Parametric Curves Explorer
Plot Lissajous figures, epicycloids, astroids, butterfly curves, and hypotrochoids. Watch the curve trace as the parameter t sweeps from 0 to 2ฯ.
Parametric Curve
Properties
| Curve | -- |
| x(t) | -- |
| y(t) | -- |
| Period | -- |
| Symmetry | -- |
The Math Behind It
Parametric Equations
- A curve is defined by separate functions x(t) and y(t)
- The parameter t usually ranges over [0, 2ฯ]
- Each value of t gives a point (x, y) on the curve
- Lissajous: the ratio a : b determines the shape โ rational ratios close the curve
- Parametric form lets us describe curves that fail the vertical line test
Famous Curves
- Lissajous: x = sin(at), y = sin(bt) โ frequency ratio a:b determines pattern
- Epicycloid: circle of radius b rolling outside a circle of radius a โ produces cusps
- Astroid: special epicycloid with 4 cusps, x = aยทcosยณt
- Butterfly: r = ecos t โ 2cos(4t) + sin&sup5;(t/12) in polar form
- Hypotrochoid: circle rolling inside another โ the Spirograph principle
You're crushing it!