So good! So now, stereographic projections. This is something I was playing with [INAUDIBLE]. So this is what [INAUDIBLE] came up with originally. And he said you have to put your projection point here minus one will still on the beta not axis. So, that's where you put your spotlight. Your hyperplane is now diagonal to beta and equal to zero. It's still plus one away from the projection point. Just instead of here, we've shifted it here. And then what happens is you take your current attitude, right? You connect the line to the projection point and intercept it with your hyper point and that set of number on that coordinate, that interjection point is the MRP definition. So let's look at it with animations. I find these a little bit easier. So this is the same thing. I have my point. I can move it around. And you can see, if I just go from this angle to this angle, I have doubled my angles and I roughly double my distance, which is what we expect, they linearized very rural. If I go towards the 180 part, so I go from here to here, it's a little bit more than double but it's still pretty close. With CRPs if I did that, that coordinate went off into infinity, alright? But it's not until you really, once I go pass vertical then my beta not goes negative, right? I flip to this side. So here is a key property of MRPs. Yes they linearize angles over four. The zero rotation is the easy 000. So from control point of view it is very nice. It just has to drive everything to zero. That's a handy property. But if I look at the norm of the MRPs I can tell right away, are we doing a small rotation or a large. In fact, if the norm of the MRPs becomes one, I am doing a 180 degree rotation. That's an important property, I'll say it again. If the norm of the MRPs becomes one, that means you're dong 180 degree rotation. If the norm is greater than 1 that means you're describing a long rotation. If I'm doing a long, there must be a short. So if you look at this, you can see this is 1, that's beta, over here, the black point is my shadow sets are written a dark color, that's the minus beta, the alternate attitude, right this right now, is the short one, this is the long one, and the long one projected gives me an MRP set that's way outside the unit sphere, because it's a long way around. As I complete, as I do a 180 degree tumble, right, at this point these, if I continue, they will grow bigger than one, I leave the unit sphere and the shadow set starts to enter the unit sphere, so by just looking at the MRP set, are they within the unit sphere I know that's the short description, outside, that's the long one. And now, we can switch between them, so the beauty here with the singularities is, right now I would use the red dot, those are the short descriptions, they behave very well. If I continue to tumble here I don't want to use 250 degrees I would rather use minus something smaller right, that's my description. So here I'm going to switch from one set to another and I'll show you the math it's almost a negative but a slight modification. That's it. As I complete a revolution as we talked about earlier the shadows set now is at the origin and it's extremely well behaved, very linear. And the original set went of the crazy land it went of to infinity right, If you continue tumbling then you basically just keep switching as you norm of the MRP's. Exceeds one this one got bad now. Then I'll switch back to the other set. So like with the Cortanians if we didn't control we might have to switch between one set to another to always have a short descriptions with MRPs. By doing that equivalent switching we actually avoid any singularity by using a single parameter, pre parameter set but to be to fair, we were actually using two, three parameters sets right. There's the MRPs and the shadow MRPs, the alternate set. But combined between them we can actually define now any attitude, any orientation without singularity but with the cost of a discontinuity. You have to go to four-dimensional space to find the nonsingular description that is continuous, the sternums on the stuff. Three-dimensional space. It is possible to define a nonsingular the description, this is an example of it. But it's always in a noncontinuous way. At some point, as you go to 180, I have to flip from 180 to minus 180 and continue my tumble. So we have to deal with the discontinuity, but it's very practical. You will see in the code and how we write this and how we use it for control. So good! MRPs. We've discovered that so the stereographic projection, in your homework two you have to solve for this. Basically think similar triangles. If you write this out and you can get the mapping between beta naught and beta one you can do it equivalently for beta two and three. So you can just draw this circle as I have it. And this is a plus 1. This coordinate is beta naught beta one so this length of the triangle would be one plus beta naught. This is beta i and you can do similar triangles and you can quickly prove this. There are other ways with vectors, and hyper dimensional space, and intersecting hyper planes works as well, actually not too hard. But this is an easy way to do it.
