenthetical remark about the calculation. The

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Approximating Earth as a sphere

Isaac Newton suggested in 1687 that the earth is not a perfectly round sphere but rather an ellipsoid, and he was right. But since our planet is roughly a sphere, it’s often useful to approximate it by a sphere. So

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Poor Mercator

This morning’s xkcd cartoon is “What your favorite map projection says about you.” It’s really funny, but poor Mercator comes off as the most boring projection. Mercator is the most familiar projection, but it has some interesting properties. The most

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Latitude doesn’t exactly mean what I thought

Don Fredkin left a comment on my previous blog post that surprised me. I found out that latitude doesn’t exactly mean what I thought. Imagine a line connecting your location with the center of the Earth. I thought that your

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Journey away from the center of the Earth

What point on Earth is farthest from its center? Mt. Everest comes to mind, but its summit is the point highest above sea level, not the point farthest from the center. These are not the same because the Earth is

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How many trig functions are there?

How many basic trigonometric functions are there? I will present the arguments for 1, 3, 6, and at least 12. The calculator answer: 3 A typical calculator has three trig functions if it has any: sine, cosine, and tangent. The

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Inverse Mercator projection

In my earlier post on the Mercator projection, I derived the function h(φ) that maps latitude on the Earth to vertical height on a map. The inverse of this function turns out to hold a few surprises. The height y

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Mercator projection

A natural approach to mapping the Earth is to imagine a cylinder wrapped around the equator. Points on the Earth are mapped to points on the cylinder. Then split the cylinder so that it lies flat. There are several ways

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Why care about spherical trig?

Last spring I wrote a post on spherical trigonometry, the study of triangles drawn on a sphere (e.g. the surface of the Earth). Mel Hagen left a comment on that post a few days ago saying I am revisiting Spherical

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Converting miles to degrees longitude or latitude

Someone sent me email regarding my online calculator for computing the distance between to locations given their longitude and latitude values. He wants to do sort of the opposite. Starting with the longitude and latitude of one location, he wants

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What is the shape of the Earth?

To first approximation, out planet is a sphere. But how accurate is that approximation? What’s a better approximation? How good is that? This post will answer these questions and some related questions. How well does a sphere describe the Earth’s

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Finding distances using latitude and longitude

I posted a page this evening that lets you calculate the distance between two locations using their latitudes and longitudes. I’ve had to do this calculation once in a while and thought I’d make it available online for anyone else

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