In the previous lecture we developed the notion of the curvature of a surface, showing a bunch of nice theoretical properties. But aside from theoretical elegance, what’s so great about curvature? Well, remember from the plane curve setting that the curvature function uniquely defines a curve parameterized by arc length up to a rigid motion. A similar result holds for surfaces, though the proof...

We discover new information about the spherical curvature of stereographically projected analytic curves. To do so, we first state formulas for the spherical curvature and spherical torsion of the curves on S2 which result after stereographically projecting the image curves of analytic, univalent functions belonging to the class . We then derive results concerning the location of the critical p...

The Gaussian curvature of the surface at the point p is the product of the maximum and minimum curvatures in the family. The objective of this paper is to provide a deeper and broader understanding of the meaning of Gaussian curvature, using some more general alternative computational methods. We define the coefficients of the expected Fisher Information Matrix as the coefficients of the first ...

In this paper we shall discuss hypersurfaces M of space forms of constant curvature; where curvature means one of the symmetric functions of curvature associated to the second fundamental form. The values of the constant will be chosen so that the linearized equation will be an elliptic equation onM . For example, for surfaces in 3 the two possible curvatures are the mean curvature H and the Ga...

In this paper, we study gradient Ricci expanding solitons (X, g) satisfying Rc = cg +Df, where Rc is the Ricci curvature, c < 0 is a constant, and Df is the Hessian of the potential function f on X . We show that for a gradient expanding soliton (X, g) with non-negative Ricci curvature, the scalar curvature R has at least one maximum point on X , which is the only minimum point of the potential...

Fascicle curvature of human medial gastrocnemius muscle (MG) was determined in vivo by ultrasonography during isometric contractions at three (distal, central, and proximal) locations (n = 7) and at three ankle angles (n = 7). The curvature significantly (P < 0.05) increased from rest to maximum voluntary contraction (MVC) (0.4-5.2 m(-1)). In addition, the curvature at MVC became larger in the ...

An approximate numerical^graphical solution was made for the motion of a human aortic valve leaflet in a meridional plane through the center of the sinus of Vabalva. The observed buckling, whipping action of the leaflet was obtained. Of primary significance in the results was the location and the determination of the maximum curvature of the leaflet in this plane, since this point corresponds t...

We present a new method for regularization of ill-conditioned problems, such as those that arise in image restoration or mathematical processing of medical data. The method extends the traditional trust-region subproblem, TRS, approach that makes use of the L-curve maximum curvature criterion, a strategy recently proposed to find a good regularization parameter. We apply a parameterized trust r...

An infrared-imaging system has been used to study the influence of gravity on the kinetics of first positive phototropism. The development of phototropic curvature of etiolated seedlings of Arabidopsis thaliana was measured in the absence of visible radiation. Following a pulse of blue light, stationary seedlings curved to a maximum of approximately 16 degrees about 80 minutes after stimulati...