Lie sphere geometry
Encyclopedia
Lie sphere geometry is a geometrical
theory of planar
or spatial geometry in which the fundamental concept is the circle
or sphere
. It was introduced by Sophus Lie
in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.
The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold
known as the Lie quadric (a quadric hypersurface in projective space
). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).
To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent space
s. This provides a natural realisation of the osculating circle
to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.
Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric
. This similarity led Lie to his famous "line-sphere correspondence" between the space of lines and the space of spheres in 3-dimensional space.
in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangent
ial contact have a more natural formulation in a more general context in which circles, line
s and points
(resp. spheres, planes
and points) are treated on an equal footing. This is achieved in three steps. First an ideal point at infinity is added to Euclidean space so that lines (or planes) can be regarded as circles (or spheres) passing through the point at infinity (i.e., having infinite radius
). This extension is known as Möbius geometry. Second, points are regarded as circles (or spheres) of zero radius. Finally, for technical reasons, the circles (or spheres), including the lines (or planes) are given orientations
.
These objects, i.e., the points, oriented circles and oriented lines in the plane, or the points, oriented spheres and oriented planes in space, are sometimes called cycles or Lie cycles. It turns out that they form a quadric hypersurface in a projective space
of dimension 4 or 5, which is known as the Lie quadric. The natural symmetries of this quadric form a group of transformations known as the Lie transformations. These transformations do not preserve points in general: they are transforms of the Lie quadric, not of the plane/sphere plus point at infinity. The point-preserving transformations are precisely the Möbius transformations. The Lie transformations which fix the ideal point at infinity are the Laguerre
transformations of Laguerre geometry. These two subgroups generate the group of Lie transformations, and their intersection are the Möbius transforms that fix the ideal point at infinity, namely the
affine conformal maps.
(3,2) symmetric bilinear form
defined by
The projective space RP4 is the space of lines through the origin
in R5 and is the space of nonzero vectors x in R5 up to scale, where x= (x0,x1,x2,x3,x4). The planar Lie quadric Q consists of the points [x] in projective space represented by vectors x with x · x = 0.
To relate this to planar geometry it is necessary to fix an oriented timelike line. The chosen coordinates suggest using the point [1,0,0,0,0] ∈ RP4. Any point in the Lie quadric Q can then be represented by a vector x = λ(1,0,0,0,0) + v, where v is orthogonal to (1,0,0,0,0). Since [x] ∈ Q, v · v = λ2 ≥ 0.
The orthogonal space to (1,0,0,0,0), intersected with the Lie quadric, is the two dimensional celestial sphere
S in Minkowski space-time. This is the Euclidean plane with an ideal point at infinity, which we take to be [0,0,0,0,1]: the finite points (x,y) in the plane are then represented by the points [v] = [0,x,y, −1, (x2+y2)/2]; note that v · v = 0, v · (1,0,0,0,0) = 0 and v · (0,0,0,0,1) = −1.
Hence points x = λ(1,0,0,0,0) + v on the Lie quadric with λ = 0 correspond to points in the Euclidean plane with an ideal point at infinity. On the other hand points x with λ nonzero correspond to oriented circles (or oriented lines, which are circles through infinity) in the Euclidean plane. This is easier to see in terms of the celestial sphere
S: the circle corresponding to [λ(1,0,0,0,0) + v] ∈ Q (with λ ≠ 0) is the set of points y ∈ S with y · v = 0. The circle is oriented because v/λ has a definite sign; [−λ(1,0,0,0,0) + v] represents the same circle with the opposite orientation. Thus the isometric
reflection map x → x + 2 (x · (1,0,0,0,0)) (1,0,0,0,0) induces an involution ρ of the Lie quadric which reverses the orientation of circles and lines, and fixes the points of the
plane (including infinity).
To summarize: there is a one to one correspondence between points on the Lie quadric and cycles in the plane, where a cycle is either an oriented circle (or straight line) or a point in the plane (or the point at infinity); the points can be thought of as circles of radius zero, but they are not oriented.
. If [x] ∈ S ≅ R2 ∪ {∞}, then this just means that [x] lies on the circle corresponding to [y]; this case is immediate from the definition of this circle (if [y] corresponds to a point circle then x · y = 0 if and only if [x] = [y]).
