# I Set of possible rotations of a 3D object and the rotation history

#### lavinia

Gold Member
Consider the simpler situation of rotations in the plane where the rotation group is (part of) $SO(2)$ which has $2(2-1)/2 = 1$ degrees of freedom, i.e. one parameter (while $SO(3)$ has $3(3-1)/2 = 3$ parameters). The parameter space is the line $[0,2 \pi]$. A continuous path in parameter space is just a parameter $\theta$ moving along the interval $[0,2\pi]$, e.g. if you rotate around a circle, the path in parameter space is just the angle $\theta$ moving from $0$ to a bunch of other angles...

Regarding "rotation space" and "orientation space", I am not sure these things make sense and again it seems like you are thinking of rigid bodies where one fixes the position of the center of mass then the orientation of the rigid body about that center of mass.
To continue this example, the loop that connects $0$ to $2π$ is a circle and cannot be shrunk to a point without breaking the circle,

Unlike in the case of $SO(3)$ no multiple of this loop is contractible. In $SO(3)$ the double of any loop is contractible ("null homotopic").

#### Spinnor

Gold Member
Regarding "rotation space" and "orientation space", I am not sure these things make sense and again it seems like you are thinking of rigid bodies where one fixes the position of the center of mass then the orientation of the rigid body about that center of mass.
I was trying I was using the wrong terminology and was confused but a little so less now, I think, and I was thinking of a physical object with a fixed point.

Say we have two right-handed rectangular coordinate systems that coincide. Now say we rotate one of the coordinate systems in some complicated manor keeping the other coordinate system fixed. The rotated coordinate system has a new orientation, relative to the the fixed coordinate system, which can be represented as a point, p, of our 3-ball and the rotation "history" of the rotated coordinate system is represented as a path in our 3-ball starting at the origin and ending at the point p. What I called the rotation history is just the orientation history. From the Wiki article on orientation it looks like there is more than one way to describe the orientation, such as the matrix of unit vectors that Lavinia listed above.

I am in a better place now in regards to rotations. Thanks.

#### lavinia

Gold Member
The diagram is just crudely representing a 'continuous' sequence of rotations in 3-D space as a continuous path in an abstract parameter space, you can try to set up a better picture with the line through a sphere (anti-podal points) idea if necessary.
From the topological point of view the representation is precise. The map from the 3 ball with antipodal points on the boundary sphere identified into the rotation group is a homeomorphism.

The point of the diagram is really to show that paths through SO(3) can be connected but not simply connected.
And also that the double of any loop is null homotopic. So the diagram implies that the fundamental group of $SO(3)$ is $Z_2$.

#### bolbteppa

I was trying I was using the wrong terminology and was confused but a little so less now, I think, and I was thinking of a physical object with a fixed point.

Say we have two right-handed rectangular coordinate systems that coincide. Now say we rotate one of the coordinate systems in some complicated manor keeping the other coordinate system fixed. The rotated coordinate system has a new orientation, relative to the the fixed coordinate system, which can be represented as a point, p, of our 3-ball and the rotation "history" of the rotated coordinate system is represented as a path in our 3-ball starting at the origin and ending at the point p. What I called the rotation history is just the orientation history. From the Wiki article on orientation it looks like there is more than one way to describe the orientation, such as the matrix of unit vectors that Lavinia listed above.

I am in a better place now in regards to rotations. Thanks.
Maybe this will help - one can take the perspective that rotations are "active" or "passive", one view is that they are rotating vectors in a fixed coordinate system, the other view is rotating the coordinate system while leaving the vector fixed.

https://en.wikipedia.org/wiki/Active_and_passive_transformation

The active 'rotation history' of rotating some vector is basically the path the tip of the vector threads out as it's rotated, or rather the inverse image of this (i.e. in 'parameter space'), while the passive 'rotation history' is the same path in parameter space, but this time the image of the path in parameter space is not a vector threading out a path, it's the coordinate system rotating around.

#### Spinnor

Gold Member
The active 'rotation history' of rotating some vector is basically the path the tip of the vector
But the vector does not record rotations about its axis so can we please attach a ridgid flag to the end of the vector? With the flag we can then record and note rotations about the length of the vector. Then it starts to look like a spinor, my favorite unknown.

Thank you.

#### lavinia

Gold Member
But the vector does not record rotations about its axis so can we please attach a ridgid flag to the end of the vector? With the flag we can then record and note rotations about the length of the vector. Then it starts to look like a spinor, my favorite unknown.

