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The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions. The area between a positive-valued curve and the horizontal axis, measured between two values a and b (b is defined as the larger of the two values) on the horizontal axis, is given by the integral from a to b of the function ...
The blue area above the x-axis may be specified as positive area, while the yellow area below the x-axis is the negative area. The integral of a real function can be imagined as the signed area between the x {\displaystyle x} -axis and the curve y = f ( x ) {\displaystyle y=f(x)} over an interval [ a , b ].
The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is (,) =, that is, ‖ ‖ = (,) = {(): =, =} = [], where the expectation is taken with respect to the probability measure on the space where (,) lives, and the infimum is taken over all such with marginals and , respectively.
The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.
An important tool for calculating the Fréchet distance of two curves is the free-space diagram, which was introduced by Alt and Godau. [4] The free-space diagram between two curves for a given distance threshold ε is a two-dimensional region in the parameter space that consists of all point pairs on the two curves at distance at most ε:
Two curves in the plane intersecting at a point p are said to have: 0th-order contact if the curves have a simple crossing (not tangent). 1st-order contact if the two curves are tangent. 2nd-order contact if the curvatures of the curves are equal. Such curves are said to be osculating. 3rd-order contact if the derivatives of the curvature are ...
Proposition 2: The area of circles is proportional to the square of their diameters. [3] Proposition 5: The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases. [4] Proposition 10: The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. [5]
Two ski tracks with different degrees of sinuosity on the same slope Sinuosity , sinuosity index , or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance ( straight line ) between the end points of the curve.