Astroid
Rolling curves (roulettes)The four-cusped star x^{2/3} + y^{2/3} = a^{2/3}, also the envelope of a sliding ladder.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Astroid?
The astroid is the four-cusped hypocycloid: roll a circle inside a circle four times its radius and a rim point draws the star x^{2/3} + y^{2/3} = a^{2/3}. The exponent 2/3 makes it the Lamé curve (superellipse) with n = 2/3 — the concave extreme of that family.
It has a second, entirely different construction: slide a ladder of fixed length down a wall, and the region the ladder can never enter is bounded by an astroid. In other words, the astroid is the envelope of all segments of length a with endpoints on the two axes — a staple of envelope and related-rates problems.
Key properties
- Perimeter 6a and area 3πa²/8 — both elementary despite the cusps.
- It is the hypocycloid with k = 4 (R = 4r).
- Envelope of a segment of length a sliding with ends on the axes.
- Every tangent line cuts the axes in a segment of constant length a.
- It is the evolute of an ellipse, stretched by the axis ratio.
Where you'll see it
The sliding-ladder problem, caustics of some reflected light patterns, and stress-line visualizations in mechanics.