In mathematics, apedal curve of a given curve results from theorthogonal projection of a fixed point on thetangent lines of this curve. More precisely, for aplane curveC and a given fixedpedal pointP, thepedal curve ofC is thelocus of pointsX so that thelinePX is perpendicular to atangentT to the curve passing through the pointX. Conversely, at any pointR on the curveC, letT be the tangent line at that pointR; then there is a unique pointX on the tangentT which forms with the pedal pointP a lineperpendicular to the tangentT (for the special case when the fixed pointP lies on the tangentT, the pointsX andP coincide) – the pedal curve is the set of such pointsX, called thefoot of the perpendicular to the tangentT from the fixed pointP, as the variable pointR ranges over the curveC.

Complementing the pedal curve, there is a unique pointY on the line normal toC atR so thatPY is perpendicular to the normal, soPXRY is a (possibly degenerate) rectangle. The locus of pointsY is called thecontrapedal curve.

Theorthotomic of a curve is its pedal magnified by a factor of 2 so that thecenter of similarity isP. This is locus of the reflection ofP through the tangent lineT.

The pedal curve is the first in a series of curvesC₁,C₂,C₃, etc., whereC₁ is the pedal ofC,C₂ is the pedal ofC₁, and so on. In this scheme,C₁ is known as thefirst positive pedal ofC,C₂ is thesecond positive pedal ofC, and so on. Going the other direction,C is thefirst negative pedal ofC₁, thesecond negative pedal ofC₂, etc.^[1]

TakeP to be the origin. For a curve given by the equationF(x,y)=0, if the equation of thetangent line atR=(x₀,y₀) is written in the form

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

then the vector (cos α, sin α) is parallel to the segmentPX, and the length ofPX, which is the distance from the tangent line to the origin, isp. SoX is represented by thepolar coordinates (p, α) and replacing (p, α) by (r, θ) produces a polar equation for the pedal curve.^[2]

For example,^[3] for the ellipse

${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$

the tangent line atR=(x₀,y₀) is

${\displaystyle \frac{x_{0}x}{a^2}+\frac{y_{0}y}{b^2}=1}$

and writing this in the form given above requires that

${\displaystyle \frac{x_0}{a^2}=\frac{\cos \alpha }{p},\frac{y_0}{b^2}=\frac{\sin \alpha }{p}.}$

The equation for the ellipse can be used to eliminatex₀ andy₀ giving

${\displaystyle a^2{\cos }^2\alpha +b^2{\sin }^2\alpha =p^2,}$

and converting to (r, θ) gives

${\displaystyle a^2{\cos }^2\theta +b^2{\sin }^2\theta =r^2,}$

as the polar equation for the pedal. This is easily converted to a Cartesian equation as

${\displaystyle a^2{x}^2+b^2{y}^2=(x^2+y^2{)}^2.}$

ForP the origin andC given inpolar coordinates byr = f(θ). LetR=(r, θ) be a point on the curve and letX=(p, α) be the corresponding point on the pedal curve. Let ψ denote the angle between the tangent line and the radius vector, sometimes known as thepolar tangential angle. It is given by

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

Then

${\displaystyle p=r\sin \psi }$

and

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

These equations may be used to produce an equation inp and α which, when translated tor and θ gives a polar equation for the pedal curve.^[4]

For example,^[5] let the curve be the circle given byr =a cos θ. Then

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

so

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

Also

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

So the polar equation of the pedal is

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

Thepedal equations of a curve and its pedal are closely related. IfP is taken as the pedal point and the origin then it can be shown that the angle ψ between the curve and the radius vector at a pointR is equal to the corresponding angle for the pedal curve at the pointX. Ifp is the length of the perpendicular drawn fromP to the tangent of the curve (i.e.PX) andq is the length of the corresponding perpendicular drawn fromP to the tangent to the pedal, then by similar triangles

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

It follows immediately that the if the pedal equation of the curve isf(p,r)=0 then the pedal equation for the pedal curve is^[6]

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

From this all the positive and negative pedals can be computed easily if the pedal equation of the curve is known.

Let${\displaystyle \overrightarrow{v}=P-R}$be the vector forR toP and write

${\displaystyle \overrightarrow{v}={\overrightarrow{v}}_{\parallel }+{\overrightarrow{v}}_{\perp }}$,

thetangential and normal components of${\displaystyle \overrightarrow{v}}$ with respect to the curve.Then${\displaystyle {\overrightarrow{v}}_{\parallel }}$ is the vector fromR toX from which the position ofX can be computed.

Specifically, ifc is aparametrization of the curve then

${\displaystyle t\mapsto c(t)+\frac{c^{\prime }(t)\cdot (P-c(t))}{|c^{\prime }(t){|}^2}{c}^{\prime }(t)}$

parametrises the pedal curve (disregarding points wherec'is zero or undefined).