It therefore remains to consider the case that neither [x] nor [y] are in S. Without loss of generality, we can then take x= (1,0,0,0,0) + v and y = (1,0,0,0,0) + w, where v and w are spacelike unit vectors in (1,0,0,0,0)⊥. Thus
v⊥ ∩ (1,0,0,0,0)⊥ and w⊥ ∩ (1,0,0,0,0)⊥ are signature (2,1) subspaces of (1,0,0,0,0)⊥. They therefore either coincide or intersect in a 2-dimensional subspace. In the latter case, the 2-dimensional subspace can either have signature (2,0), (1,0), (1,1), in which case the corresponding two circles in S intersect in zero, one or two points respectively. Hence they have first order contact if and only if the 2-dimensional subspace is degenerate (signature (1,0)), which holds if and only if the span of v and w is degenerate. By Lagrange's identity, this holds if and only if (v · w)2 = (v · v)(w · w) = 1, i.e., if and only if v · w = ± 1, i.e., x · y = 1 ± 1. The contact is oriented if and only if v · w = – 1, i.e., x · y = 0.
The incidence of cycles in Lie sphere geometry provides a simple solution to the problem of Apollonius
. This problem concerns a configuration of three distinct circles (which may be points or lines): the aim is to find every other circle (including points or lines) which is tangent to all three of the original circles. For a generic configuration of circles, there are at most eight such tangent circles.
The solution, using Lie sphere geometry, proceeds as follows. Choose an orientation for each of the three circles (there are eight ways to do this, but there are only four up to reversing the orientation of all three). This defines three points [x], [y], [z] on the Lie quadric Q. By the incidence of cycles, a solution to the Apollonian problem compatible with the chosen orientations is given by a point [q] ∈ Q such that q is orthogonal to x, y and z. If these three vectors are linearly dependent, then the corresponding points [x], [y], [z] lie on a line in projective space. Since a nontrivial quadratic equation has at most two solutions, this line actually lies in the Lie quadric, and any point [q] on this line defines a cycle incident with [x], [y] and [z]. Thus there are infinitely many solutions in this case.
If instead x, y and z are linearly independent then the subspace V orthogonal to all three is 2-dimensional. It can have signature (2,0), (1,0), or (1,1), in which case there are zero, one or two solutions for [q] respectively. (The signature cannot be (0,1) or (0,2) because it is orthogonal to a space containing more than one null line.) In the case that the subspace has signature (1,0), the unique solution q lies in the span of x, y and z.
The general solution to the Apollonian problem is obtained by reversing orientations of some of the circles, or equivalently, by considering the triples (x,ρ(y),z), (x,y,ρ(z)) and (x,ρ(y),ρ(z)).
Note that the triple (ρ(x),ρ(y),ρ(z)) yields the same solutions as (x,y,z), but with an overall reversal of orientation. Thus there are at most 8 solution circles to the Apollonian problem unless all three circles meet tangentially at a single point, when there are infinitely many solutions.
O(3,2) of orthogonal transformations
of R3,2 maps any null one dimensional subspaces of R3,2 to another such subspace. Hence the group O(3,2) acts
on the Lie quadric. These transformations of cycles are called "Lie transformations". They preserve the incidence relation between cycles. The action is transitive and so all cycles are Lie equivalent. In particular, points are not preserved by general Lie transformations. The subgroup of Lie transformations preserving the point cycles is essentially the subgroup of orthogonal transformations which preserve the chosen timelike direction. This subgroup is isomorphic to the group O(3,1) of Möbius transformations of the sphere. It can also be characterized as the centralizer of the involution ρ, which is itself a Lie transformation.
Lie transformations can often be used to simplify a geometrical problem, by transforming circles into lines or points.
.
An oriented contact element in the plane is a pair consisting of a point and an oriented
(i.e., directed) line through that point. The point and the line are incident cycles. The key observation is that the set of all cycles incident with both the point and the line is a Lie invariant object: in addition to the point and the line, it consists of all the circles which make oriented contact with the line at the given point. It is called a pencil
of Lie cycles, or simply a contact element.
Note that the cycles are all incident with each other as well. In terms of the Lie quadric, this means that a pencil of cycles is a (projective) line lying entirely on the Lie quadric, i.e., it is the projectivization of a totally null two dimensional subspace of R3,2: the representative vectors for the cycles in the pencil are all orthogonal to each other.
The set of all lines on the Lie quadric is a 3-dimensional manifold
called the space of contact elements Z3. The Lie transformations preserve the contact elements, and act transitively on Z3. For a given choice of point cycles (the points orthogonal to a chosen timelike vector v), every contact element contains a unique point. This defines a map from Z3 to the 2-sphere S2 whose fibres are circles. This map is not Lie invariant, as points are not Lie invariant.
