Tendency of a Curve

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The tendency of a curve is a discrete parameter with possible values , and is determined according to the tendency indicator vector of the curve, , where

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and

are the one-sided detachments (if the limits exist) of the functions , that define the curve.

In this Demonstration, all possible cases for the tendency indicator vector are depicted and the tendency is shown once the vector is selected. You can choose to set the tendency indicator vector either in a discrete or a continuous manner. The calculation of each of the parameters refers to the highlighted point (the bold pink point in the middle of the figure). The classification of the point according to the tendency indicator vector (slanted corner, perpendicular edge, etc.) is also updated.

If the "show tendency" checkbox is selected, in addition to the notation of the tendency in the upper part of the graph, the geometric interpretation of the tendency is depicted: dashed corners appear on the left-hand side of the curve at the point, and the sum of the numbers inside the corners is the tendency at the point.

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Contributed by: Amir Finkelstein (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Thumbnail above caption: tendency at perpendicular corners agrees with the definition of the parameter from the discrete Green's theorem

Snapshot 1: tendency at acute corners whose angle is less than can be zero: in this case, the geometric interpretation shows why: no dashed corners can be contained in the left-hand side of the curve at the pink point, hence the sum of the matching numbers is vacuously zero, and so is the tendency at the point

Snapshot 2: tendency is defined only in cases where the curve is defined for left and right neighborhoods of the given point

Snapshot 3: tendency at slanted corners is independent of the orientation of the curve, hence the bidirectional arrow; in this case, we see that the sum of the numbers in the dashed corners is , and so 1 is the tendency; the endpoints were chosen in "continuous" mode, which enabled the shorter edge (although the edges' length is not a critical parameter in this Demonstration)

The discrete Green's theorem points out the connection between the double integral of a function over a domain that is the union of rectangles and a linear combination of the values of the given function's cumulative distribution function at the corners of the domain. The coefficients in this linear combination are uniquely determined according to the corner type, which is denoted by in the formulation of the theorem. The definition of tendency agrees with the definition of , gives a rigorous interpretation to it via the tendency indicator vector at a point, and extends the definition of to any kind of point (not only perpendicular corners as in the formulation of the discrete Green's theorem). The definition of tendency hence enables the discussion regarding an extended version of the discrete Green's theorem, which holds for more general types of domains.

For a more detailed discussion, please refer to the preprint.



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