Curves | Affine geometry | Differential geometry
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. (Wikipedia).
The all important concept of curvature. We look at two equations for curvature and introduce the radius of curvature.
From playlist Life Science Math: Vectors
Reference: Differential Geometry by Do Carmo My first video! Thank you for coming and any suggestion is very welcomed! #some2
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(PP 6.5) Affine property, Constructing Gaussians, and Sphering
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In this lecture, we define the exponential mapping, the Riemannian curvature tensor, Ricci curvature tensor, and scalar curvature. The focus is on an intuitive explanation of the curvature tensors. The curvature tensor of a Riemannian metric is a very large stumbling block for many student
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6C Second equation for curvature on the blackboard
In this lecture I show you a second equation for curvature.
From playlist Life Science Math: Vectors
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This video explains how to determine curvature using short cut formula for a function of x.
From playlist Vector Valued Functions
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This video explains how to determine curvature using short cut formula for a vector function in 2D.
From playlist Vector Valued Functions
Ben Andrews: Limiting shapes of fully nonlinear flows of convex hypersurfaces
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From playlist MATRIX-SMRI Symposium: Singularities in Geometric Flows
Holomorphic Cartan geometries on simply connected manifolds by Sorin Dumitrescu
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From playlist Global Noncommutative Geometry Seminar (Europe)
Affinity Designer - Create complex shapes with the Corner and Pen Tool
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New Methods in Finsler Geometry - 22 May 2018
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Quantum Finite Elements: Lattice Field Theory on Curved Manifolds by Richard Brower
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How to Trace a Character in Affinity Designer
:: Support Me :: https://www.patreon.com/alecaddd http://www.alecaddd.com/support-me/ https://amzn.to/2Hcp5mo In this video I'm showing you my process of creating a vector character. Probably it's wrong, but hey, it works for me... :: Support Me :: http://www.alecaddd.com/support-me/ ::
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Lec 10. Einstein's General Relativity and Gravitation: General Relativity 6
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Multivariable Calculus | Curvature
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From playlist Multivariable Calculus