WOLFRAM|DEMONSTRATIONS PROJECT

Tendency of a Curve

​
edges choice method
discrete
discrete
right detachment of x
1
-1
0
left detachment of x
1
-1
0
right detachment of y
1
-1
0
left detachment of y
1
-1
0
show tendency
continuous
first endpoint
second endpoint
The tendency of a curve is a discrete parameter with possible values
{+1,0,-1}
, and is determined according to the tendency indicator vector of the curve,
t

;
x
+
(
t
0
),
;
x
-
(
t
0
),
;
y
+
(
t
0
),
;
y
-
(
t
0
)
, where
;
x
±
(
t
0
)≡
lim
h0±
sgn[x(
t
0
+h)-x(
t
0
)]
and
;
y
±
(
t
0
)≡
lim
h0±
sgn[y(
t
0
+h)-y(
t
0
)]
are the one-sided detachments (if the limits exist) of the functions
x
,
y
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.