Equations Pedal curve

1 equations

1.1 cartesian equation
1.2 polar equation
1.3 pedal equation
1.4 parametric equations

equations
from cartesian equation

take p origin. curve given equation f(x, y)=0, if equation of tangent line @ r=(x0, y0) written in form

cos
⁡
α
x
+
sin
⁡
α
y
=
p


{\displaystyle \cos \alpha x+\sin \alpha y=p}

then vector (cos α, sin α) parallel segment px, , length of px, distance tangent line origin, p. x represented polar coordinates (p, α) , replacing (p, α) (r, θ) produces polar 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}.\,}




from polar equation

for p origin , c given in polar coordinates r = f(θ). let r=(r, θ) point on curve , let x=(p, α) corresponding point on pedal curve. let ψ denote angle between tangent line , radius vector, known polar tangential angle. given by

r
=



d
r


d
θ



tan
⁡
ψ
.


{\displaystyle r={\frac {dr}{d\theta }}\tan \psi .}

then

p
=
r
sin
⁡
ψ


{\displaystyle p=r\sin \psi }

and

α
=
θ
+
ψ
−


π
2


.


{\displaystyle \alpha =\theta +\psi -{\frac {\pi }{2}}.}

these equations may used produce equation in p , α which, when translated r , θ gives polar equation pedal curve.

for example, let curve circle given r = cos θ. then

a
cos
⁡
θ
=
−
a
sin
⁡
θ
tan
⁡
ψ


{\displaystyle a\cos \theta =-a\sin \theta \tan \psi }

so

tan
⁡
ψ
=
−
cot
⁡
θ
,

ψ
=


π
2


+
θ
,
α
=
2
θ
.


{\displaystyle \tan \psi =-\cot \theta ,\,\psi ={\frac {\pi }{2}}+\theta ,\alpha =2\theta .}

also

p
=
r
sin
⁡
ψ
 
=
r
cos
⁡
θ
=
a

cos

2


⁡
θ
=
a

cos

2


⁡


α
2


.


{\displaystyle p=r\sin \psi \ =r\cos \theta =a\cos ^{2}\theta =a\cos ^{2}{\alpha \over 2}.}

so polar equation of pedal is

r
=
a

cos

2


⁡


θ
2


.


{\displaystyle r=a\cos ^{2}{\theta \over 2}.}



from pedal equation

the pedal equations of curve , pedal closely related. if p taken pedal point , origin can shown angle ψ between curve , radius vector @ point r equal corresponding angle pedal curve @ point x. if p length of perpendicular drawn p tangent of curve (i.e. px) , q length of corresponding perpendicular drawn p tangent pedal, similar triangles

p
r


=


q
p


.


{\displaystyle {\frac {p}{r}}={\frac {q}{p}}.}

it follows if pedal equation of curve f(p,r)=0 pedal equation pedal curve is

f
(
r
,



r

2


p


)
=
0


{\displaystyle f(r,{\frac {r^{2}}{p}})=0}

from positive , negative pedals can computed if pedal equation of curve known.

from parametric equations

contrapedal of same ellipse



pedal of evolute of ellipse : same contrapedal of original ellipse

let






v
→



=
p
−
r


{\displaystyle {\vec {v}}=p-r}

vector r p , write

v
→



=




v
→




∥


+




v
→




⊥




{\displaystyle {\vec {v}}={\vec {v}}_{\parallel }+{\vec {v}}_{\perp }}

,

the tangential , normal components of






v
→





{\displaystyle {\vec {v}}}

respect curve.







v
→




∥




{\displaystyle {\vec {v}}_{\parallel }}

vector r x position of x can computed.

specifically, if c parametrization of curve then

t
↦
c
(
t
)
+




c
′

(
t
)
⋅
(
p
−
c
(
t
)
)



|


c
′

(
t
)


|


2






c
′

(
t
)


{\displaystyle t\mapsto c(t)+{c (t)\cdot (p-c(t)) \over |c (t)|^{2}}c (t)}

parametrises pedal curve (disregarding points c 0 or undefined).

for parametrically defined curve, pedal curve pedal point (0;0) defined as

x
[
x
,
y
]
=



(
x

y
′

−
y

x
′

)

y
′




x

′

2



+

y

′

2








{\displaystyle x[x,y]={\frac {(xy -yx )y }{x ^{2}+y ^{2}}}}








y
[
x
,
y
]
=



(
y

x
′

−
x

y
′

)

x
′




x

′

2



+

y

′

2






.


{\displaystyle y[x,y]={\frac {(yx -xy )x }{x ^{2}+y ^{2}}}.}

the contrapedal curve given by:

t
↦
p
−




c
′

(
t
)
⋅
(
p
−
c
(
t
)
)



|


c
′

(
t
)


|


2






c
′

(
t
)


{\displaystyle t\mapsto p-{c (t)\cdot (p-c(t)) \over |c (t)|^{2}}c (t)}

with same pedal point, contrapedal curve pedal curve of evolute of given curve.