Editing Pedal curve (section)

==Geometrical properties==
Consider a right angle moving rigidly so that one leg remains on the point ''P'' and the other leg is tangent to the curve. Then the vertex of this angle is ''X'' and traces out the pedal curve. As the angle moves, its direction of motion at ''P'' is parallel to ''PX'' and its direction of motion at ''R'' is parallel to the tangent ''T'' = ''RX''. Therefore, the [[instant center of rotation]] is the intersection of the line perpendicular to ''PX'' at ''P'' and perpendicular to ''RX'' at ''R'', and this point is ''Y''. It follows that the tangent to the pedal at ''X'' is perpendicular to ''XY''.

Draw a circle with diameter ''PR'', then it circumscribes rectangle ''PXRY'' and ''XY'' is another diameter. The circle and the pedal are both perpendicular to ''XY'' so they are tangent at ''X''. Hence the pedal is the [[envelope (mathematics)|envelope]] of the circles with diameters ''PR'' where ''R'' lies on the curve.

The line ''YR'' is normal to the curve and the envelope of such normals is its [[evolute]]. Therefore, ''YR'' is tangent to the evolute and the point ''Y'' is the foot of the perpendicular from ''P'' to this tangent, in other words ''Y'' is on the pedal of the evolute. It follows that the contrapedal of a curve is the pedal of its evolute.

Let ''C′'' be the curve obtained by shrinking ''C'' by a factor of 2 toward ''P''. Then the point ''R′'' corresponding to ''R'' is the center of the rectangle ''PXRY'', and the tangent to ''C′'' at ''R′'' bisects this rectangle parallel to ''PY'' and ''XR''. A ray of light starting from ''P'' and reflected by ''C′'' at ''R' ''will then pass through ''Y''. The reflected ray, when extended, is the line ''XY'' which is perpendicular to the pedal of ''C''. The envelope of lines perpendicular to the pedal is then the envelope of reflected rays or the [[catacaustic]] of ''C′''. This proves that the catacaustic of a curve is the evolute of its orthotomic.
<!-- Note, this exposition differs slightly from Greenhill's in that his construction is magnified by a factor of 2. -->

As noted earlier, the circle with diameter ''PR'' is tangent to the pedal. The center of this circle is ''R′'' which follows the curve ''C′''.

Let ''D′'' be a curve congruent to ''C′'' and let ''D′'' roll without slipping, as in the definition of a [[roulette (curve)|roulette]], on ''C′'' so that ''D′'' is always the reflection of ''C′'' with respect to the line to which they are mutually tangent. Then when the curves touch at ''R′'' the point corresponding to ''P'' on the moving plane is ''X'', and so the roulette is the pedal curve. Equivalently, the orthotomic of a curve is the roulette of the curve on its mirror image.

===Example===
[[Image:PedalCurve2.gif|500px|right|thumb|[[Limaçon]]&nbsp;— pedal curve of a [[circle]]]]When ''C'' is a circle the above discussion shows that the following definitions of a [[limaçon]] 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 the [[Roulette (curve)|roulette]] 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.
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