In differential geometry, affine differential geometry is the study of differential invariants of curves, surfaces, and higher-dimensional submanifolds under affine transformations. Its equiaffine form, often called equiaffine differential geometry, studies invariants under affine transformations that preserve a fixed volume form.
Classically, the subject concerns nondegenerate hypersurfaces in affine space. In contrast with Euclidean differential geometry, which uses a metric and a unit normal vector, equiaffine differential geometry uses the affine or Blaschke normal, the induced affine connection, the affine fundamental form, the affine shape operator, and related invariants such as the cubic form.
The name affine differential geometry reflects Klein's Erlangen program, in which geometries are studied through the invariants of transformation groups.
Affine geometry in the plane This section based on.
The simplest case of affine differential geometry studies differential properties of curves in the affine plane . That is, it studies properties of a curve in that are invariant under an affine transformation of . Many properties are only invariant under an area-preserving affine transformation. These properties are equi-affine, not affine. However, equiaffine properties are essentially affine properties with a fixed length scale, which allows us to convert between equiaffine geometry and affine geometry. For example, since the equiaffine arclength of a strictly convex and smooth curve well-defined, the affine arclength-ratio between two strictly convex and smooth curves is also well-defined.
For the rest of the section, we only consider curves that have continuous 4th derivatives.
Affine normal
An affine normal line of a parabola is its axis.The affine normal line through a strictly convex point on a curve is the axis of the osculating parabola. Equivalently, it is constructed by midpoints.
Let be a curve with a strictly convex point at , then the affine normal at that point is obtained by cutting the curve near using lines parallel to the tangent line . This produces a family of line segments. Their midpoints trace out another curve. The affine normal line is the tangent line of the midpoint locus at .Davis, D. (2006), Generic Affine Differential Geometry of Curves in Rn, Proc. Royal Soc. Edinburgh, 136A, 1195−1205.
In detail, if we have a curve specified by then the slope of the midpoint curve at is determined by , and its curving is determined by higher-order coefficients, etc. By an affine transformation of the parabola, the axis of the parabola can be matched to the curve.
The envelope curve of the affine normals is the affine focal set. The affine focal set of a parabola is the point at infinity of its axis. Similarly, for both the ellipse and the hyperbola, the affine focal set degenerates to a single point. In general, the affine focal set is a single point iff the curve is a conic section.
Note that in affine geometry, the focus of a conic section is not a meaningful concept, since it is not invariant under equiaffine transformation. Therefore, when the word "focus" is used, it refers to the affine focal set.
In general Similarly, for a smooth surface in 3-space, consider its Dupin indicatrix at a point as we shift its tangent plane. If the point is an elliptic point, then the tangent plane cuts a series of ellipses, and the affine normal line is spanned by the centers of the ellipses. Similarly if it is a hyperbolic point. This generalizes to a hypersurface in n-space.
Line elements We define a line element to be a line with a distinguished point. It can be represented by , where , and is only defined up to a multiplication by a nonzero real number. Now we show that the line elements in the 2-dimensional plane can be regarded as points in a 3-dimensional space .
For any two parabolas in the , there exists an equiaffine transformation mapping one to the other. Thus, many constructions using tangent lines in Euclidean differential geometry can be translated to constructions in affine differential geometry using osculating parabolas instead.
Through any 4 lines in general position, there exists a unique parabola tangent to all 4 of them. Thus, for any 2 line elements , there exists a unique parabola tangent to both of at the points . The 1-parameter family of all line elements tangent to it is a straight line in .
Given a parameterized curve , at any point , there is a Euclidean transformation that changes the curve to the form of . In other words, the curve has a standard form of with of length 1. The vector is unique up to a sign.
This is because at , there exists a unique straight line that osculates with degree 2, thus eliminating 2 terms in the power expansion of .
Similarly, we can find a parabola that is tangent to at two infinitesimally close points, which means it is an osculation of degree 4, thus eliminating 4 terms in the power expansion of . Therefore, there is an equiaffine transformation that changes the curve to the form of . In other words, there exists two vectors such that , and span a parallelogram of area 1. The pair is unique up to a sign of .
We will subsequently call this the equiaffine parameterization.
Equiaffine arclength
Since areas are invariant under equiaffine geometry, given 2 line elements not parallel to each other, they form a triangle. We define the distance between them to be of the area of the triangle. In the degenerate case where the two line elements are parallel, their distance is undefined. Since areas in are invariant under equiaffine transformations, the metric on is too. Note that in , the exponent of is necessary by an additive property of parabolas, but the factor of is merely conventional to make the equations appear simpler.
Therefore, we can speak of an equiaffine arclength on a strictly convex curve by integrating over the infinitesimal triangles formed by its successive tangent lines, which then allows us to define an equiaffine parameterization.
