Affine focal set

Affine focal set

In mathematics, and especially affine differential geometry, the affine focal set of a smooth submanifold "M" embedded in a smooth manifold "N" is the caustic generated by the affine normal lines. It can be realised as the bifurcation set of a certain family of functions. The bifurcation set is the set of parameter values of the family which yield functions with degenerate singularities. Please note that this is not the same as the bifurcation diagram in dynamical systems.

Let us assume that "M" is an "n"-dimensional smooth hypersurface in real ("n"+1)-space. We assume that "M" has no points where the second fundamental form is degenerate. We recall from the article affine differential geometry that there is a unique transverse vector field over "M". This is the affine normal vector field, or the Blaschke normal field. The key thing to note is that a special (i.e. det = 1) affine transformation of real ("n" + 1)-space will carry the affine normal vector field of "M" onto the the affine normal vector field of the image of "M" under the transformation.

Geometric interpretation

Let us consider a local parametrisation of "M". Let U subset mathbb{R}^{n} be an open neighbourhood of 0 with coordinates old{u} = (u_1,ldots,u_n), and let old{X} : U o mathbb{R}^{n+1} be a smooth parametrisation of "M" in a neighbourhood of one of its points.

The affine normal vector field will be denoted by old{A}. At each point of "M" it is transverse to the tangent space of "M", i.e.

: old{A} : U o T_{old{X}(U)}mathbb{R}^{n+1}. ,

For a fixed old{u}_0 in U the affine normal line to "M" at old{X}(old{u}_0) may be parametrised by "t" where: t mapsto old{X}(old{u}_0) + t old{A}(old{u}_0). The affine focal set is given geometrically as the infinitesimal intersections of the "n"-parameter family of affine normal lines. To calculate this we choose an affine normal line, say at point "p"; then we look at the affine normal lines at points infinitesimally close to "p" an see if any intersect the one at "p". If we choose a point infinitesimally close to old{u} in U, then it may be expressed as old{u} + dold{u} where dold{u} represents the infinitesimal difference. Thus old{X}(old{u}) and old{X}(old{u} + dold{u}) will be our "p" and its neighbour.

For "t" and dold{u} we try to solve: old{X}(old{u}) + t old{A}(old{u}) = old{X}(old{u} + dold{u}) + t old{A}(old{u} + dold{u}). This can be done by using power series expansions, and is not too difficult; it is lengthy and has thus been omitted.

We recall from the article affine differential geometry that the affine shape operator "S" is a type (1,1)-tensor field on "M", and is given by Sv = D_vold{A}, where "D" is the covariant derivative on real ("n" + 1)-space (for those well read: it is the usual flat and torsion free connexion).

We find that the solutions to old{X}(old{u}) + t old{A}(old{u}) = old{X}(old{u} + dold{u}) + t old{A}(old{u} + dold{u}) are when 1/"t" is an eigenvalue of "S" and that dold{u} is a corresponding eigenvector. Note that the eigenvalues of "S" are not always distinct: there may be repeated roots, there may be complex roots, and "S" may not always be diagonalisable. For 0 le k le [n/2] , where [-] denotes the greatest integer function, there will generically be ("n" − 2"k")-pieces of the affine focal set above each point "p". The −2"k" corresponds to pairs of eigenvalues becoming complex (like the solution to x^2 + a = 0 as "a" changes from negative to positive).

The affine focal set need not be made up of smooth hypersurfaces. In fact, for a generic hypersurface "M", the affine focal set will have singularities. The singularities could be found by calculation, but that may be difficult, and we still have no idea of what the singularity looks like up to diffeomorphism. If we use some singularity theory then we get much more information.

Singularity theory aproach

The idea here is to define a family of functions over "M". The family will have the ambient real ("n" + 1)-space as its parameter space, i.e. for each choice of ambient point we will get a function defined over "M". This family is the family of affine distance functions:

: Delta : mathbb{R}^{n+1} imes M o mathbb{R}.,

Given an ambient point old{x} and a surface point "p", we can decompose the chord joining "p" to old{x} as a tangential component and a transverse component parallel to old{A}. The value of Δ is given implicitly in the equation

: old{x} - p = Z(old{x},p) + Delta(old{x},p) old{A}(p)

where "Z" is a tangent vector. We now seek the bifurcation set of the family Δ, i.e. the ambient points for which the restricted function: Delta : {old{x}} imes M o mathbb{R} has degenertate singularity at some "p". A function has degenerate singularity if both the Jacobian matrix of first order partial derivatives and the Hessian matrix of second order partial derivatives have zero determinant.

To discover if the Jacobian matrix has zero determinant we differentiate the equation "x - p = Z + ΔA". Let "X" be a tangent vector to "M", and differentiate in that direction:: D_X(old{x}-p) = D_X(Z + Delta old{A}),

: -X = abla_XZ + h(X,Z)old{A} + d_XDelta old{A} - Delta SX ,

: ( abla_XZ + (I - Delta S)X) + (h(X,Z) + d_XDelta)old{A} = 0 ,

where "I" is the identity. This tells us that abla_XZ = (Delta S - I)X and h(X,Z) = -d_XDelta. The last equality says that we have the following equation of differential one-forms h(-,Z) = dDelta. The Jacobian matrix will have zero determinant if, and only if, dDelta is degenerate as a one-form, i.e. d_XDelta = 0 for all tangent vectors "X".Since h(-,Z) = dDelta it follows that dDelta is degenerate if, and only if, h(-,Z) is degenerate. Since "h" is a non-degenerate two-form it follows that "Z = 0". Notice that since "M" has a non-degenerate second fundamental form it follows that "h" is a non-degenerate two-form. Since "Z = 0" the set of ambient points "x" for which the restricted function Delta : {old{x}} imes M o mathbb{R} has a singularity at some "p" is the affine normal line to "M" at "p".