Thank you.
Right. The motion of a single vector does not determine a rotation. It merely describes a path on a sphere centered at the origin.
A rotation is described by a rigid motion of an orthonormal frame.

A rotation of coordinates is a rotation of basis frames of the coordinate system.

I do not see what this has to do with spinors, Can you explain?

#### Spinnor

Gold Member
I do not see what this has to do with spinors, Can you explain?
I think I read that spinors don't live in our 3 dimensional space but they can be represented in our 3 dimensional space by a flagpole and flag (vector and flag) plus a sign,

but the flagpole and flag is mathematically the same as a rectangular coordinate system, we need a direction and an angle to determine their orientations relative to some fixed coordinate system as above?

Thanks.

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#### bolbteppa

But the vector does not record rotations about its axis so can we please attach a ridgid flag to the end of the vector? With the flag we can then record and note rotations about the length of the vector. Then it starts to look like a spinor, my favorite unknown.

Thank you.
Unfortunately, spinors are not expressing the idea of a vector rotating about it's own axis, they are related to connectivity of paths in e.g. $SO(3)$ and the necessity of (in this case) dealing with $SU(2)$ rather than $SO(3)$ as the pdf of the OP is trying to explain. What I have read of this flagpole stuff seems to assume what a spinor is and then randomly uses flags to represent it.

#### Spinnor

Gold Member
Unfortunately, spinors are not expressing the idea of a vector rotating about it's own axis
The flag is supposed to be rigidly attached to the flagpole. So if we rotate the vector about the vector axis the flag will rotate as well. But the flag and flagpole convey no more information than a right handed coordinate frame so I am not sure why it is used( edit, the spinor "flagpole" can have a length, an extra bit of information that the coordinate frame does not have)?

What I have read of this flagpole stuff seems to assume what a spinor is and then randomly uses flags to represent it.
From the little I understand the flag is a necessary part of the representation. See,

https://arxiv.org/pdf/1312.3824.pdf

Thank you.

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#### lavinia

Gold Member
The flag is supposed to be rigidly attached to the flagpole. So if we rotate the vector about the vector axis the flag will rotate as well. But the flag and flagpole convey no more information than a right handed coordinate frame so I am not sure why it is used( edit, the spinor "flagpole" can have a length, an extra bit of information that the coordinate frame does not have)?

From the little I understand the flag is a necessary part of the representation. See,

https://arxiv.org/pdf/1312.3824.pdf

View attachment 238886

Thank you.
@bolbteppa is correct. The flag just gives the rotation amount around the axis. This is not a spinor. As the book says there is also a sign. But this method of visualization does not give a clear definition IMO. For every rotation there are 2 spinors that correspond to it. These have opposite sign in the group of unit quaternions.

#### Spinnor

Gold Member
The idea is used by several authors, those that are still alive could better defend its use, not I. I think they all mention the need for another bit of information, the sign. Thank you.

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#### bolbteppa

If the flag is to not be completely superfluous it should at the very least capture that a $2 \pi$ rotation brings in the minus sign, however (page 6):

"The angle doubling leads to the curious feature that when $\theta= 2 \pi$ (a single full rotation) the spin rotation matrices all give $-I$. It is not that the flagpole reverses direction — it does not, and neither does the flag — but rather, the spinor picks up an overall sign that has no ready representation in the flagpole picture."

My reading of MTW (page 1157 on from your last post) is again that they do not give you spinors from this idea of flags, they try to describe the already-established notion of spinors using flags, and earlier (very badly) motivate spinors by arguments that amount to the arguments of the PDF of the OP.

#### Spinnor

Gold Member
All right then, into the trashcan of science with the flagpole and flag. I wish there was something just as intuitive to replace it.

Thank you.

#### bolbteppa

I would be shocked if one could get more intuitive than the notes of the OP for this stuff

This perspective of spinors from the POV of connectivity of Lie groups is very important for understanding why (finite dimensional) spinors exist for the subgroup SO(n) of GL(n) but not GL(n) itself

A nice simple proof of this would be welcome (hint hint readers)

#### lavinia

Gold Member
I thought it would be helpful to elaborate on the parameterization of $SO(3)$ given in post #1. This is the 3 dimensional ball $B^3$ with antipodal points on its boundary identified.

Topologically $B^3$ modulo these identifications is homeomorphic to the real projective 3 space $RP^3$. The book is showing that $SO(3)$ viewed as a topological space is homeomorphic to $RP^3$.

In topology, the real $n$ dimensional projective space $RP^{n}$ is the quotient space of the $n$ dimensional unit sphere in $R^{n+1}$ with its antipodal points identified. Antipodal points come in pairs and are the intersections of lines through the origin with the unit $n$ sphere. They may be thought of as opposite geographic poles.

