2D Curves







Parametrisation of curve :
* closed curve can be parametrised by angle : c = f(angle)
* ray ( half-line) can be parametrised by radius : c = f(radius)

Rasterisarion of curve :
* closed curve ( circle or cardioid)
* ray ( half-line)


Scale :
* linear
* nonlinear


Rasterisation of radius R using linear scale :
/* code in Maxima CAS */
R_Max:4;
R_Min:1.1;
for R:R_Max step -0.5 thru R_Min do disp(R);
/* Maxima allows non-integer values in for statement */ (%o1) 4
(%o2) 1.1
4
3.5
3.0
2.5
2.0
1.5
(%o3) done



Method by Curtis T McMullen
Rasterisation of radius R using nonlinear scale :
/* R:2^r ; code in Maxima CAS */
(%i1) rMax:2;
rMin:0.1; /* R tends to 1 as r tends to 0 because R=2^r */
caution:0.93;
r:rMax;
unless r < rMin do (r:r*caution, R:2^r, disp(R));
(%o1) 2
(%o2) 0.1
(%o3) 0.93
(%o4) 2
3.630076621268644
3.316818342728249
3.049798654210589
2.820799296565715
2.623288972344463
2.452036351025536
2.302820150013453
2.172210053964319
2.057399794458432
1.956078968662366
1.866333866345315
1.786570189926742
1.715452421936768
1.651855942297139
1.594828977653743
1.54356218274762
1.497364183535279
1.455641805533936
1.417884005591913
1.383648747345652
1.352552229006523
1.324260000584035
1.29847960624068
1.274954463566127
1.253458750610672
1.233793117588228
1.215781076285127
1.199265948678497
1.184108278807163
1.170183629865262
1.157380702810876
1.145599724272923
1.134751060796999
1.124754023958772
1.115535836953474
1.107030738225175
1.099179201752541
1.091927256934423
1.085225893758314
1.079030541198431
1.073300608666395
1.067999081897454
(%o5) done



Rasterisation of radius R using nonlinear scale :
/* R = r^(1/(2^D)) ; code in Maxima CAS */
r: 2000;
D_Max:20;
D_Min:0.1;
for D:D_Min step 1 thru D_Max do disp(float(r^(1/2^D)));
(%o9) 2000
(%o10) 20
(%o11) 0.1
1202.181089063019
34.67248316840054
5.888334498684713
2.426589066711691
1.557751285254386
1.24809906868581
1.117183542971257
1.05696903595671
1.028089994094248
1.013947727495973
1.006949714482294
1.003468840812855
1.001732918902466
1.000866084400139
1.000432948477877
1.000216450813461
1.000108219550995
1.000054108311643
1.000027053789868
1.000013526803447
(%o12) done



The simple trick for drawing arcs: 2 coats of paint, along x axis and along y axis
Try only one plot to see the differance
The same trick is in drawing M-set with escape lines method.

Cardioid:
Cardioid by Paul Bourke


Delphi program for drawing above images (see main cardioid). In status bar you can read period of orbit





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Feel free to e-mail me!

Author: Adam Majewski adammaj1-at-o2-dot-pl

http://republika.pl/fraktal/

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