To compute the Hessian matrix we consider the differential two-form (X,Y) mapsto d_Y(d_XDelta). This is the two-form whose matrix representation is the Hessian matrix. We have already seen that h(X,Z) = -d_XDelta we see that d_Y(d_XDelta) = -d_Y(h(X,Z)). We have: (X,Y) mapsto -d_Y(h(X,Z)) = -( abla_Yh)(X,Z) - h( abla_YX,Z) - h(X, abla_YZ) .Now assume that Δ has a singularity at "p", i.e. Z = 0, then we have the two-form: (X,Y) mapsto - h(X, abla_YZ) . We have also seen that abla_XZ = (Delta S - I)X, and so the two-form becomes: (X,Y) mapsto h(X,(I- Delta S)Y) . This is degenerate as a two-form if, and only if, there exists non-zero "X" for which it is zero for all "Y". Since "h" is non-degenerate it must be that det(I- Delta S) = 0 and Y in ker(I- Delta S). So the singularity is degenerate if, and only if, the ambient point "x" lies on the affine normal line to "p" and the reciprocal of its distance from "p" is an eigenvalue of "S", i.e. points old{x} = p + told{A} where 1/"t" is an eigenvalue of "S". The affine focal set!

Singular points

The affine focal set can be the following:: { p + t old{A}(p) : p in M, det(I - tS) = 0} . To find the singular points we simply differentiate "p + tA" in some tangent direction "X":: D_X(p + t old{A}) = (I-tS)X + d_Xt old{A}. The affine focal set is singular if, and only if, there exists non-zero "X" such that D_X(p + t old{A}) = 0, i.e. if, and only if, "X" is an eigenvector of "S" and the derivative of "t" in that direction is zero. This means that the derivative of an affine principal curvature in it own affine principal direction is zero.

Local structure

We can use the standard ideas in singularity theory to classify, up to local diffeomorphism, the affine focal set. If the family of affine distance functions can be shown to be a certain kind of family then the local structure is known. We want the family of affine distance functions to be a versal unfolding of the singularities which arrise.

The affine focal set of a plane curve will generically consist of smooth pieces of curve and ordinary cusp points (semi-cubical palabara|semi-cubical parabolae).

The affine focal set of a surface in three-space will generically consist of smooth pieces of surface, cuspidal cylinder points (A_3), swallowtail points (A_4), purse points (D_4^+), and pyramid points (D_4^-).The A_k and D_k series are as in Arnold's list.

The question of the local structure in much higher dimension is of great interest. For example, we were able to construct a discrete list of singularity types (up to local diffeomprhism). In much higher dimensions no such discrete list can be costructed, there are functional modulii.

References

* V. I. Arnold, S. M. Gussein-Zade and A. N. Varchenko, "Singularities of differentiable maps", Volume 1, Birkhäuser, 1985.
* J. W. Bruce and P. J. Giblin, "Curves and singularities", Second edition, Cambridge University press, 1992.
* T. E. Cecil, "Focal points and support functions", Geom. Dedicada 50, No. 3, 291 – 300, 1994.
* D. Davis, "Affine differential geometry and singularity theory", PhD thesis, Liverpool, 2008.
* K. Nomizu and Sasaki, "Affine differential geometry", Cambridge university press, 1994.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • Projective space — In mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non zero vectors which are equal up to a multiplication by a non zero scalar. A formal… …   Wikipedia

  • Ellipse — Elliptical redirects here. For the exercise machine, see Elliptical trainer. This article is about the geometric figure. For other uses, see Ellipse (disambiguation). Not to be confused with ellipsis. An ellipse obtained as the intersection of a… …   Wikipedia

  • Conic section — Types of conic sections: 1. Parabola 2. Circle and ellipse 3. Hyperbola …   Wikipedia

  • Sprite (computer graphics) — Part of a series on …   Wikipedia

  • Bargaining problem — The two person bargaining problem is a problem of understanding how two agents should cooperate when non cooperation leads to Pareto inefficient results. It is in essence an equilibrium selection problem; Many games have multiple equilibria with… …   Wikipedia

  • Bargaining Problem — The two person bargaining problem is a problem of understanding how two agents should cooperate when non cooperation leads to Pareto inefficient results. It is in essence an equilibrium selection problem. Many games have multiple equilibria with… …   Wikipedia

  • List of Latin words with English derivatives — This is a list of Latin words with derivatives in English (and other modern languages). Ancient orthography did not distinguish between i and j or between u and v. Many modern works distinguish u from v but not i from j. In this article both… …   Wikipedia

  • Color balance — This article is about the process applied to still images. For the equivalent process applied to video, see Color grading. The left half shows the photo as it came from the digital camera. The right half shows the photo adjusted to make a gray… …   Wikipedia

  • Triangulation (computer vision) — In computer vision triangulation refers to the process of determining a point in 3D space given its projections onto two, or more, images. In order to solve this problem it is necessary to know the parameters of the camera projection function… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”