The quotient mapping $S^{n}→RP^{n}$ that identifies antipodal points is 2 to 1, is continuous and any small enough ball in $S^{n}$ - a ball that contains no antipodal points e.g. a polar ice cap - is mapped homomorphically into a small ball in $RP^{n}$.This follows from the definition of the quotient topology and shows that $RP^{n}$ is a closed $n$ dimensional manifold and that $S^{n}$ is a 2 fold covering space of it.

Consider now the northern hemisphere of $S^{n}$ together with the equator - which is the $n-1$ dimensional sphere $S^{n-1}$. This space is an $n$ dimensional topological ball as can be seen by projecting it vertically onto the $n$ dimensional plane containing the equator. Under the quotient map $S^{n}→RP^{n}$ antipodal points on the equatorial $n-1$ sphere are identified and in the northern hemisphere proper, each point is mapped to a unique point. Every point in $RP^{n}$ is represented either by a unique point if the northern hemisphere or by a pair of antipodal points on the equator. In the case of $S^3$ the northern hemisphere is a three dimensional ball So its image in $RP^3$ is a three dimensional ball with antipodal points on its boundary 2-sphere identified. This is exactly the same topological description and in post #1 and shows why $SO(3)$ is topologically the same as the real projective space $RP^3$. So $SO(3)$ is more than just a group of matrices. It is a closed three dimensional manifold.

The only difference between this description and the description in post #1 is the interpretation of parameters. Instead of axes of rotation, one has directions along great circles emanating from the north pole. Instead of an angle of rotation, one has the distance from the north pole. The angle $θ$ lies between $0$ and $π/2$ rather than between $0$ and $π$. Points along the equator all have an angle of $π/2$. The curve illustrated in post #1, the straight line through the center of the ball with end points on the boundary, corresponds now to a half great circle through the north pole. Its end points are antipodal so it projects to a closed loop $γ$ in projective space.

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#### lavinia

Gold Member
Closed loops in projective space

As was shown in post #40 projective space $RP^{n}$ is the quotient of the $n$-sphere $S^{n}$ with antipodal points identified. A loop in $RP^{n}$ lifts to two antipodal paths on the sphere and beginning and ending points on one path are antipodal to beginning and end points on the other. If the paths are also closed loops the lift is two antipodal closed loops on the sphere. If the paths are not loops then the end points of each are antipodal. In this second case, the two paths fit together to make a single loop.

So every closed loop in projective space is covered twice by its preimage on the sphere. The preimage is either two antipodal loops or two antipodal paths that join together at their antipodal ends to form a single loop. In this second case, the closed loop does not project back onto the original loop but instead projects to its double, the loop that wraps around it twice. This is because each piece wraps around it once.

Each of these loops is contractible - in fact every closed loop on a sphere of dimension greater than 1 is contractible. - and any contraction projects to a contraction in projective space. A contraction of either of the antipodal closed loops projects back to a contraction of the original loop that they come from, while a contraction of the. spliced together loop projects to a contraction of the double of the original loop.

So every closed loop in a projective space of dimension greater than one is either contractible or its double is contractible. The double of the diameter line in post #1 is contractible but since it is not closed in the 3 ball, it may not be contractible by itself.

The idea of the proof that every closed loop on the sphere is contractible.

If a closed loop on the sphere misses at least one point, then it can be contracted along great circles through one of the missing points. If the loop is space filling, then it can be first continuously deformed into a loop that is not, then contracted. Proving that a space filling loop can be deformed to a non-space filling loop requires a little work and is the only hard part of the proof.

Notes:

- A space filling curve is a continuous path that completely covers a region of space. Every point in the region - for instance of a square or of a cube. - is crossed by the path. Such paths can be shown to exist as the uniform limits of certain sequences of continuous paths.

- The formal definition of a contraction of a loop

One imagines a contraction of a loop as a stretched rubber band that shrinks as it releases tension. At each point in time, the band forms a smaller loop until finally it has zero tension. Formally this is a continuous 1 parameter family of loops and may be described as a continuous map from $H:S^1×[0,1]→X$ from a circle Cartesian product an interval into a topological space $X$ which at time zero is the starting loop and at time one is the constant loop. The map $H$ is called a homotopy and is similar to a variation except that it is only required to be continuous rather than smoothly differentiable and end points are allowed to wander. The idea of a homotopy is not restricted to $S^1$ and makes sense for any topological space.

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"Set of possible rotations of a 3D object and the rotation history"

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