Not to be confused with the curvature of an affine connection.
Special affine curvature, also known as the equiaffine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine transformation (an affine transformation that preserves area). The curves of constant equiaffine curvature k are precisely all non-singular plane conics. Those with k > 0 are ellipses, those with k = 0 are parabolae, and those with k < 0 are hyperbolae.
The usual Euclidean curvature of a curve at a point is the curvature of its osculating circle, the unique circle making second order contact (having three point contact) with the curve at the point. In the same way, the special affine curvature of a curve at a point P is the special affine curvature of its hyperosculating conic, which is the unique conic making fourth order contact (having five point contact) with the curve at P. In other words, it is the limiting position of the (unique) conic through P and four points P1, P2, P3, P4 on the curve, as each of the points approaches P:
In some contexts, the affine curvature refers to a differential invariant κ of the general affine group, which may readily obtained from the special affine curvature k by κ = k−3/2dk/ds, where s is the special affine arc length. Where the general affine group is not used, the special affine curvature k is sometimes also called the affine curvature.[1]
Special affinecurvature, also known as the equiaffine curvature or affinecurvature, is a particular type of curvature that is defined on a plane curve...
connection on the frame bundle. The main invariants of an affine connection are its torsion and its curvature. The torsion measures how closely the Lie bracket...
& Sasaki 1994). If ∇ {\displaystyle \nabla } denotes an affine connection, then the curvature tensor R {\displaystyle R} is the (1,3)-tensor defined by...
differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is bilinear map of two input vectors X ,...
with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity. The curvature of spacetime...
complete set of affine structural invariants of the curve. In the plane, this gives a single scalar invariant, the affinecurvature of the curve. The...
Riemannian manifolds; the curvature of an affine connection or covariant derivative (on tensors); the curvature form of an Ehresmann connection: see Ehresmann...
characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve...
curvature tensor or the Ricci tensor, the scalar curvature cannot be defined for an arbitrary affine connection, for the reason that the trace of a (0...
geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using...
Yoshinori (2006). "Singularities of improper affine spheres and surfaces of constant Gaussian curvature". International Journal of Mathematics. 17 (3):...
{\displaystyle CC(M)} and may be infinite-dimensional. Every affine vector field is a curvature collineation. Conformal vector field Homothetic vector field...
Cartesian y = y ( x ) {\displaystyle y=y(x)} x = t {\displaystyle x=t} Affinecurvature F [ x , y ] = x ″ y ‴ − x ‴ y ″ ( x ′ y ″ − x ″ y ′ ) 5 / 3 − 1 2 [...
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian...
along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle)...
principal bundle with the curvature form of the connection. To make this theorem plausible, consider the familiar case of an affine connection (or a connection...
In the fields of computer vision and image analysis, the Harris affine region detector belongs to the category of feature detection. Feature detection...
also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the...
the study of differential geometry, describing surfaces of constant affinecurvature. The Tzitzeica equation has also been used in nonlinear physics, being...
An affine vector field (sometimes affine collineation or affine) is a projective vector field preserving geodesics and preserving the affine parameter...
the spherical and cylindrical coordinates for three-dimensional space. An affine line with a chosen Cartesian coordinate system is called a number line....
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