Question

For the pedal equation of a curve p​

Answer 1

Answer:

For a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point.

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Epi- and hypocycloids.

n Curve Pedal eq.

−4, −4⁄3 Astroid p 2 = − 1 3 ( r 2 − a 2 )

Step-by-step explanation:

Hope this helps

Answer 2

Answer:

To measure the distance of O to the normal p_{c} (the contrapedal coordinate) even though it is not an independent quantity and it relates to $(r,p) as {\displaystyle p_{c}:={\sqrt {r^{2}-p^{2.$

Step-by-step explanation:

For a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point.

Refer fig.

1. Rectangular cordinates.

For C given in rectangular coordinates by$f(x, y)=0$, and with $O$ taken to be the origin, the pedal coordinates of the point $(x, y)$ are given by:

$r=\sqrt{x^{2}+y^{2}}\\p=\frac{x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}}{\sqrt{\left(\frac{\partial f}{\partial x}\right)^{2}+\left(\frac{\partial f}{\partial y}\right)^{2}}}$

The pedal equation can be found by eliminating $x and y$ from these equations and the equation of the curve.

The expression for $p$ may be simplified if the equation of the curve is written in homogeneous coordinates by introducing a variable $z$, so that the equation of the curve is $g(x, y, z)=0$. The value of $p$ is then given by

$p=\frac{\frac{\partial g}{\partial z}}{\sqrt{\left(\frac{\partial g}{\partial x}\right)^{2}+\left(\frac{\partial g}{\partial y}\right)^{2}}}$

z=1

2. Polar coordinates

For C given in polar coordinates by  $r=f(\theta)$, then

$p=r \sin \phi$

where $\phi$ is the polar tangential angle given by

$r=\frac{d r}{d \theta} \tan \phi$

The pedal equation can be found by eliminating $\theta$ from these equations.

Alternatively, from the above we can find that

$\left|\frac{d r}{d \theta}\right|=\frac{r p_{c}}{p}$

where $p_{c}:=\sqrt{r^{2}-p^{2}}$ is the "contrapedal" coordinate, i.e. distance to the normal. This implies that if a curve satisfies an autonomous differential equation in polar coordinates of the form:

$f\left(r,\left|\frac{d r}{d \theta}\right|\right)=0$

equation becomes

$f\left(r, \frac{r p_{c}}{p}\right)=0$