2DPlot - parametric math curves in AutoCAD

1) Sinusoid

A trivial function to demonstrate a simple equation. The math definition used for this function:

X=u, Y=sin(u) -- in AutoLISP notation:

(defun fXYt1(u)(list u (sin u)))

2) Damped cosine

A combination of multiple curves. The math definition used for this function:

(defun fXYinterf (x / offset)
(defun dampcos (x / dist omag sfreq decfr)
   (setq
     omag  2.0  ; Overall magnitude scale factor
     sfreq 8.0  ; Spatial frequency factor
     decfr 1.5  ; Exponential decay spatial frequency
   )
   (setq dist (sqrt (+ (* x x))))
   (* omag
     (cos (* dist sfreq))
     (exp (- (* decfr dist)))
   )
);defun
  (setq offset 0.9) ; Offset of centres from origin
  (list x (+ (dampcos (- x offset)) (dampcos (+ x offset))))
)

3) Hypotrochoid 1

A roulette traced by a point attached to a smaller circle rolling around the inside of a fixed bigger circle - see Wikipedia. r1 is the smaller radius, r2 is the larger radius, d is the distance of the "pen", th is the current angle from horizontal. The math definition used for this curve:

(defun fXYhypotr1 (th / r1 r2)
 (setq r1 5.0  r2 3.0  d 5.0) ; 3 rotations
 (list
  (+ (* (- r1 r2)(cos th)) (* d (cos (* (/ (- r1 r2) r2) th))))
  (- (* (- r1 r2)(sin th)) (* d (sin (* (/ (- r1 r2) r2) th))))
 )
)

4) Hypotrochoid 2

Another example of the hypotrochoid curve, this time with the parameters of:

(setq r1 100.0  r2 2.0  d 80.0)

This plot (2000 points) is invoked by:

(2DPlot fXYhypotr2 0 (* pi 2) (/ (* 2 pi) 2000.0))

5) Epitrochoid 1

Epitrochoid is similar to hypotrochoid, only the inner circle rolls outside the control circle. This example uses:

(setq r1 5.0  r2 2.0  d 2.5)

6) Epitrochoid 2

You can control the shape and complexity of the curves by changing the parameters of the circles (wheels):

(setq r1 5.0  r2 2.25  d 2.5)

7) Farrel wheels

Farrel wheels is a structure of 3 wheels rolling on each other by the principles of the hypotrochoid. You can specify the size (radius) of all 3 wheels (can also be the same) and their speed (frequency). Farrel wheels are used in a Spirograph or in ancient machines used to generate Guilloches.

(setq fR1 10.0  fR2 10.0  fR3 10.0) ; radii
(setq fn1 13  fn2 -7  fn3 -3) ; speeds
(2DPlot fXYfarrel 0.0 1.0 (/ 1.0 1600.0) )

8) Farrel wheels (parametric)

The most interesting function - a playground for your own Farrel wheels - specify parameters for the 3 interlocked wheels and watch the results. You can enter the radii for all 3 wheels and their frequency (number of rotations per 0-1 change, can be negative).

Try e.g. the parameters (radius+freq):

• 12/10/8, 120/-77/-34, 6000 nodes
• 11/10/9, 180/-37/-53, 6000 nodes
• 10/10/10, 200/-51/-17, 6000 nodes
• 10/10/10, 256/-64/-32, 6000 nodes

9) Guilloche

A combination (superposition) of two hypotrochoids with the parameters of [5 3 5] (with phase offset 1.5x). By combining multiple hypotrochoids or epitrochoids you can generate pleasant and symmetrical curves often used on banknotes, printed stocks, diplomas, etc.

10) Guilloche (parametric)

This is a series of guilloche curves from the example above. Each next curve has its parameters scaled down by 10%.

11) Epitrochoid (slow)

This is a series of epitrochoids with scaled-down parameters similar to the example above. It uses the epitrochoid from the case #5 but repeats it (and scales to 90%) 10-times. Also uses a different method of plotting the polyline - 2DPlotP. This function plots the curve by calling the PLINE command.

12) Catenary (øetìzovka)

This is a simple parametric curve defined by:

(defun fXYcatenary (x)
 (defun cosh (x)  (/ (1+ (exp (* 2.0 x))) (* 2.0 (exp x))))
 (list x (* fCa (cosh (/ x fCa))))
)

13) Fermat's spiral (parametric)

This is a simple parametric curve defined by:

(defun fXYfermat (x)
 (list
  (* fCa (sqrt f)(cos f))
  (* fCa (sqrt f)(sin f))
 )
)

14) Lissajous curve

This is well-known parametric curve defined by:

(defun fXYlissajous (f)
 (setq a 9  b 7  delta (/ pi 2.0))
 (list
  (* 10.0 (sin (+ (* a f) delta)))
  (* 10.0 (sin (* b f)))
 )
)