For a parametrically defined curve, its pedal curve with pedal point (0;0) is defined as

${\displaystyle X[x,y]=\frac{(x{y}^{\prime }-y{x}^{\prime })y^{\prime }}{x^{\prime 2}+y^{\prime 2}}}$

${\displaystyle Y[x,y]=\frac{(y{x}^{\prime }-x{y}^{\prime })x^{\prime }}{x^{\prime 2}+y^{\prime 2}}.}$

The contrapedal curve is given by:

${\displaystyle t\mapsto P-\frac{c^{\prime }(t)\cdot (P-c(t))}{|c^{\prime }(t){|}^2}{c}^{\prime }(t)}$

With the same pedal point, the contrapedal curve is the pedal curve of theevolute of the given curve.

Consider a right angle moving rigidly so that one leg remains on the pointP and the other leg is tangent to the curve. Then the vertex of this angle isX and traces out the pedal curve. As the angle moves, its direction of motion atP is parallel toPX and its direction of motion atR is parallel to the tangentT =RX. Therefore, theinstant center of rotation is the intersection of the line perpendicular toPX atP and perpendicular toRX atR, and this point isY. It follows that the tangent to the pedal atX is perpendicular toXY.

Draw a circle with diameterPR, then it circumscribes rectanglePXRY andXY is another diameter. The circle and the pedal are both perpendicular toXY so they are tangent atX. Hence the pedal is theenvelope of the circles with diametersPR whereR lies on the curve.

The lineYR is normal to the curve and the envelope of such normals is itsevolute. Therefore,YR is tangent to the evolute and the pointY is the foot of the perpendicular fromP to this tangent, in other wordsY is on the pedal of the evolute. It follows that the contrapedal of a curve is the pedal of its evolute.

LetC′ be the curve obtained by shrinkingC by a factor of 2 towardP. Then the pointR′ corresponding toR is the center of the rectanglePXRY, and the tangent toC′ atR′ bisects this rectangle parallel toPY andXR. A ray of light starting fromP and reflected byC′ atR'will then pass throughY. The reflected ray, when extended, is the lineXY which is perpendicular to the pedal ofC. The envelope of lines perpendicular to the pedal is then the envelope of reflected rays or thecatacaustic ofC′. This proves that the catacaustic of a curve is the evolute of its orthotomic.

As noted earlier, the circle with diameterPR is tangent to the pedal. The center of this circle isR′ which follows the curveC′.

LetD′ be a curve congruent toC′ and letD′ roll without slipping, as in the definition of aroulette, onC′ so thatD′ is always the reflection ofC′ with respect to the line to which they are mutually tangent. Then when the curves touch atR′ the point corresponding toP on the moving plane isX, and so the roulette is the pedal curve. Equivalently, the orthotomic of a curve is the roulette of the curve on its mirror image.

WhenC is a circle the above discussion shows that the following definitions of alimaçon are equivalent:

• It is the pedal of a circle.
• It is the envelope of circles whose diameters have one endpoint on a fixed point and another endpoint which follow a circle.
• It is the envelope of circles through a fixed point whose centers follow a circle.
• It is theroulette formed by a circle rolling around a circle with the same radius.

We also have shown that the catacaustic of a circle is the evolute of a limaçon.

Pedals of some specific curves are:^[7]

Curve	Equation	Pedal point	Pedal curve
Circle		Point on circumference	Cardioid
Circle		Any point	Limaçon
Parabola		Focus	The tangent line at the vertex
Parabola		Vertex	Cissoid of Diocles
Deltoid		Center	Trifolium
Central conic		Focus	Auxiliary circle
Central conic	${\displaystyle \frac{x^2}{a^2}\pm \frac{y^2}{b^2}=1}$	Center	${\displaystyle a^2{\cos }^2\theta \pm b^2{\sin }^2\theta =r^2}$ (ahippopede)
Rectangular hyperbola		Center	Lemniscate of Bernoulli
Logarithmic spiral		Pole	Logarithmic spiral
Sinusoidal spiral	${\displaystyle r^n=a^n\cos n\theta }$	Pole	${\displaystyle r^{\frac{n}{n+1}}=a^{\frac{n}{n+1}}\cos \frac{n}{n+1}\theta }$ (another Sinusoidal spiral)

• List of curves

Notes

Sources

Wikimedia Commons has media related toPedal curves.

• Weisstein, Eric W."Pedal Curve".MathWorld.
• Weisstein, Eric W."Contrapedal Curve".MathWorld.
• Weisstein, Eric W."Orthotomic".MathWorld.

• Differential geometry
• Curves

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