Snapshot 1: Since the option "skip the point
" was chosen and the curve
is uniformly tended, then
is evaluated via the double integral over the domain
on the left of the curve
. The colors of the edges show their tendencies for uniformly tended edges (green for positively tended, red for negatively tended, and black for zero tended). The only role of the blue edges
is to determine the pointwise tendencies at the points
, respectively; these edges' tendencies do not affect the slanted line integral over
. The vertices
are hollow since the cumulative distribution function
is not evaluated there; the vertex
is dark green since the coefficient of
, and the disks of the vertices
are light red to denote that the coefficient of
at those points is
Snapshot 2: The change with respect to the first snapshot is that the "skip the point
" option is unchecked, which lets us illustrate the additivity of the slanted line integral operator; the discrete Green's theorem can be used to prove this.
Snapshot 3: When compared to the first snapshot, this one illustrates another property of the slanted line integral, which holds for any uniformly tended curve whose tendency is never zero:
; again the discrete Green's theorem can be used to prove this.
Snapshot 4: Here
merge, so they cancel:
; hence the vertex is black, which implies that the coefficient of
at the point is zero. The curvature of the lines
only affects the domains
at the definition of the slanted line integral, as long as their tendency is unchanged.
Snapshot 5: The intersection of the domains
is colored in dark yellow; in both the third and the fourth snapshots, although the "skip the point
" option was selected, both integrals
are depicted because the curve
is not uniformly tended; hence in order to calculate the slanted line integral over
, the definition insists on going through the point where the curve ceases to be uniformly tended (the tendency indicator vector changes), that is, the point
Snapshot 6: Here we see in what sense this integration method is a natural extension to the one in the discrete Green's theorem: the slanted line integral over a perpendicular corner agrees with the parameter
from that theorem
. In this case, the slanted line integral is
, and indeed the parameter
for such corners is -1
The name "slanted" is due to the fact that this integration method is affected by the rotation angle of the coordinate system, since the domain
and the calculation of the cumulative distribution function both depend on that angle. Hence, it is assumed that the coordinate system is static and given in advance, which is a reasonable assumption in computer applications. In this Demonstration the axes are parallel to the edges of the figure. The definition of the slanted line integral might seem awkward at first glance, but this integration method has interesting properties that resemble those of the line integral and it enables a broader and a more rigorous discussion of the integration method used in the discrete Green's theorem: it extends the integration method found in the discrete Green's theorem to more general types of domains, those that are not necessarily formed by a finite union of rectangles. In turn, this semidiscrete integration method results in a theorem that extends the discrete Green's theorem to more general domains.
More details regarding the definition of tendency are also available in a previously uploaded Demonstration
The theory of semidiscrete calculus is given in .
 A. Finkelstein. "The Theory behind the 'Summed Area Tables' Algorithm: A Simple Approach to Calculus." (May 25, 2010) www.arxiv.org/abs/1005.1418