In a previous blog post, I discussed the recent result of Guth and Katz obtaining a near-optimal bound on the Erdos distance problem. One of the tools used in the proof (building upon the earlier work of Elekes and Sharir) was the observation that the incidence geometry of the Euclidean group {SE(2)} of rigid motions of the plane was almost identical to that of lines in the Euclidean space {{\bf R}^3}:

Proposition 1 One can identify a (Zariski-)dense portion of {SE(2)} with {{\bf R}^3}, in such a way that for any two points {A, B} in the plane {{\bf R}^2}, the set {l_{AB} := \{ R \in SE(2): RA = B \}} of rigid motions mapping {A} to {B} forms a line in {{\bf R}^3}.

Proof: A rigid motion is either a translation or a rotation, with the latter forming a Zariski-dense subset of {SE(2)}. Identify a rotation {R} in {SE(2)} by an angle {\theta} with {|\theta| < \pi} around a point {P} with the element {(P, \cot \frac{\theta}{2})} in {{\bf R}^3}. (Note that such rotations also form a Zariski-dense subset of {SE(2)}.) Elementary trigonometry then reveals that if {R} maps {A} to {B}, then {P} lies on the perpendicular bisector of {AB}, and depends in a linear fashion on {\cot \frac{\theta}{2}} (for fixed {A,B}). The claim follows. \Box

As seen from the proof, this proposition is an easy (though ad hoc) application of elementary trigonometry, but it was still puzzling to me why such a simple parameterisation of the incidence structure of {SE(2)} was possible. Certainly it was clear from general algebraic geometry considerations that some bounded-degree algebraic description was available, but why would the {l_{AB}} be expressible as lines and not as, say, quadratic or cubic curves?

In this post I would like to record some observations arising from discussions with Jordan Ellenberg, Jozsef Solymosi, and Josh Zahl which give a more conceptual (but less elementary) derivation of the above proposition that avoids the use of ad hoc coordinate transformations such as {R \mapsto (P, \cot\frac{\theta}{2})}. The starting point is to view the Euclidean plane {{\bf R}^2} as the scaling limit of the sphere {S^2} (a fact which is familiar to all of us through the geometry of the Earth), which makes the Euclidean group {SE(2)} a scaling limit of the rotation group {SO(3)}. The latter can then be lifted to a double cover, namely the spin group {Spin(3)}. This group has a natural interpretation as the unit quaternions, which is isometric to the unit sphere {S^3}. The analogue of the lines {l_{AB}} in this setting become great circles on this sphere; applying a projective transformation, one can map {S^3} to {{\bf R}^3} (or more precisely to the projective space {{\bf P}^3}), at whichi point the great circles become lines. This gives a proof of Proposition 1.

Details of the correspondence are provided below the fold. One by-product of this analysis, incidentally, is the observation that the Guth-Katz bound {g(N) \gg N / \log N} for the Erdos distance problem in the plane {{\bf R}^2}, immediately extends with almost no modification to the sphere {S^2} as well (i.e. any {N} points in {S^2} determine {\gg N/\log N} distances), as well as to the hyperbolic plane {H^2}.

— 1. Euclidean geometry as the scaling limit of spherical geometry —

Euclidean geometry and spherical geometry are examples of Kleinian geometries: the geometry of a space {X} with a group of symmetries {G} acting transitively on it. In the case of Euclidean plane geometry, the space {X} is the plane {{\bf R}^2} and the symmetry group is the special Euclidean group {SE(2) = SO(2) \ltimes {\bf R}^2}; in the case of spherical geometry, the space {X} is the unit sphere {S^2} and the symmetry group is the special orthogonal group {SO(3)}. According to the Kleinian way of thinking (as formalised by the Erlangen program), explicit coordinates on {X} should be avoided if possible, with a preference instead for only using concepts (e.g. congruence, distance, angle) that are invariant with respect to the group of symmetries {G}.

As we all know from the geometry of the Earth (and the Greek root geometria literally means “Earth measurement”), the geometry of the sphere {S^2} resembles the geometry of the plane {{\bf R}^2} at scales that are small compared to the radius of the sphere. There are at least two ways to make this intuitive fact more precise. One is to make the radius {R} of the sphere go to infinity, and perform a suitable limit (e.g. a Gromov-Hausdorff limit). A dual approach is to keep the radius of the sphere fixed (e.g. considering only the unit sphere), but making the scale {\epsilon} being considered on the sphere shrink to zero. The two approaches are of course equivalent, but we will consider the latter.