Let γ:[a,b] → R2 be an oriented curve. Then γ determines a map λ from the interval [a,b] to Z3 by sending t to the contact element corresponding to the point γ(t) and the oriented line tangent to the curve at that point (the line in the direction γ '(t)). This map λ is called the contact lift of γ.
In fact Z3 is a contact manifold, and the contact structure is Lie invariant. It follows that oriented curves can be studied in a Lie invariant way via their contact lifts, which may be characterized, generically as Legendrian curves in Z3. More precisely, the tangent space to Z3 at the point corresponding to a null 2-dimensional subspace π of R3,2 is the subspace of those linear maps (A mod π):π → R3,2/π with
and the contact distribution is the subspace Hom(π,π⊥/π) of this tangent space in the space Hom(π,R3,2/π) of linear maps.
It follows that an immersed Legendrian curve λ in Z3 has a preferred Lie cycle associated to each point on the curve: the derivative of the immersion at t is a 1-dimensional subspace of Hom(π,π⊥/π) where π=λ(t); the kernel of any nonzero element of this subspace is a well defined 1-dimensional subspace of π, i.e., a point on the Lie quadric.
In more familiar terms, if λ is the contact lift of a curve γ in the plane, then the preferred cycle at each point is the osculating circle
. In other words, after taking contact lifts, much of the basic theory of curves in the plane is Lie invariant.
The Lie quadric Qn is again defined as the set of [x] ∈ RPn+2 = P(Rn+1,2) with x · x = 0. The quadric parameterizes oriented (n – 1)-spheres
in n-dimensional space, including hyperplane
s and point spheres as limiting cases. Note that Qn is an (n + 1)-dimensional manifold (spheres are parameterized by their center and radius).
The incidence relation carries over without change: the spheres corresponding to points [x], [y] ∈ Qn have oriented first order contact if and only if x · y = 0. The group of Lie transformations is now O(n + 1, 2) and the Lie transformations preserve incidence of Lie cycles.
The space of contact elements is a (2n – 1)-dimensional contact manifold Z2n – 1: in terms of the given choice of point spheres, these contact elements correspond to pairs consisting of a point in n-dimensional space (which may be the point at infinity) together with an oriented hyperplane
passing through that point. The space Z2n – 1 is therefore isomorphic to the projectivized cotangent bundle
of the n-sphere. This identification is not invariant under Lie transformations: in Lie invariant terms, Z2n – 1 is the space of (projective) lines on the Lie quadric.
Any immersed oriented hypersurface in n-dimensional space has a contact lift to Z2n – 1 determined by its oriented tangent space
s. There is no longer a preferred Lie cycle associated to each point: instead, there are n – 1 such cycles, corresponding to the curvature spheres in Euclidean geometry.
The problem of Apollonius has a natural generalization involving n + 1 hyperspheres in n dimensions.
Suppose [x], [y] ∈ RP3, with homogeneous coordinates
(x0,x1,x2,x3) and (y0,y1,y2,y3). Put pij = xiyj - xjyi. These are the homogeneous coordinates of the projective line
joining x and y. There are six independent coordinates and they satisfy a single relation, the Plücker relation
It follows that there is a one to one correspondence between lines in RP3 and points on the Klein quadric
, which is the quadric hypersurface of points [p01, p23, p02, p31, p03, p12] in RP5 satisfying the Plücker relation.
The quadratic form
defining the Plücker relation comes from a symmetric bilinear form of signature (3,3). In other words the space of lines in RP3 is the quadric in P(R3,3). Although this is not the same as the Lie quadric, a "correspondence" can be defined between lines and spheres using the complex number
s: if x = (x0,x1,x2,x3,x4,x5) is a point on the (complexified) Lie quadric (i.e., the xi are taken to be complex numbers), then
defines a point on the complexified Klein quadric (where i2 = –1).