There is another way to produce equiaffine parameterization. A parameterization is Euclidean-arclength iff the vector is unit length. Analogously, a parameterization is equiaffine-arclength iff the two vectors span unit-area parallelogram for all .
Additivity on a parabola
In coordinates, consider the following parameterized parabola:It passes a point , with tangent in the direction of , and with axis in the direction of .
Consider two points on the parabola . The two tangent lines form a triangle. By direct computation, the triangle has area . This then means that given two line elements along the parabola , we haveWe see that the equiaffine arclength is invariant under subdivisions of the same parabola, meaning the equiaffine arclength converges to a well-defined limit under increasingly fine subdivisions.
In particular, we see that this is an affine parameterization of the parabola. It becomes an equiaffine parameterization after an affine scaling of .
In particular, is an equiaffine parameterization.
Equiaffine curvature The equiaffine arclength construction relies on approximating a curve as a sequence of osculating parabolas. Obviously, osculating parabolas of a parabola are just the parabola itself. However, for a generic strictly convex curve, its osculating parabolas change over the arclength. This can be used to define curvature in equiaffine geometry.
In a Euclidean parameterization, is perpendicular to , so it has the natural parametrization, defined by where is the rate at which its unit-velocity vector is turning. That is, , where is rotated clockwise by a right angle.
Similarly, in an equiaffine parameterization, is parallel to , so it has the natural parametrization, defined by where is the rate at which its affine-normal is shearing: . The quantity is the equiaffine curvature.
The parabola is the only curve with . Its equiaffine parameterization is . This shows that the parabola is analogous to the straight line, since it has zero curvature.
The ellipse is the only curve with constant positive curvature . Its equiaffine parameterization isor equivalently, where is its area.
The hyperbola with is the only curve with constant negative curvature . Its equiaffine parameterization is The Euclidean curvature of a curve at a point is obtained by finding the unique degree-3 osculating circle. Similarly, the affine curvature is obtained by finding the unique degree-5 osculating conic section.
Where the equiaffine curvature reaches a generic local maximum or minimum, the corresponding focal point reaches a semicubical cusp.
Affine geometry in space Let be a parameterized surface .
Normal form
At a generic elliptic point , the surface can be transformed via an equiaffine transformation into a normal form: Or in cylindrical coordinates, .
To obtain the transformation, first perform a Euclidean transformation, so that the tangent plane is moved to the plane, with the two axes of the Dupin indicatrix pointing in the directions, then perform a volume-preserving stretch in the directions. In other words, locally the surface is a paraboloid modified by a monkey saddle.
Since rotating around the axis paraboloid axis by 60° flips the sign of , only is a well-defined quantity in equiaffine geometry.
there is a 2-parameter family of planes. Generically, each plane intersects the surface at a curve, which has an equiaffine curvature at . This is the corresponding equiaffine sectional curvature.
For each unit vector , there is a 2-parameter family of lines in the same direction near . Generically, each line intersects at two points, and their midpoints make up a curved surface passing . Its tangent plane at is the Transon plane corresponding to the line element . As the unit vector rotates, it sweeps out a 1-parameter family of Transon planes. The envelope surface of these planes is a deltoid curve cone apexed at , and has an in-ellipse. This is related to the focal surface of the elliptical umbilic.
The central line of the deltoid cone is the affine normal of the surface at .
On the deltoid cone, 6 points are special: the 3 cusps, and the 3 tangents to its in-ellipse. They correspond to 6 Transon planes, which correspond to the 3 Darboux tangents (for the cusps), and the 3 Segre tangents of the surface at .
The 3 Darboux tangents can also be found by intersecting with the most osculating paraboloid . The intersection looks like an umbrella with 3 pairs of spines, which are the 3 Darboux tangents.
Blaschke metric Since at each elliptical point, there is only one equiaffine transformation (up to rotation by 120°) that transforms the surface to its normal formthe positive-definite quadratic form is well-defined in equiaffine geometry. This is the Blaschke metric, or Berwald–Blaschke metric.
Given a curve on the surface, integration of the Blaschke metric along it produces the same result as the previously constructed equiaffine arclength. Thus, the metric structure of equiaffine geometry of the surface is the same as the metric structure of a Riemannian geometry, though this does not reduce equiaffine geometry to Riemannian geometry (discussed below).
The Blaschke metric can also be defined without first transforming the surface to its normal form. Let be a smooth and locally strictly convex hypersurface in , and suppose locally near a point, the surface is the graph of a function , then the Blaschke metric iswhere ranges over . This definition is invariant under equiaffine transformations of the ambient space .
The immersed manifold perspective Riemannian geometry can be studied by immersing manifolds in , regarded as a Riemannian manifold. Similarly, affine differential geometry can be studied by immersing manifolds in , regarded as an affine manifold.
Affine fundamental form