Posts tagged with coordinate systems

As to longitude, I declare that I found so much difficulty in determining it that I was put to great pains to ascertain the east-west distance I had covered. The final result of my labours was that I found nothing better to do than to watch for and take observations at night of the conjunction of one planet with another, and especially of the conjunction of the moon with the other planets….

After I had made experiments many nights, one night, the twenty-third of August 1499, there was a conjunction of the moon with Mars, which according to the almanac was to occur at midnight or a half hour before. I found that…at midnight Mars’s position was three and a half degrees to the east.

Amerigo Vespucci

Good gosh. Can you imagine having travelled so far on the globe—without a swift means of return, of course—that you literally had no idea where you were?

And what’s more, science to save you. You can’t ask anyone around you for the answer. Many of the people around you not only have never heard of Europe, but can’t even conceive of such a thing.

Nobody knows the answer. ∄ books that purport to have the answer. ∄ communication channels back to home. You’re all alone, mentally. To figure out what’s going on all you have to go on is reason and facts. And if you get the answer right, whom are you going to tell?

(Source: Wikipedia)

[G]eometry and number[s]…are unified by the concept of a coordinate system, which allows one to convert geometric objects to numeric ones or vice versa. …

[O]ne can view the length ❘AB❘ of a line segment AB not as a number (which requires one to select a unit of length), but more abstractly as the equivalence class of all line segments that are congruent to AB.

With this perspective, ❘AB❘ no longer lies in the standard semigroup ℝ⁺, but in a more abstract semigroup (the space of line segments quotiented by congruence), with addition now defined geometrically (by concatenation of intervals) rather than numerically.

A unit of length can now be viewed as just one of many different isomorphisms Φ: ℒ → ℝ⁺ between and ℝ⁺, but one can abandon … units and just work with directly. Many statements in Euclidean geometry … can be phrased in this manner.

(Indeed, this is basically how the ancient Greeks…viewed geometry, though of course without the assistance of such modern terminology as “semigroup” or “bilinear”.)
Terence Tao

(Source: terrytao.wordpress.com)