Thus, we view {S^2} as the unit sphere in {{\bf R}^3}. With an eye to using the quaternionic number system later on, we will denote the standard basis of {{\bf R}^3} as {i,j,k}, thus in particular {i} is a point on the sphere {S^2} which we will view as an “origin” for this sphere. The tangent plane to {S^2} at this point is then

\displaystyle  \{ i + y j + z k: y,z \in {\bf R}^2 \}.

This plane is tangent to the sphere to second order. In particular, if {(y, z) \in {\bf R}^2}, and {\epsilon > 0} is a small parameter (which we think of as going to zero eventually), then we can find a point on {S^2} of the form {i + \epsilon y j + \epsilon z k + O(\epsilon^2)}. (If one wishes, one can enforce the {O(\epsilon^2)} error to lie in the {i} direction, in order to make the identification uniquely well-defined, although this is not strictly necessary for the discussion below.) Thus, we can view the {\epsilon}-neighbourhood of the origin {i} as being approximately identifiable with a bounded neighbourhood of the origin {0} in the plane {{\bf R}^2} via the identification

\displaystyle  (y,z) \mapsto i + \epsilon y j + \epsilon z k + O(\epsilon^2).

With this identification, one can see various structures in spherical geometry correspond (up to errors of {O(\epsilon)}) to analogous structures in planar geometry. For instance, a great circle in {S^2} is of the form

\displaystyle  \{ p \in S^2: p \cdot \omega = 0 \}

for some {\omega \in S^2}, where {\cdot} is the usual dot product. In order for this great circle to intersect the {O(\epsilon)} neighbourhood of the origin {i}, one must have {i \cdot \omega = O(\epsilon)}, and so we have

\displaystyle  \omega = \epsilon a i + (\cos \theta) j + (\sin \theta) k + O(\epsilon^2)

for some bounded quantity {a} and some angle {\theta}. If one then restricts the great circle to points {p = i + \epsilon y j + \epsilon z k + O(\epsilon^2)}, the constraint {p \cdot \omega = 0} then becomes

\displaystyle  a + (\cos \theta) y + (\sin \theta) z = O(\epsilon),

which is within {O(\epsilon)} of the equation of a typical line in {{\bf R}^2},

\displaystyle  a + (\cos \theta) y + (\sin \theta) z = 0.

This formalises the geometrically intuitive fact that great circles resemble lines at small scales.

Remark 1 One can also adopt a more “intrinsic” Riemannian geometry viewpoint to see {{\bf R}^2} as the limit of rescaled versions of {S^2}. Indeed, for each real number {\kappa}, there is a unique (up to isometry) simply connected Riemannian surface {S_\kappa} of constant scalar curvature {\kappa}. For {\kappa > 0}, this is the unit sphere {S^2} (rescaled by {\sqrt{\kappa/2}}); for {\kappa=0}, this is the Euclidean plane {{\bf R}^2}; and for {\kappa < 0}, it is the hyperbolic plane {H^2} (rescaled by {\sqrt{|\kappa|/2}}). Sending {\kappa} to zero, we thus see the sphere (or hyperbolic plane) converging to the Euclidean plane.

We now apply a similar analysis to a rotation matrix {R \in SO(3)} acting on the unit sphere {S^2}. In order for this rotation matrix to map an {O(\epsilon)} neighbourhood of the origin {i} to another such neighbourhood, the rotation matrix {R} must be of the form {R = (1 + \epsilon T + O(\epsilon^2)) S}, where {S} is a rotation that fixes {i}, thus

\displaystyle  S (x i + y j + z k) = xi + (y \cos \theta - z \sin \theta) j + (y \sin \theta + z \cos \theta) k

for some angle {\theta}, and {T} is an infinitesimal rotation (i.e. an element of the Lie algebra {\mathfrak{so}(3)}), thus {Ti = aj + bk} for some reals {a,b}. We then have

\displaystyle  R ( i + \epsilon yj + \epsilon zk + O(\epsilon^2) ) = i + \epsilon (y \cos \theta - z \sin \theta + a)j

\displaystyle  + \epsilon (y \sin \theta + z \cos \theta+b) + O(\epsilon^2),

so in {(y,z)} coordinates, {R} becomes the map

\displaystyle  (y,z) \mapsto (y \cos \theta - z \sin \theta + a, y \sin \theta + z \cos \theta+b) + O(\epsilon),

which is within {\epsilon} of the Euclidean rigid motion

\displaystyle  (y,z) \mapsto (y \cos \theta - z \sin \theta + a, y \sin \theta + z \cos \theta+b).

