Hi all, > And, thinking about it a bit, I guess the proper rule is that (10, > 10) -> (30, 30) passes through (20, 20), since it's completely > unrealistic to assume that the basic renderers will do otherwise. And this is where you are wrong. On zoomlevel 0 (one tile for the whole earth) (10,10) ends up on (135.11, 135.15) and (30,30) ends up on (149.33, 150.38).
One thing that should be stand out is the fact that x!=y even for points which have lat==long. The halfway point between the two is (142.22, 142.77) while (20,20) is projected to (142.22, 142.52) or about a quarter of a pixels off. Other zoom levels or other triplets of points could expose much bigger deviations but I wanted to prove you wrong using your very own example. > My understanding is that this is equivalent to say saying that the > line is "straight in the Mercator projection", as my understanding is > that the Mercator projection represents each pixel as a fixed length > and width in degrees. Nope it does not. Each pixel represents the same width in degrees, but the height in degress increases as you go away from the equator. > And what that also means is that a straight line on earth which is > more than a certain length is not properly represented by a way with > two points. THAT depends on your definition of "straight line". > One thing I can't quite get my mind wrapped around is whether or not > a geodesic is what we'd call a straight line on the earth. If we put > a few million (?) rulers end-to-end as best we could, would that > form a geodesic, and if not, what would it form? I'm fairly certain > it wouldn't pass (10, 10) -> (30, 30) through (20, 20), since 20 > degrees of longitude does not (generally) equal 20 degrees of > latitude in length. But I'm not sure if it'd be a geodesic or not. > I'd love for someone to answer that question and provide a link or > source to back up their answer. Well. There isn't one single definition of "straight line" here. We are used to the fact that straight lines are the shortest line between two points. Geodesics are the more general form of that. They connect two points in with the shortest way possible and are "straight" in that sense. One a flat surface geodesics are just straight lines. One a sphere the are segments of great circles. If your metric gets more complicated geodesics get more complicated too. On the other hand we are used to the fact that a straight line always intersects lines which are parallel to the y-axis at a constant angle. This is not (necessarily) true for geodesics. The compass heading along the great circle route from (10,10) to (30,30) is NOT going to be NE all the time! So in that sense geodesics are NOT straight. A line which follow a constant heading is called a "loxodrome". As you can see now geodesics and loxodromes are two different lines which both might be considered straight by some definition. There are other definitions of straightness and "straight lines" which are defined by them. In a flat plane they all coincide, but for a sphere or our not-quite-spherical earth the don't. So don't stop assuming that is a simple topic, everybody was just a lazy bum or that you know it all (tm). HTH, Patrick "Petschge" Kilian _______________________________________________ talk mailing list [email protected] http://lists.openstreetmap.org/listinfo/talk
