Math
Envelopes
Compare a sliding ladder with the family of tangents to a parabola. The contact point traces an astroid in the first mode; focus and directrix distances agree in the second.
What happens
Drag A or B along its rail. The brass bar keeps its length; the red contact point moves along it. Move D along the directrix. The brass line bisects FD at a right angle. At E, the two gold segments have equal lengths.
Try this
Drag a blue handle in either view, or select a point on the green curve in the diagram. Right-drag rotates the scene; the wheel zooms. Sweep records a full pass; Pause freezes it.
Equations used in the module
Equations x = Lcos³φ; y = Lsin³φ
