From the Cartesian equation Pedal curve
pedal curve (red) of ellipse (black). here a=2 , b=1 equation of pedal curve 4x+y=(x+y)
for example, ellipse
x
2
a
2
+
y
2
b
2
=
1
{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}
the tangent line @ r=(x0, y0) is
x
0
x
a
2
+
y
0
y
b
2
=
1
{\displaystyle {\frac {x_{0}x}{a^{2}}}+{\frac {y_{0}y}{b^{2}}}=1}
and writing in form given above requires that
x
0
a
2
=
cos
α
p
,
y
0
b
2
=
sin
α
p
.
{\displaystyle {\frac {x_{0}}{a^{2}}}={\frac {\cos \alpha }{p}},\,{\frac {y_{0}}{b^{2}}}={\frac {\sin \alpha }{p}}.}
the equation ellipse can used eliminate x0 , y0 giving
a
2
cos
2
α
+
b
2
sin
2
α
=
p
2
,
{\displaystyle a^{2}\cos ^{2}\alpha +b^{2}\sin ^{2}\alpha =p^{2},\,}
and converting (r, θ) gives
a
2
cos
2
θ
+
b
2
sin
2
θ
=
r
2
,
{\displaystyle a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta =r^{2},\,}
as polar equation pedal. converted cartesian equation as
a
2
x
2
+
b
2
y
2
=
(
x
2
+
y
2
)
2
.
{\displaystyle a^{2}x^{2}+b^{2}y^{2}=(x^{2}+y^{2})^{2}.\,}
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