Misplaced Pages

Negative pedal curve

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Mathematical plane curve
This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Negative pedal curve" – news · newspapers · books · scholar · JSTOR (June 2012) (Learn how and when to remove this message)
Circle — negative pedal curve of a limaçon

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 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

X [ x , y ] = ( y 2 − x 2 ) y ′ + 2 x y x ′ x y ′ − y x ′ {\displaystyle X={\frac {(y^{2}-x^{2})y'+2xyx'}{xy'-yx'}}}
Y [ x , y ] = ( x 2 − y 2 ) x ′ + 2 x y y ′ x y ′ − y x ′ {\displaystyle Y={\frac {(x^{2}-y^{2})x'+2xyy'}{xy'-yx'}}}

Properties

The negative pedal curve of a pedal curve with the same pedal point is the original curve.

See also

  • Fish curve, the negative pedal curve of an ellipse with squared eccentricity 1/2

External links

Differential transforms of plane curves
Unary operations
Unary operations defined by a point
Unary operations defined by two points
Binary operations defined by a point
Operations on a family of curves
Categories:
Negative pedal curve Add topic