From parametric equations Pedal curve

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.