Pedal Curve - Geometrical Properties

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. If 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 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.

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′. It follows that the envelope of circles through a fixed point and whose centers lie on a given curve is the orthotomic of the curve.

Let D′ be a curve congruent to C′ and let D′ roll without slipping, as in the definition of a 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.