Thus we see the Euclidean rigid motion group {SE(2)} emerging as a scaling limit of the orthogonal rotation group {SO(3)} (or alternatively, as the normal bundle of the stabiliser of the origin, which is a copy of {SO(2)}).

Remark 2 One can also analyse the situation from a Lie algebra perspective. As is well known, one can equip the three-dimensional Lie algebra {\mathfrak{so}(3)} with a basis {X,Y,Z} obeying the commutation relations

\displaystyle  [X,Y] = Z; [Y,Z] = X; [Z,X] = Y,

corresponding to infinitesimal rotations around the {x,y,z} axes respectively. If we then rescale {X_\epsilon := X, Y_\epsilon := \epsilon Y, Z_\epsilon := \epsilon Z} (which morally corresponds to looking at rotation matrices that almost fix {i}, as above), the commutation relations rescale to

\displaystyle  [X_\epsilon,Y_\epsilon] = Z_\epsilon; [Y_\epsilon,Z_\epsilon] = \epsilon^2 X_\epsilon; [Z_\epsilon,X_\epsilon] = Y_\epsilon.

Sending {\epsilon \rightarrow 0}, the Lie algebra degenerates to the solvable Lie algebra

\displaystyle  [X_0,Y_0] = Z_0; [Y_0,Z_0] = 0; [Z_0,X_0] = Y_0,

which is the Lie algebra {\mathfrak{se}(2)} of the Euclidean group {SE(2)}.

There is an exact analogue of this phenomenon for the isometry group {SO(2,1) \equiv SL_2({\bf R})} of the hyperbolic plane {H^2} (which one can think of as one sheet of the unit sphere in Minkowski space {{\bf R}^{2+1}}, just as {S^2} is the unit sphere in {{\bf R}^3}). The Lie algebra here can be equipped with a basis {X,Y,Z} (which one can interpret as infinitesimal rotations and Lorentz boosts in Minkowski space) with the relations

\displaystyle  [X,Y] = Z; [Y,Z] = -X; [Z,X] = Y,

and the same scaling argument as before gives {SE(2)} as a scaling limit of {SO(2,1)}.

— 2. Lifting to the quaternions —

The quaternions are a number system, defined formally as the set {{\bf H}} of numbers of the form

\displaystyle  t + xi + yj + zk

with {t,x,y,z \in {\bf R}}. This a four-dimensional vector space; it can be turned into an algebra (and into a division ring) by enforcing Hamilton’s relations

\displaystyle i^2=j^2=k^2=ijk=-1.

The quaternions also come equipped with a conjugation operation

\displaystyle  (t+xi + yj + zk)^* := t - xi - yj - zk

and a norm

\displaystyle  |\alpha| := (\alpha \alpha^*)^{1/2} = (\alpha^* \alpha)^{1/2},

thus

\displaystyle  |(t+xi+yj+zk)| = \sqrt{t^2+x^2+y^2+z^2}.

The conjugation operation is an anti-automorphism, and the norm is multiplicative: {|\alpha \beta| = |\alpha| |\beta|}. The quaternions also have a trace

\displaystyle  \hbox{tr}(t+xi+yj+zk) = t

(in particular, {\hbox{tr}(\alpha^*) = \hbox{tr}(\alpha)} and {\hbox{tr}(\alpha \beta) = \hbox{tr}(\beta \alpha)}), giving rise to a dot product

\displaystyle  \alpha \cdot \beta := \hbox{tr}( \alpha \beta^* )

which (together with the norm) gives a Hilbert space structure on the quaternions.

