Calculus How To

Pedal Curve

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A pedal curve is a new curve derived from a given curve and a point.

Positive Pedal Curve

pedal curve

Construction of a Limaçon as a pedal curve of a circle [1].

General steps to creating a positive pedal curve [2]:

  1. Choose a curve, with fixed point O (called the pedal origin), from which you want to create a pedal curve. For the above example, the choice is a circle centered at the origin.
  2. Draw a tangent line at any point P on the curve. In the above image, point P is the black dot on the circumference (outer edge) of the circle; the tangent line is shown in blue.
  3. Mark point Q on the tangent line (shown as a red dot in the above image) so that OQ and PQ are perpendicular (i.e., at right angles). If you look closely at the above image, you’ll see that the tangent line (blue) and perpendicular (green) are always at right angles with respect to each other.
  4. Repeat steps 2 and 3 for more points P on the curve; Technically you could repeat these steps an infinite number of times (as in the image). However, if drawing by hand you’ll want to stop when you see the general shape of the new curve.

The locus* of point Q (on the tangent line) is called the pedal of the given curve with respect to point O. The curve that results from this procedure is sometimes called a primitive curve.

*A locus (plural loci) is the set of points with a location that satisfies or is determined by specified conditions. For example, the locus of the above image is the set of points Q that satisfy the conditions laid out in the above steps.

Negative Pedal Curve

negative pedal curve

A negative pedal curve of a Limaçonresults in a circle [1].

A negative pedal curve is the inverse of the pedal curve. It can be constructed as follows [2]:

  1. Choose a curve, with fixed point O, from which you want to create a pedal curve. The above image shows a Limaçon.
  2. Draw a line from O to any point P on the curve. In this example, point O is the black point at the top of the Limaçon. Point P is the red dot; the line OP is shown in green.
  3. Draw a line perpendicular to OP, passing point P. The perpendicular is shown in blue in the image above.
  4. Repeat Steps 2 and 3 ad infinitum.

As you can tell from the image, the result is the circle. In other words, the negative pedal curve inverses (reverses) the pedal curve procedure and vice versa,


[1] Sam Derbyshire at the English Wikipedia, CC BY-SA 3.0 , via Wikimedia Commons
[2] Pedal Curve. Retrieved March 6, 2021 from:

Stephanie Glen. "Pedal Curve" From Calculus for the rest of us!

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