Pedal curve explorer
Given a curve and a fixed point, the pedal curve is the locus of the feet of perpendiculars dropped from that point onto every tangent line of the curve. As the tangent sweeps around the base curve, each perpendicular foot traces a new curve whose shape depends on where you place the pedal point. Drag the gold point to see the pedal transform in real time.
Pedal point P, tangent at α(t): foot F = P + [(⟨α − P, T⟩) · T] | T = unit tangent
What is a pedal curve?
The pedal curve of a given curve C with respect to a point P is constructed as follows: for each point on C, draw the tangent line; then drop a perpendicular from P to that tangent line. The foot of the perpendicular is a point on the pedal curve. As the tangent sweeps around C, the foot traces the pedal. The pedal transformation is a fundamental operation in classical differential geometry, and it reveals deep connections between seemingly unrelated curves.
Famous pedal relationships
The pedal of a circle with respect to a point on the circle is a cardioid. The pedal of an ellipse with respect to a focus is a circle — this is connected to the reflection property of ellipses and the equal-area law in orbital mechanics. The pedal of a parabola from its focus is a straight line (the tangent at the vertex). These relationships are not coincidence: they encode the geometric optics and reflection properties of the conic sections.
Duality and the inverse pedal
Every pedal transformation has an inverse. Given a pedal curve, you can reconstruct the original curve by drawing tangent lines to the pedal at each point and taking the envelope. This duality means that many classical curves come in pedal pairs. The astroid, for instance, is the pedal of a certain deltoid. Moving the pedal point continuously deforms the pedal curve, creating a one-parameter family of curves. When the pedal point is at a special position — such as the center, a focus, or a point on the curve — the pedal often degenerates into a simpler or more symmetric shape.
Applications
Pedal curves appear in cam design (where a follower traces a pedal of the cam profile), in geometric optics (the caustic of a reflector is related to the pedal of the mirror curve), and in kinematics (the path traced by a point on a rolling wheel is related to pedal curves of the wheel's profile). The pedal with respect to the origin of a curve given in polar form r = f(θ) has an especially clean formula: the pedal's equation is p = r · sin(ψ), where ψ is the angle between the radius vector and the tangent.