The unit sphere {S^3 = \{ \alpha \in {\bf H}: \alpha \alpha^* = 1 \}} of the quaternions forms a group, which is a model for the spin group {Spin(3)} (thus giving rise to an interpretation of the quaternions {{\bf H}}, which are of course acted upon by {S^3} by left-multiplication, as spinors for this group). This group acts on itself isometrically by conjugation, with an element {\alpha \in S^3 \equiv Spin(3)} mapping {\beta \in S^3} to {\alpha \beta \alpha^*}. As {\alpha \alpha^*=1}, this action preserves the quaternionic identity {1}, and thus preserves the orthogonal complement {\{ xi+yj+zk: x^2+y^2+z^2\}} of that identity in {S^3}, which we can of course identify with {S^2}. Thus {S^3 \equiv Spin(3)} acts on {S^2} isometrically by conjugation, thus providing a map from {Spin(3)} to {SO(3)}. One can verify that this map is surjective (indeed, conjugation by the quaternion {e^{i\theta}} corresponds to a rotation around the {i}-axis by {2\theta}, and similarly for rotations around other axes) and is a double cover (since the center of {S^3} is {\{-1,+1\}}), with the pre-image of any rotation in {SO(3)} being a pair {\{\alpha,-\alpha\}} of diametrically opposed points in {S^3}. Thus we see that {SO(3)} can be identified (in either the topological or Riemannian geometrical senses) with the projective sphere {S^3/\pm \equiv {\bf P}^3}. As {SO(3)} acts transitively on {S^2}, we see that {S^3 \equiv Spin(3)} does also.

The stabiliser {l_{AA} := \{ \alpha \in S^3: \alpha A \alpha^* = A \}} of a point {A \in S^2} is easily seen to be a great circle in {S^3} (being the intersection of {S^3} with the center of {\alpha}, which is a plane). For instance, the stabiliser {l_{ii}} of the origin {i} is the circle {\{ e^{i\theta}: \theta \in {\bf R} \}}. (This, incidentally, gives an explicit geometric manifestation of the Hopf fibration.) By transitivity (and the isometric nature of the {S^3} action), we conclude that the sets {l_{AB} := \{ \alpha \in S^3: \alpha A \alpha^* = B \}} are also great circles in {S^3} for any pair of points {A, B}. Conversely, as all great circles are isometric to each other, we see that all great circles are of the form {l_{AB}}. One also sees that two great circles {l_{AB}, l_{CD}} coincide only when {A, C} have the same stabiliser, and when {B, D} have the same stabiliser, which forces {C, D} to either equal {A, B}, or {-A, -B}.

Remark 3 Using a projective transformation, one can identify (a hemisphere of) {S^3} with {{\bf R}^3}, with (most) great circles becoming lines in {{\bf R}^3}. Thus, we see that the incidence geometry of the {l_{AB}} in {S^3} is essentially equivalent to the incidence geometry of lines in {{\bf R}^3}. Because of this, the Guth-Katz argument to establish the bound {g(N) \gg N / \log N} for the number of distances determined by {N} points in the plane {{\bf R}^2}, also extends to {N} points in the sphere {S^2}. Indeed, as in the Guth-Katz paper, it suffices to show that the number of congruent line segments {AB, CD} in these {N} points is {O( N^3 \log N )}. For each such pair of line segments, there is a unique element of {SO(3)} (and thus two antipodal elements of {S^3 \equiv Spin(3)}) that maps {AB} to {CD}; these two antipodal points are also the intersection of {l_{AC}} with {l_{BD}}. Applying the projective transformation, one arrives at exactly the same incidence problem between points and lines considered by Guth-Katz (and in particular, one can apply Theorems 2.4, 2.5 from their paper as a black box, after verifying that at most {O(N)} lines of the form {l_{AB}} project into a plane or regulus, which is proven in the {S^2} case in much the same way as it is in the {{\bf R}^2} case). We omit the details.

A similar argument (changing the signatures in various metrics, and in the Clifford algebra underlying the quaternions) also allows one to establish the same results in the hyperbolic plane {H^2}; again, we omit the details.

If we restrict attention to an {\epsilon}-neighbourhood of the origin {i} in the sphere {S^2}, and similarly restrict to an {\epsilon}-neighbourhood of the stabiliser of {i} in the spin group {S^3 \equiv Spin(3)}, we can use the correspondences from the previous section to convert {S^2} into {{\bf R}^2} in the limit, and {Spin(3)} in the limit into a double cover of the rotation group {SE(2)} (which ends up just being isomorphic to {SE(2)} again). The great circles {l_{AB}} in {Spin(3)} then, in the limit, become the analogous sets {l_{AB} = \{ R \in SE(2): RA = B \}} in {SE(2)}, and the above correspondences can then be used to map (most of) {SE(2)} to {{\bf R}^3}, and (most) {l_{AB}} to lines, giving Proposition 1.