# Archive for August 5th, 2015

### HP 7475A Plotter: Superformula Demo

Posted by Ed in Machine Shop, Software on 2015-08-05

Setting `n2=n3=1.5`

generates smoothly rounded shapes, rather than the spiky ones produced by `n2=n3=1.0`

, so I combined the two into a single demo routine:

A closer look shows all the curves meet at the points, of which there are 37:

The spikes suffer from limited resolution: each curve has 10 k points, but if the extreme end of a spike lies between two points, then it gets blunted on the page. Doubling the number of points would help, although I think this has already gone well beyond the, ah, point of diminishing returns.

I used the three remaining “disposable” liquid ink pens for the spiked curves; the black pen was beyond repair. They produce gorgeous lines, although the magenta ink seems a bit thinned out by the water I used to rinse the remains of the last refill out of the spiral vent channel.

I modified the Chiplotle `supershape()`

function to default to my choices for `point_count`

and `travel`

, then copied the `superformula()`

function and changed it to return polar coordinates, because I’ll eventually try scaling the linear value as a function of the total angle, which is *much* easier in polar coordinates.

The demo code produces the patterns in the picture by iterating over interesting values of `n1`

and `n2=n3`

, stepping through the pen carousel for each pattern. As before, `m`

should be `prime/10`

to produce a `prime`

number of spikes / bumps. You could add more iteration values, but six of ’em seem entirely sufficient.

A real demo should include a large collection of known-good parameter sets, from which it can pick six sets to make a plot. A legend documenting the parameters for each pattern, plus the date & time, would bolster the geek cred.

The Python source code with the modified Chiplotle routines:

from chiplotle import * from math import * def superformula_polar(a, b, m, n1, n2, n3, phi): ''' Computes the position of the point on a superformula curve. Superformula has first been proposed by Johan Gielis and is a generalization of superellipse. see: http://en.wikipedia.org/wiki/Superformula Tweaked to return polar coordinates ''' t1 = cos(m * phi / 4.0) / a t1 = abs(t1) t1 = pow(t1, n2) t2 = sin(m * phi / 4.0) / b t2 = abs(t2) t2 = pow(t2, n3) t3 = -1 / float(n1) r = pow(t1 + t2, t3) if abs(r) == 0: return (0,0) else: # return (r * cos(phi), r * sin(phi)) return (r,phi) def supershape(width, height, m, n1, n2, n3, point_count=10*1000, percentage=1.0, a=1.0, b=1.0, travel=None): '''Supershape, generated using the superformula first proposed by Johan Gielis. - `points_count` is the total number of points to compute. - `travel` is the length of the outline drawn in radians. 3.1416 * 2 is a complete cycle. ''' travel = travel or (10*2*pi) ## compute points... phis = [i * travel / point_count for i in range(1 + int(point_count * percentage))] points = [superformula_polar(a, b, m, n1, n2, n3, x) for x in phis] ## scale and transpose... path = [ ] for r, a in points: x = width * r * cos(a) y = height * r * sin(a) path.append(Coordinate(x, y)) return Path(path) ## RUN DEMO CODE if __name__ == '__main__': paperx = 8000 papery = 5000 if not False: plt=instantiate_plotters()[0] plt.set_origin_center() plt.write(hpgl.VS(10)) pen = 1 for m in [3.7]: for n1 in [0.20, 0.60, 0.8]: for n2 in [1.0, 1.5]: n3 = n2 e = supershape(paperx, papery, m, n1, n2, n3) plt.select_pen(pen) if pen < 6: pen += 1 else: pen = 1 plt.write(e) plt.select_pen(0) else: e = supershape(paperx, papery, 1.9, 0.8, 3, 3) io.view(e)

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