0:27

In essence, what I'm doing is doing a one D rotation.

Â If you only rotate about b1 for example, then only the Beta 0 and

Â the Beta 1 parameters will be non zero.

Â The other two will just be all zero.

Â So you can think of this at interpretation just doing a one dimensional rotation

Â then, your four dimensional hyper surface just boils down to a circle and

Â circles are so much easier to draw in 2D than 4D surfaces.

Â But it really, I'm trying Beta I, everyone of these axis are actually orthogonal and

Â that four dimensional space so once you've figured out two of them in a plane

Â the same math holds across the other planes as well the geometry.

Â So this is interesting.

Â Actually for the MRPs to [INAUDIBLE] credit he is now professor

Â at Georgia tech but he and I co-wrote some papers when he was on the post doc and

Â I was just finishing up my PhD.

Â And in one of his papers,

Â he actually interpreted the MRP's as being a stereographic projection and

Â he put a certain point, and that's what this math means.

Â I thought, that's kind of interesting.

Â But what does that really mean?

Â And so I started drawing it out.

Â So, there is a whole family of parameters that are developed way back then,

Â that people have been following up on now.

Â And they're all based on the same principle.

Â So as CK was saying, your quaternion has to resolve itself to be on the surface.

Â You don't just have the attitude that all of a sudden, leaves the surface,

Â goes inside, and wanders around.

Â You have to be on that surface so when we integrate,

Â we always have to make sure to map position.

Â We stay on that surface.

Â Good, but this is a four dimensional manifold.

Â If you want to reduce this down to a three dimensional amount of space,

Â we have to do a mapping.

Â And so for cartography we use the Earth, all right?

Â And you could take different maps but we take the Earth's surface and

Â you essentially peel off that surface and we stretch it onto a plane.

Â 2:24

>> Poles.

Â >> The poles, those areas look huge, right?

Â Greenland looks amazingly big and then you see it actually on a globe,

Â and that's still big but not quite that big, right?

Â Things get stretched way outta proportion.

Â There are more area preserving projections that happen but somewhere there's always

Â a single layer, there were things go a little bit wonky, right?

Â So there's different mathematical procedures on how did map

Â the earth surface into a plane.

Â This is one of them and it's called the Stereographic Projection.

Â [INAUDIBLE] recently published a paper looking at whole series

Â of geographic projection methods and how this map, it applies itself to coordinates

Â as well kind of based on this work which is kind of cool, very, very elegant.

Â So what we do here?

Â With a stereographic projection you have to have essentially a projection point.

Â Think of it as a point light source.

Â You've got your point here,

Â that's my point on the surface that represents, let's say this is 45 degrees,

Â this is always angles over two, that's how we define datas.

Â So that would be like a 90 degree rotation, that's one point on that

Â surface, I want to project that point, I make a line between that

Â projection point and the attitude of interest, that's my projection line.

Â And let it intercept a projection plane.

Â This plane is define by Beta Naught equal to plus one.

Â 3:43

So that actually that means Beta one, Beta two, Beta three can be anything.

Â It's a plane orthogonal to your first axis, the Beta Naught axis.

Â That's it.

Â So this is one D rotation that plane just becomes a line.

Â Instead of a three dimensional hyper space, but

Â you could imagine two other dimensions sticking out of here.

Â Just very hard to draw.

Â So if you do this, you can show, in the homework actually you do,

Â you can use geometry and symmetry triangles, it's not very hard actually.

Â Once you see the pattern, that this mathematics is true.

Â This projection, this coordinate, in this case, if I have Beta 1,

Â I would get q 1 here.

Â I would get my classical Rodrigues parameters.

Â 4:40

There we go.

Â So hopefully, yep, this works.

Â So on my website, if you go to classes 5010,

Â you can get to these links they're also in the slides.

Â I've got a little JavaScript exported a while back.

Â So I've got two points I'm tracking here.

Â The red point is my current attitude and then this point up here.

Â This is the projected attitude onto that hyper plane.

Â That's the CRP coordinate.

Â So what happens now is, I can make rotations and you can see for

Â small rotations, as I double my angle, I roughly double my distance.

Â That's good.

Â I double my angle, I still double my distance.

Â But as you get further away, if I double my angle from here to here,

Â I haven't exactly doubled my distance, right?

Â That one goes off to infinity.

Â This angle is the singularity point that you would have.

Â I can't move it.

Â Darn, it will be nice.

Â So 90 times 2, this are all half angle.

Â That's the upside down condition that you were identifying earlier to be

Â singularity.

Â So, geometrically, that projection line basically became parallel with your

Â projection hyperplane and they will never intercept.

Â And that's the cue.

Â They went off to infinity in that direction.

Â That's what happened there.

Â Good.

Â Any questions in this geometry?

Â When you see this a few times and you'll play with it in the homework and you see

Â some other coordinates arrived on this, there's some interesting stuff you can do.

Â So what we like is for smaller angles, and it's not just ten,

Â 20 degrees like with older angles, you can do extra 30 large angles.

Â They behave quite nearly, but if you go close past 90 to 180,

Â very non-linear, Kayleigh.

Â >> [INAUDIBLE] >> Yes.

Â >> [INAUDIBLE] >> Okay.

Â >> [INAUDIBLE] >> I'm getting in to that,

Â I havent talked about that yet.

Â >> Okay.

Â >> I'm glad you're paying attention because we're about to get to that.

Â What is that q S, right?

Â So I'm just talking about one set right now.

Â So lets also remember, right, quatronions are not unique.

Â There's always two sets of quatronions.

Â There's the Beta set and the minus Beta, so

Â whatever point on that sphere, the opposite is equally valid and

Â sometimes preferred depending on how far you've tumbled.

Â One is long, one is short.

Â And in this case,

Â everything that's the short location in quaternions is the stuff on this side of

Â this sphere because I'm always drawing the Beta Naught axis on the horizontal, right.

Â So this is always the short rotation and

Â then when the other expand out of this plane.

Â 7:22

If I do the same projection, you can see because of this two points on

Â opposite locations and the projection point is in the middle of this sphere,

Â it doesn't matter if I project Beta or minus Beta.

Â They all project on to the same point.

Â So these ultimate set for Rodrigues parameters, I call the Shadow Sets.

Â Here the shadow set doesn't really do anything, they are the same.

Â So thus the CRPs are unique.

Â Quaternions were not unique, that is two possible sets.

Â But the CRP with just one free parameter set and its a unique set.

Â There's no way to avoid the singularity.

Â That will be different with the MRPs that we'll talk about shortly.

Â So that's where the s comes from.

Â So in this case you can see geometrically, yep, they always will project.

Â And if I go to 180, it goes to infinity.

Â If I go to infinity and beyond as, who quoted that?

Â >> [LAUGH] Lightyear.

Â >> Thank you.

Â Good. Buzz Lightyear.

Â Infinity and beyond that's a wonderful statement.

Â Definitely works here.

Â Cause the coordinates literally do good to infinity and beyond and

Â then it come back in the minus sign and the form of the infinity and continue.

Â It's just we can't handle infinity very well in computer.

Â Well that's just the challenge.

Â So good!

Â Moving on, so we've got that and that's a shadow set.

Â If you go back to the definition of them, your Betas, this is very apparent.

Â Since it's Beta I over Beta Naught,

Â the alternate quaternion set is minus Beta I over minus Beta Naught, and

Â those minus signs cancel, so that mapping gives you the same set.

Â So CRPs unique at singular at 180.

Â