Wednesday, 9 April 2014

Straight Lines, Straight Lines and Straight Lines

If you wanted to travel from point A to point B (there's ice-cream at point B) what route would you take? You're real hungry so you may choose the green route because the shortest distance between two points is a straight line. Or you may be a daydreamer like me and choose the red route and see some cool clouds along the way. 

That's all fine for the math textbook, but let's name points A and B. Say there is REALLY good ice-cream in Hawaii. So point A we designate to Sydney, Australia and Point B is Waikiki, Hawaii. 

Straight line between the two points gets us there right?

Google, what are you doing? I put the red dotted line in myself. Are Google daydreamers like me? Maybe. But I certainly don't think that airlines are. Longer routes equal more fuel used equal less profits. Maybe there is something about Australian planes, or pilots. 

Nope. It seems that everyone is flying on curved routes. What the hell is going on? 

We all live on a sphere.

Firstly, a straight line across the surface of a sphere is always curved. Put a string along the surface of a globe and you have a curved string. There's just no way to put a length of string across the surface of a globe and have it completely straight (let's not go into calculus people).

Secondly the maps that I am showing and the Google uses are flat. They do not show the real world, they show a projection, parts of the globe are distorted. Therefore straight lines will also be distorted and curved. 

The trouble all happens when we take a straight line on along the surface of a sphere, a three-dimensional object, and try to draw it on a flat two-dimensional sheet. It just doesn't work properly.

One of the greatest examples of straight lines appearing curved is one that is constantly in our skies, the moon. Have you ever thought about the phases of the moon? We know that one part is light, and one part is dark and they intersect in a curved line apart from the half moon.

The line between light and dark (known as the terminator) is a straight line, well as straight as you can get on the surface of a sphere. But it is a straight line from the north to the south pole of the moon as the moon is always half-lit by the sun. We just see it as curved when we see it at an angle, we see differing amounts of the part in shadow throughout the month. But the line is always straight!

Taken to the extremes straight lines around a globe can look quite curved. Have you ever watched a movie with the NASA control centre in it, like 'Apollo 13'? They often have a large map up on the front wall with the orbital path plotted over a projection of the surface. It's always a wave isn't it? It appears to be very far from a straight line. But it is!

If we use the International Space Station for example, it has an orbit which when viewed in it's side is a straight line. It goes in a circle around the earth in a plane.

But project this onto a two-dimensional surface and you get a lovely sinusoidal wave. And because the Earth turns relative to the International Space Station, the next orbit is slightly offset and it passes over different territory.

So you can see that straight lines are not that straight sometimes, and even curved lines can be straight in three-dimensions. In a recent blog article, the wonderful Ken Jennings shows us another straight line across the globe.

Did you know that you can sail from Pakistan to Kamchatka in a straight line? 

 Just like the orbits, it certainly doesn't look straight. And it is an awfully long way, it's definitely the scenic route. Don't believe me, check out the video animation.

So is the shortest route between two points a straight line? I hear you your brain groaning over that question now. But the answer is still yes. It's only no when we put a restriction on it. And what is the one restriction that has been put in place in all these above examples? That you travel on or above the surface of a sphere.

Theoretically you could tunnel through the earth from one point to another in a straight line that is the shortest distance between these two points. 

You don't need to pass through the centre of the earth unless you are going to the exact opposite side of the globe. And if you were you're likely to come out awfully wet. With 70.8% of the surface of the earth being water you are more than likely have to pack a pair of trunks. The lucky and dry centre-earth tunnelers live in Chile and Argentina, Malaysia, Philippines, China, Indonesia, Greenland, far north Canada and some of Antarctica. Oh and some of New Zealand corresponds with Spain.

So what route will you now choose?

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