MASTERCLASS

# 2D Geometry

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In this lecture, we want to talk about how we describe the position of things in the world. In robotics, this is really, really important.

For instance, in our robotics problem, we might want to describe the position of a robot. We might also want to describe the position of something that the robot wants to interact with. For instance, it might want interact with my water bottle. So I need to describe the position of the robot, the position of the water bottle and also interested in describing the relative position.

Where is the water bottle with respect to the robot? And that's clearly something that the robot would like to know if it wanted to go and pick up the water bottle.

When it comes to describing the position of things, we need some kind of frame of reference. This is a pretty important aspect of describing the position of things in the world. Here is a picture of our world. And one of the things we need to keep in mind of course is that our world, the surface of our world is moving. A point on the surface at the earth – the equator is moving nearly 500 meters per second in space. The earth itself is moving at quite a high speed around the sun. And then our solar system is moving at very high speed around the centre of the galaxy. And the galaxy itself is moving at incredible speed towards the constellation Leo. So, everything is in motion.

If we're trying to describe where things are, it's really important that we consider the frame of reference.

Everything is moving so we just have to put effectively a stake in the ground and say, "This is the reference point and we'll decide all positions with respect to that reference point". That's exactly what we do. We're going to put a stake in the ground. For instance, like this sign post here and say, "This is the origin. This is the coordinate, zero zero and we're going to measure all distances from that.

And we're going to create a coordinate frame. Effectively two orthogonal axes. An x-axis, and a y-axis, sorts of things that you learn at school. And we're going to use those two distances, the distance along the x-axis, the distance along the y-axis, in order to describe the position of an object in two-dimensional space.

In this lecture, we're going to keep things simple. We're only going to consider the position in two dimensions. In the next lecture, I will kick it up a notch and talk about position in 3 dimensions.

Now, reference points are entirely arbitrary. So, let's talk for a moment about the reference system that we use on our own planet. So, here we have a depiction of planet earth. And this is a projection, the earth of course is a sphere. And to make a rectangular map like this we have to do is cut the sphere unroll it, stretch it out so that it looks like a rectangle. And we put a couple of reference lines on our globe. The most important one, perhaps that most obvious one, is the equator.

Because the earth is a sphere, we have an obvious axis which is that which the planet rotates around. And you may create a plane orthogonal to this rotational axis that cuts the sphere into two equal hemispheres, and we call that line the equator and it shown here in black. And you can go and visit the equator when you can stand with your leg one in the Northern Hemisphere and one in the Southern Hemisphere.

Now there's another line, which we call the Prime meridian and it passes orthogonal to the equator, and it passes through the Greenwich observatory in London.

This line is completely arbitrary. It doesn't have to be through the Greenwich observatory but the convention on our planet is that we used the Greenwich observatory line as the Prime meridian, and you can go visit Greenwich you can put your feet on either side of the Prime meridian in the West and the Eastern Hemispheres of the planet.

The distance in the horizontal direction, we refer to as longitude. And the direction, the distance in the other direction, we refer to as latitude. So we have lines of longitude and lines of latitude that we inscribe onto the sphere. And those two numbers, longitude and latitude, are sufficient to describe the position of any point, any place on the surface of our planet.

Now, we have this arbitrary line as I've mentioned before which we called the Prime meridian which goes through the Greenwich observatory. If history had been different, we would have had a different Prime meridian. There was a proposal for the Prime meridian to go through the Paris observatory. And if you visit Paris today, you can see a line referred to as to Arago line which is where this non-existent Prime meridian passes through the city of Paris. And it's named after a French astronomer and mathematician, Arago, who proposed that particular Prime meridian.

So, the take home message here is that this Prime meridian is completely arbitrary. It is a convention that we've all agreed upon. It's useful to have this convention and that's how we describe points on the planet of the earth.

We learn how to describe the position and orientation of objects on a 2-dimensional plane. We introduce the notion of reference frames as a basis for describing the position of objects in two dimensions.

### Skill level

High school mathematics

This content assumes an understanding of high school-level mathematics, e.g. trigonometry, algebra, calculus, physics (optics) and some knowledge/experience of programming (any language).