Preset
Circle
a:b = 1:1, δ = 90°
Figure Eight
a:b = 1:2, δ = 90°
Three Loops
a:b = 1:3, δ = 90°
Classic Knot
a:b = 3:4, δ = 45°
Phase Sweep
a:b = 3:2, δ rotating
X frequency ratio a
1
Y frequency ratio b
1
Phase δ (°)
90
Speed (Hz)
0.50
↻ Animate δ
◻ Ghost
⏸ Pause
↺ Reset
Harmonically related oscillations
trace closed curves — the shape
encodes the ratio a:b and the
phase offset δ.
a:b =
1:1
δ =
90°
T =
2.000
s
t =
0.00
s
x = sin(aωt + δ) y = sin(bωt)
Sliders
— tune frequency ratios & phase
Animate δ
— watch the figure morph
Space
— pause / play
G
— ghost