Lie sphere geometry provides a natural description of Dupin cyclides. These are characterized as the common envelope of two one parameter families of spheres S(s) and T(t), where S and T are maps from intervals into the Lie quadric. In order for a common envelope to exist, S(s) and T(t) must be incident for all s and t, i.e., their representative vectors must span a null 2-dimensional subspace of R4,2. Hence they define a map into the space of contact elements Z5. This map is Legendrian if and only if the derivatives of S (or T) are orthogonal to T (or S), i.e., if and only if there is an orthogonal decomposition of R4,2 into a direct sum of 3-dimensional subspaces σ and τ of signature (2,1), such that S takes values in σ and T takes values in τ. Conversely such a decomposition uniquely determines a contact lift of a surface which envelops two one parameter families of spheres; the image of this contact lift is given by the null 2-dimensional subspaces which intersect σ and τ in a pair of null lines.
Such a decomposition is equivalently given, up to a sign choice, by a symmetric endomorphism of R4,2 whose square is the identity and whose ±1 eigenspaces are σ and τ. Using the inner product on R4,2, this is determined by a quadratic form on R4,2.
To summarize, Dupin cyclides are determined by quadratic forms on R4,2 such that the associated symmetric endomorphism has square equal to the identity and eigenspaces of signature (2,1).
This provides one way to see that Dupin cyclides are cyclides, in the sense that they are zero-sets of quartics of a particular form. For this, note that as in the planar case, 3-dimensional Euclidean space embeds into the Lie quadric Q3 as the set of point spheres apart from the ideal point at infinity. Explicitly, the point (x,y,z) in Euclidean space corresponds to the point
in Q3. A cyclide consists of the points [0,x1,x2,x3,x4,x5] ∈ Q3 which satisfy an additional quadratic relation
for some symmetric 5 × 5 matrix A = (aij). The class of cyclides is a natural family of surfaces in Lie sphere geometry, and the Dupin cyclides form a natural subfamily.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
theory of planar
Plane geometry
In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...
or spatial geometry in which the fundamental concept is the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
or sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. It was introduced by Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.
The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
known as the Lie quadric (a quadric hypersurface in projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).
To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
s. This provides a natural realisation of the osculating circle
Osculating circle
In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...
to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.
Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric
Klein quadric
The lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent a line in S lie on a hyperbolic quadric, Q known as the Klein quadric....
. This similarity led Lie to his famous "line-sphere correspondence" between the space of lines and the space of spheres in 3-dimensional space.
Basic concepts
The key observation that leads to Lie sphere geometry is that theorems of Euclidean geometryEuclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
ial contact have a more natural formulation in a more general context in which circles, line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
s and points
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
(resp. spheres, planes
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
and points) are treated on an equal footing. This is achieved in three steps. First an ideal point at infinity is added to Euclidean space so that lines (or planes) can be regarded as circles (or spheres) passing through the point at infinity (i.e., having infinite radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
). This extension is known as Möbius geometry. Second, points are regarded as circles (or spheres) of zero radius. Finally, for technical reasons, the circles (or spheres), including the lines (or planes) are given orientations
Orientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
.
These objects, i.e., the points, oriented circles and oriented lines in the plane, or the points, oriented spheres and oriented planes in space, are sometimes called cycles or Lie cycles. It turns out that they form a quadric hypersurface in a projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of dimension 4 or 5, which is known as the Lie quadric. The natural symmetries of this quadric form a group of transformations known as the Lie transformations. These transformations do not preserve points in general: they are transforms of the Lie quadric, not of the plane/sphere plus point at infinity. The point-preserving transformations are precisely the Möbius transformations. The Lie transformations which fix the ideal point at infinity are the Laguerre
Edmond Laguerre
Edmond Nicolas Laguerre was a French mathematician, a member of the Académie française . His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials...
transformations of Laguerre geometry. These two subgroups generate the group of Lie transformations, and their intersection are the Möbius transforms that fix the ideal point at infinity, namely the
affine conformal maps.
The Lie quadric
The Lie quadric of the plane is defined as follows. Let R3,2 denote the space R5 of 5-tuples of real numbers, equipped with the signatureSymmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
(3,2) symmetric bilinear form
Symmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
defined by
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
in R5 and is the space of nonzero vectors x in R5 up to scale, where x= (x0,x1,x2,x3,x4). The planar Lie quadric Q consists of the points [x] in projective space represented by vectors x with x · x = 0.
To relate this to planar geometry it is necessary to fix an oriented timelike line. The chosen coordinates suggest using the point [1,0,0,0,0] ∈ RP4. Any point in the Lie quadric Q can then be represented by a vector x = λ(1,0,0,0,0) + v, where v is orthogonal to (1,0,0,0,0). Since [x] ∈ Q, v · v = λ2 ≥ 0.
