Shortest Path in R2 versus R3 space


#1

In the discussion about conversion of R1 and R2 points to R3 points, the RhinoPython primer (p.54) describes an example in which the shortest path in R2 space is not the shortest path in the R3 space.

I don’t understand that example because the straight path is clearly shorter than the red path and falls within the surface domain. Can anybody explain how this works?


#2

Somebody can correct me if I’m wrong but here is how I see it:

As far as the surface is concerned (in R2 space), it is a flat square (with u and v axis instead of x and y). If you were to remap this as a square where the u and v axis were all straight like in a cartesian grid, the shortest point from A to B is the red line.
Looking at the surface in R3 space however we can see that the u and v coordinates are not straight lines and thus the shortest path in R3 space may look quite different if it were remapped into our flat, square, R2 space.

Another example is a map of the earth. If you look at a typical, Mercator projection map of the earth (the earth in R2 space), you may think that the shortest path for a plane to take from Sydney to Buenos Aires is to travel Eastwards. If you look at it in R3 space however you can easily see that the shortest route is to go south, almost over Antartica:

So, whilst the shortest route still lies in the surface domain (we dont tunnel through the earth!), the path looks quite different in R3 space than it does in R2 space.


(Dale Fugier) #3

Hi Bhupati,

Surface have a 3-D space (what you see) and a 2-D (parameter) space (what you don’t see.). And, it is possible to evaluate a surface in either 3-D space or 2-D space.

What the author is saying is that if you determine the shortest path between 2 2-D points (in 2-D space) and then project (pull back) the results to 3-D space, the resulting curve will not necessarily be the shortest path in 3-D space.

Not sure I’ve helped…


#4

Hello @wattzie, thanks for your explanation. I was (finally) able to wrap my head around this concept (visualizing myself tunneling through the Earth definitely helped here). Not vital, but I’m not sure what you meant by ‘traveling Eastwards to go to Beunos Aires’. But thanks much, I understood the concept!


#5

Hi @dale! Sorry, I’m just logging in to see your replies. I was a bit concerned because I was expecting a reply notification to show up in my email (I’m new to this forum). Thanks for your reply. Yes, it helps. I’m very new to 3-d modelling so it takes a while to get used to thinking of distances and, more pertinently, computing them, in 3-d.


#6

In 2D space, the flight path would look like this:

If you flatten out the image of the actual shortest path in 3d space to 2d space, it looks like this:


#7

Oh, silly me. ‘Eastward’ seems quite evident from this perspective. Thanks! P.S. I’m not sure how map printing standards differ according to country, but (at least) where I come from, I’ve only seen world maps with this perspective :-] (…on which we’d have to say ‘westward’)


#8

on the map, it’s shorter to go to the right/east (green) than west (red)… but look at the globe and see the shortest route is actually (blue)




draw a sphere in rhino… put two points on the sphere… use _InterpCrvOnSurface to connect the two points with the shortest route… in a projected viewport, the red line looks to be the shortest route but it’s actually the blue one:


#9

Yep, the map projection shows the kind of distortion you can get in trying to flatten a sphere into a rectangular plane. Greenland is bigger than Canada; Alaska and Australia are nearly the size of the continental US.

–Mitch


#10

Australia IS larger than the contiguous US :wink:

[Of this area, 2,959,064.44 square miles (7,663,941.7 km2) is land, composing 83.65% of U.S. land area, just slightly smaller than that of Australia.][1]

(Australia is 7 692 024 km²)
[1]: https://en.wikipedia.org/wiki/Contiguous_United_States


#11

Whoaa, no way! That’s some new information to me.


#12

Thanks so much @jeff_hammond . Your explanation makes this concept clear to me. After looking at your drawings, I am wondering if this could be the reason why if you take a transatlantic flight from the US to Germany, the flight path runs almost over Greenland.


#13

maybe most map makers tend to just fill in the gaps with more ocean? (though possibly not on the map posted in the thread? there’s supposed to be ~75% water but the map is showing basically equal amounts land/ocean… also, i don’t think greenland is as big as it looks on that map?)…

i guess the flat maps should look more like this:

…but it’s probably either ugly or confusing to most people.


i wish the world was still flat… would be so much easier back then :wink:


#14

If you are flying “westward” towards the US from northern Europe (especially towards the west coast) you fly very far north, definitely over Greenland and northern Canada. This is in general a “great circle” route, but also designed to avoid the following:

When you fly back to Europe eastward from the west coast of the US, in general the flight path is much further south, to take advantage of the 100mph+ jet-stream winds that blow from west to east at lower latitudes. The flight in the eastward direction is usually an hour and a half shorter despite the fact that the actual distance may be slightly longer.

Are there winds in Rhino r3 space? :smile:

–Mitch


#15

We are doomed! :open_mouth:


#16

Well, that “orange-slice” map cuts the US in half about at the Mississippi, which is not too bad I guess, and it definitely severs northern California from the rest just like the San Andreas fault is supposed to… :smile:


#17

it’s the flyover states map :smile:

but yeah, the jet stream is sweet if going from LA to NYC… not so sweet going the other way… sometimes nearly an hour longer to fly east->west

depending on how far it’s dipping, you can sometimes get a fast flight from miami to NYC as well.

(but you already know this stuff …just babbling is all)


#18

These path maps are definitely changing my “worldview”. Flight path from Hawaii to the UK!


(Pascal Golay) #19

I guess $7.2 million was considered cheap, to be bigger then Austraila.

-Pascal


#20

heh… approximation of how that looks on the flat map:

:slight_smile: