```

,Z8ZXSSX888888888880SXSSSSX888ZW:,,,
M,                                 ,,,SSX888XSS082XXX
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a0X8Xa                i                               :8
i80XS8X7        ,                                 Z
XSaXaS,                                 XZ
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iZa2aX              .2SZ
;          ;Z2Xar    r2XZi
.               rMB0M;
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i0SS0aX888X,                                           Ba
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```

Parametric Cartesian equation: x = asin(nt + c), y = bsin(t)
Can also be written in the form: x = acos(wxt - dx), y = acos(wyt - dy)

Lissajous curves or Lissajous figures are sometimes called Bowditch curves after Nathaniel Bowditch who studied them in 1815. They were studied in more detail (independently) by Jules-Antoine Lissajous in 1857.

Lissajous curves tend to look somewhat like the figures projected during laser light shows, and there is good reason for that. Lissajous would use sounds of different frequencies to vibrate a mirror. A beam of light reflected from the mirror would trace patterns which depended on the frequencies of the sounds. This is quite similar to the laser light shows of today, except Lissajous' setup used a simple beam of light instead of a powerful laser.

Before the days of digital frequency meters and phase-locked loops, Lissajous figures were used to determine the frequencies of sounds or radio signals. A signal of known frequency was applied to the horizontal axis of an oscilloscope, and the signal to be measured was applied to the vertical axis. The resulting Lissajous figure was a function of the ratio of the two frequencies. Compare the resulting figure to a figure from known frequencies, and you could then extrapolate the unknown frequency.

Lissajous figures often appeared as props in early science fiction movies. One of the most obvious examples can be found in the opening sequence of "The Outer Limits". The pattern of criss-cross lines is actually a Lissajous figure.

The curves can either loop back upon themselves, or repeat forever. They close upon themselves if and only if wx/wy is rational.

A few more examples of Lissajous curves...

```
,iii          :          .iii.        i                 :       77r
2i   X2        i         ai   r;       B                 i     ;W   W
7,      Z:      i       ;2      2       2                 i    ;Z     0
0        7i     i      Si        B       8                i    W       2
X.         7;    i     2,         2       0                i   M        8
0           rr   i    Z.           2      B                i  W          a
0            7i  i   X.            0      Z                i  @          2
a             S, i  ;r             0       Z               : M           Z
.i              2 . iX              0       0                iS            0
,.               a  X               a       8                M             8
;: ,,,,,,,,,,,,,  MS  ,,,,,,,,,,,,, 8    ., 0  ,,,,,,,,,,,. M  ,,,,,,,,,,, 8  ,
i.                BS                Z        Z              @               0
,,               a  a               Z        0             M                W
.;              8 , .2              8        X            Z,.               M
a             a. i  iX             0        ,r           M :               a
0            S:  i   ;:            0         B          M  i                M
0           Xi   i    2:           S         2         .a  i                W
r,         2:    i     Xi         a           M        M   i                W
0        S,     i      rX        0           Z.      M    i                Z
;i      0       i       .8      a             M     0i    i                 Z
Xr   aS        i         SX   Xi              M   M.     i                 B
.77X          ;           X7X                 2Sa       ;                 ;

7:    X7r       rar;       77r    ;i     ;         r7XX7:  :            7aa:
Xr: ,a   Zi   XZ .  Zr   72   0  2 0     i7      ZB.    .27r          2B.  7Z
,. a 2     i; Z   i   :7 a.     8 7 a      8    iM          aS       :M      W
i.  M       ,M    i     M        M  a       W  Xa           , X;    2S        M
i. .0       2ii   i    Si;       M  a        r;X            i  .8  @:         M
.: 8 8     X  S   i   i, 7:     8 X 8        7Z             i    S@           M
: S X     Z   B  i   Z   0    r: B 8        0X             i    M7           M
XX   B   B     8 i  B     8   B  i Z       ri X            i  :M  a.         W
SS   r: X      i.. 8      ;i 2    XS       M  X:           i .B    Z         M
ar    8 8       Z .;       S Z    SS      iX   W           ..W      0       iX
@      M         07         M     2:      M     M          S2        2      M
M .ii. M ,iiiii, MX :iiiii..M ,ii 8r      M iii. M .iiii, 2W ,iiiii:.iB ,i: M .
Zi    Z 2       7.:        7 X    X7     :7       M      Xr           ,r    S
SS   :r Z       ;  B      .7 8    X2     M        ;Z    Z7 :           7i  M
XS   B   B     B :  0     B   0  :.a     @         ;X  8;  i            Z  Z
i 7 7.    S   8  i  ,X   Z    a  0 8     M          ,i0,   i             ;M
.: 8 Z     2  ri  i   X  :7     0 X 8     M           MZ    i             0r
i. ;X       Z X   i    Z S      iX. a     M          M  0   i            :rS
i.  M        M    i     M        M  a     M        X0    8S ,           ;0  W
:. S.r     ,S Z   i    2 8      S i Z     .8      8r      ;0r          8S    2
;.r r;   r7   aX .  XS   ai   a: Z 8      W    aW          27S      Z0      2.
r:   ,ii;       i7:;      .;i;    ii       rrr7            .  riiii7         ;

```