The orthogonal space to (1,0,0,0,0), intersected with the Lie quadric, is the two dimensional celestial sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
S in Minkowski space-time. This is the Euclidean plane with an ideal point at infinity, which we take to be [0,0,0,0,1]: the finite points (x,y) in the plane are then represented by the points [v] = [0,x,y, −1, (x2+y2)/2]; note that v · v = 0, v · (1,0,0,0,0) = 0 and v · (0,0,0,0,1) = −1.
Hence points x = λ(1,0,0,0,0) + v on the Lie quadric with λ = 0 correspond to points in the Euclidean plane with an ideal point at infinity. On the other hand points x with λ nonzero correspond to oriented circles (or oriented lines, which are circles through infinity) in the Euclidean plane. This is easier to see in terms of the celestial sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
S: the circle corresponding to [λ(1,0,0,0,0) + v] ∈ Q (with λ ≠ 0) is the set of points y ∈ S with y · v = 0. The circle is oriented because v/λ has a definite sign; [−λ(1,0,0,0,0) + v] represents the same circle with the opposite orientation. Thus the isometric
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
reflection map x → x + 2 (x · (1,0,0,0,0)) (1,0,0,0,0) induces an involution ρ of the Lie quadric which reverses the orientation of circles and lines, and fixes the points of the
plane (including infinity).
To summarize: there is a one to one correspondence between points on the Lie quadric and cycles in the plane, where a cycle is either an oriented circle (or straight line) or a point in the plane (or the point at infinity); the points can be thought of as circles of radius zero, but they are not oriented.
Incidence of cycles
Suppose two cycles are represented by points [x], [y] ∈ Q. Then x · y = 0 if and only if the corresponding cycles "kiss", that is they meet each other with oriented first order contactContact (mathematics)
In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...
. If [x] ∈ S ≅ R2 ∪ {∞}, then this just means that [x] lies on the circle corresponding to [y]; this case is immediate from the definition of this circle (if [y] corresponds to a point circle then x · y = 0 if and only if [x] = [y]).
It therefore remains to consider the case that neither [x] nor [y] are in S. Without loss of generality, we can then take x= (1,0,0,0,0) + v and y = (1,0,0,0,0) + w, where v and w are spacelike unit vectors in (1,0,0,0,0)⊥. Thus
v⊥ ∩ (1,0,0,0,0)⊥ and w⊥ ∩ (1,0,0,0,0)⊥ are signature (2,1) subspaces of (1,0,0,0,0)⊥. They therefore either coincide or intersect in a 2-dimensional subspace. In the latter case, the 2-dimensional subspace can either have signature (2,0), (1,0), (1,1), in which case the corresponding two circles in S intersect in zero, one or two points respectively. Hence they have first order contact if and only if the 2-dimensional subspace is degenerate (signature (1,0)), which holds if and only if the span of v and w is degenerate. By Lagrange's identity, this holds if and only if (v · w)2 = (v · v)(w · w) = 1, i.e., if and only if v · w = ± 1, i.e., x · y = 1 ± 1. The contact is oriented if and only if v · w = – 1, i.e., x · y = 0.
The problem of Apollonius
Problem of Apollonius
In Euclidean plane geometry, Apollonius' problem is to construct circles that are tangent to three given circles in a plane . Apollonius of Perga posed and solved this famous problem in his work ; this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived...
. This problem concerns a configuration of three distinct circles (which may be points or lines): the aim is to find every other circle (including points or lines) which is tangent to all three of the original circles. For a generic configuration of circles, there are at most eight such tangent circles.
The solution, using Lie sphere geometry, proceeds as follows. Choose an orientation for each of the three circles (there are eight ways to do this, but there are only four up to reversing the orientation of all three). This defines three points [x], [y], [z] on the Lie quadric Q. By the incidence of cycles, a solution to the Apollonian problem compatible with the chosen orientations is given by a point [q] ∈ Q such that q is orthogonal to x, y and z. If these three vectors are linearly dependent, then the corresponding points [x], [y], [z] lie on a line in projective space. Since a nontrivial quadratic equation has at most two solutions, this line actually lies in the Lie quadric, and any point [q] on this line defines a cycle incident with [x], [y] and [z]. Thus there are infinitely many solutions in this case.
If instead x, y and z are linearly independent then the subspace V orthogonal to all three is 2-dimensional. It can have signature (2,0), (1,0), or (1,1), in which case there are zero, one or two solutions for [q] respectively. (The signature cannot be (0,1) or (0,2) because it is orthogonal to a space containing more than one null line.) In the case that the subspace has signature (1,0), the unique solution q lies in the span of x, y and z.
