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A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and ...
The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere. In this example, the curvature of a 2D-surface is mapped onto a 1D unit circle. In differential geometry , the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the ...
Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature ...
A curvature vector is a normal vector whose length is the curvature of the object. Multiplying a normal vector by −1 results in the opposite vector, which may be used for indicating sides (e.g., interior or exterior). In three-dimensional space, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the ...
The total curvature of a closed curve is always an integer multiple of 2 π, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point.
At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the asymptotic curves for that point.
The local (non-unit) basis vector is b 1 (notated h 1 above, with b reserved for unit vectors) and it is built on the q 1 axis which is a tangent to that coordinate line at the point P. The axis q 1 and thus the vector b 1 form an angle α {\displaystyle \alpha } with the Cartesian x axis and the Cartesian basis vector e 1 .
The Berry curvature per solid angle is given by ¯ = / = /. In this case, the Berry phase corresponding to any given path on the unit sphere S 2 {\displaystyle {\mathcal {S}}^{2}} in magnetic-field space is just half the solid angle subtended by the path.