In
geometry, a negative pedal curve is a
plane curve that can be constructed from another plane curve ''C'' and a fixed point ''P'' on that curve. For each point ''X'' ≠ ''P'' on the curve ''C'', the negative pedal curve has a
tangent that passes through ''X'' and is
perpendicular to line ''XP''. Constructing the negative pedal curve is the
inverse operation
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\ ...
to constructing a
pedal curve.
Definition
In the plane, for every point ''X'' other than ''P'' there is a unique line through ''X'' perpendicular to ''XP''. For a given curve in the plane and a given fixed point ''P'', called the pedal point, the negative pedal curve is the
envelope of the lines ''XP'' for which ''X'' lies on the given curve.
Parameterization
For a parametrically defined curve, its negative pedal curve with pedal point (0; 0) is defined as
:
:
Properties
The negative pedal curve of a
pedal curve with the same pedal point is the original curve.
See also
*
Fish curve
A fish curve is an ellipse negative pedal curve that is shaped like a fish. In a fish curve, the pedal point is at the focus for the special case of the squared eccentricity e^2=\tfrac. The parametric equations for a fish curve correspond to t ...
, the negative pedal curve of an ellipse with squared eccentricity 1/2
External links
Negative pedal curve on Mathworld
{{DEFAULTSORT:Negative Pedal Curve
Curves
Differential geometry