The general solution to the Apollonian problem is obtained by reversing orientations of some of the circles, or equivalently, by considering the triples (x,ρ(y),z), (x,y,ρ(z)) and (x,ρ(y),ρ(z)).
Note that the triple (ρ(x),ρ(y),ρ(z)) yields the same solutions as (x,y,z), but with an overall reversal of orientation. Thus there are at most 8 solution circles to the Apollonian problem unless all three circles meet tangentially at a single point, when there are infinitely many solutions.
Lie transformations
Any element of the groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
O(3,2) of orthogonal transformations
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
of R3,2 maps any null one dimensional subspaces of R3,2 to another such subspace. Hence the group O(3,2) acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on the Lie quadric. These transformations of cycles are called "Lie transformations". They preserve the incidence relation between cycles. The action is transitive and so all cycles are Lie equivalent. In particular, points are not preserved by general Lie transformations. The subgroup of Lie transformations preserving the point cycles is essentially the subgroup of orthogonal transformations which preserve the chosen timelike direction. This subgroup is isomorphic to the group O(3,1) of Möbius transformations of the sphere. It can also be characterized as the centralizer of the involution ρ, which is itself a Lie transformation.
Lie transformations can often be used to simplify a geometrical problem, by transforming circles into lines or points.
Contact elements and contact lifts
The fact that Lie transformations do not preserve points in general can also be a hindrance to understanding Lie sphere geometry. In particular, the notion of a curve is not Lie invariant. This difficulty can be mitigated by the observation that there is a Lie invariant notion of contact elementContact (mathematics)
In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...
.
An oriented contact element in the plane is a pair consisting of a point and an oriented
Orientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
(i.e., directed) line through that point. The point and the line are incident cycles. The key observation is that the set of all cycles incident with both the point and the line is a Lie invariant object: in addition to the point and the line, it consists of all the circles which make oriented contact with the line at the given point. It is called a pencil
Pencil (mathematics)
A pencil in projective geometry is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a projective plane....
of Lie cycles, or simply a contact element.
Note that the cycles are all incident with each other as well. In terms of the Lie quadric, this means that a pencil of cycles is a (projective) line lying entirely on the Lie quadric, i.e., it is the projectivization of a totally null two dimensional subspace of R3,2: the representative vectors for the cycles in the pencil are all orthogonal to each other.
The set of all lines on the Lie quadric is a 3-dimensional manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
called the space of contact elements Z3. The Lie transformations preserve the contact elements, and act transitively on Z3. For a given choice of point cycles (the points orthogonal to a chosen timelike vector v), every contact element contains a unique point. This defines a map from Z3 to the 2-sphere S2 whose fibres are circles. This map is not Lie invariant, as points are not Lie invariant.
Let γ:[a,b] → R2 be an oriented curve. Then γ determines a map λ from the interval [a,b] to Z3 by sending t to the contact element corresponding to the point γ(t) and the oriented line tangent to the curve at that point (the line in the direction γ '(t)). This map λ is called the contact lift of γ.
In fact Z3 is a contact manifold, and the contact structure is Lie invariant. It follows that oriented curves can be studied in a Lie invariant way via their contact lifts, which may be characterized, generically as Legendrian curves in Z3. More precisely, the tangent space to Z3 at the point corresponding to a null 2-dimensional subspace π of R3,2 is the subspace of those linear maps (A mod π):π → R3,2/π with
- A(x) · y + x · A(y) = 0
and the contact distribution is the subspace Hom(π,π⊥/π) of this tangent space in the space Hom(π,R3,2/π) of linear maps.
It follows that an immersed Legendrian curve λ in Z3 has a preferred Lie cycle associated to each point on the curve: the derivative of the immersion at t is a 1-dimensional subspace of Hom(π,π⊥/π) where π=λ(t); the kernel of any nonzero element of this subspace is a well defined 1-dimensional subspace of π, i.e., a point on the Lie quadric.
In more familiar terms, if λ is the contact lift of a curve γ in the plane, then the preferred cycle at each point is the osculating circle
Osculating circle
In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...
. In other words, after taking contact lifts, much of the basic theory of curves in the plane is Lie invariant.
