In mathematics, and especially

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, an affine sphere is a

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

for which the affine normals all intersect in a single point. The term affine sphere is used because they play an analogous role in

affine differential geometry Affine differential geometry is a type of differential geometry which studies invariants of volume-preserving affine transformations. The name ''affine differential geometry'' follows from Klein's Erlangen program. The basic difference between aff ...

to that of ordinary spheres in Euclidean differential geometry. An affine sphere is called improper if all of the affine normals are constant. In that case, the intersection point mentioned above lies on the

hyperplane at infinity In geometry, any hyperplane ''H'' of a projective space ''P'' may be taken as a hyperplane at infinity. Then the set complement is called an affine space. For instance, if are homogeneous coordinates for ''n''-dimensional projective space, then t ...

. Affine spheres have been the subject of much investigation, with many hundreds of

research article Academic publishing is the subfield of publishing which distributes academic research and scholarship. Most academic work is published in academic journal articles, books or theses. The part of academic written output that is not formally pub ...

s devoted to their study.

Examples

* All

quadric In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hype ...

s are affine spheres; the quadrics that are also improper affine spheres are the

paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axial symmetry, axis of symmetry and no central symmetry, center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar p ...

s. * If ƒ is a

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

on the plane and the

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

of the

Hessian matrix In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a functio ...

is ±1 then the graph of ƒ in three-space is an improper affine sphere.

References

Differential geometry {{differential-geometry-stub