General theory
Lie sphere geometry in n-dimensions is obtained by replacing R3,2 (corresponding to the Lie quadric in n = 2 dimensions) by Rn + 1, 2. This is Rn + 3 equipped with the symmetric bilinear form
The Lie quadric Qn is again defined as the set of [x] ∈ RPn+2 = P(Rn+1,2) with x · x = 0. The quadric parameterizes oriented (n – 1)-spheres
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
in n-dimensional space, including hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
s and point spheres as limiting cases. Note that Qn is an (n + 1)-dimensional manifold (spheres are parameterized by their center and radius).
The incidence relation carries over without change: the spheres corresponding to points [x], [y] ∈ Qn have oriented first order contact if and only if x · y = 0. The group of Lie transformations is now O(n + 1, 2) and the Lie transformations preserve incidence of Lie cycles.
The space of contact elements is a (2n – 1)-dimensional contact manifold Z2n – 1: in terms of the given choice of point spheres, these contact elements correspond to pairs consisting of a point in n-dimensional space (which may be the point at infinity) together with an oriented hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
passing through that point. The space Z2n – 1 is therefore isomorphic to the projectivized cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
of the n-sphere. This identification is not invariant under Lie transformations: in Lie invariant terms, Z2n – 1 is the space of (projective) lines on the Lie quadric.
Any immersed oriented hypersurface in n-dimensional space has a contact lift to Z2n – 1 determined by its oriented tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
s. There is no longer a preferred Lie cycle associated to each point: instead, there are n – 1 such cycles, corresponding to the curvature spheres in Euclidean geometry.
The problem of Apollonius has a natural generalization involving n + 1 hyperspheres in n dimensions.
Three dimensions and the line-sphere correspondence
In the case n=3, the quadric Q3 in P(R4,2) describes the (Lie) geometry of spheres in Euclidean 3-space. Lie noticed a remarkable similarity with the Klein correspondence for lines in 3-dimensional space (more precisely in RP3).Suppose [x], [y] ∈ RP3, with homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
(x0,x1,x2,x3) and (y0,y1,y2,y3). Put pij = xiyj - xjyi. These are the homogeneous coordinates of the projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
joining x and y. There are six independent coordinates and they satisfy a single relation, the Plücker relation
Plücker coordinates
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogenous coordinates to each line in projective 3-space, P3. Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines...
- p01 p23 + p02 p31 + p03 p12 = 0.
It follows that there is a one to one correspondence between lines in RP3 and points on the Klein quadric
Klein quadric
The lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent a line in S lie on a hyperbolic quadric, Q known as the Klein quadric....
, which is the quadric hypersurface of points [p01, p23, p02, p31, p03, p12] in RP5 satisfying the Plücker relation.
The quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
defining the Plücker relation comes from a symmetric bilinear form of signature (3,3). In other words the space of lines in RP3 is the quadric in P(R3,3). Although this is not the same as the Lie quadric, a "correspondence" can be defined between lines and spheres using the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s: if x = (x0,x1,x2,x3,x4,x5) is a point on the (complexified) Lie quadric (i.e., the xi are taken to be complex numbers), then
- p01 = x0 + x1, p23 = –x0 + x1
- p02 = x2 + ix3, p31 = x2 – ix1
- p03 = x4 , p12 = x5
defines a point on the complexified Klein quadric (where i2 = –1).
Dupin cyclides
Such a decomposition is equivalently given, up to a sign choice, by a symmetric endomorphism of R4,2 whose square is the identity and whose ±1 eigenspaces are σ and τ. Using the inner product on R4,2, this is determined by a quadratic form on R4,2.
To summarize, Dupin cyclides are determined by quadratic forms on R4,2 such that the associated symmetric endomorphism has square equal to the identity and eigenspaces of signature (2,1).
This provides one way to see that Dupin cyclides are cyclides, in the sense that they are zero-sets of quartics of a particular form. For this, note that as in the planar case, 3-dimensional Euclidean space embeds into the Lie quadric Q3 as the set of point spheres apart from the ideal point at infinity. Explicitly, the point (x,y,z) in Euclidean space corresponds to the point
- [0, x, y, z, –1, (x2 + y2 + z2)/2]
in Q3. A cyclide consists of the points [0,x1,x2,x3,x4,x5] ∈ Q3 which satisfy an additional quadratic relation
for some symmetric 5 × 5 matrix A = (aij). The class of cyclides is a natural family of surfaces in Lie sphere geometry, and the Dupin cyclides form a natural subfamily.