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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

583.0. "Piet Hein design" by 26205::YARBROUGH () Mon Sep 22 1986 13:13

    Scandinavian designer/poet Piet Hein used a curve that he called a
    Supercircle in several of his designs. Its equation is 
    
    		x^4 + y^4 = 1,
    
    scaled appropriately. This curve he describes as 'half-way between
    a circle and a square', since the circle is x^2 + y^2 = 1, and the
    limit of x^2n + y^2n = 1 as n->infinity is the square. Aesthetically
    it is quite pleasing.
    
    I recently built a couple of card tables using this design for the
    top, inscribing the supercircle in a 4-foot square. I found that
    I needed a veneer molding to apply around the edge of the table.
    How much molding did I need for each table; i.e., what is [4 times]
    the arc-length of the largest supercircle inscribed in a [unit] 
    square?
    
    (This is NOT an easy problem.)

T.R	Title	User	Personal Name	Date	Lines
583.1	superspheres	CACHE::MARSHALL	beware the fractal dragon	`Tue Sep 23 1986 04:59`	16
	avoiding the question completely.... Hein also made some superspheres, three-dimensional supercircles that had some interesting properties. One of these was the property of "stackability", that is, you can stack them quite high even though they don't have any truly "flat" surfaces. / ( ___ ) /// /
583.2	approx numerical answer	ENGINE::ROTH		`Tue Sep 23 1986 11:19`	10
	You'll need about 14.03539588712808 feet, a bit below the 16 feet necessary for a square table... Off hand I don't know a closed form for the integral, but it should be possible to bring into closed form in terms of an elliptic integral. I suppose the easiest way would be to see what MACSYMA has to say about it. A series solution should also be possible, by expressing the integrand as a series and integrating term by term. - Jim
583.3	Good guess	26205::YARBROUGH		`Tue Sep 23 1986 12:59`	4
	That's the same answer I got with my tape measure, although I was only able to get 12 places of accuracy... ;-) Anyone want to discuss how to attack the problem?
583.4	Anybody have a ReGis plot of this curve?	SQM::HALLYB	Free the quarks!	`Tue Sep 23 1986 18:28`	1

583.5	Do-it-yourself kit	26205::YARBROUGH		`Tue Sep 23 1986 21:03`	18
	Here's a BASIC program to draw it FAIRLY accurately. 10 dim y (101) 20 esc$=chr$(27%) 30 for x=1 to 101 40 y(x)=180sqr(1-((x-51)/50)^4) 50 next x 60 print esc$+'Pp'; 65 print 'P['; 200+4; ','; 200+y(1); ']'; 70 for x = 2 to 101 80 print 'V['; 200+4x; ','; 200+y(x); ']' 90 next x 95 print 'P['; 200+4; ','; 200-y(1); ']'; 100 for x = 2 to 101 110 print 'V['; 200+4*x; ','; 200-y(x); ']' 120 next x 200 print esc$+'\' 250 input z$
583.6	your wish is my command	CACHE::MARSHALL	beware the fractal dragon	`Tue Sep 23 1986 21:10`	204
	a sixel image of the supercircle follows the <FF>. if your interested EXTRACT now. 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583.7	but where'd you get the pattern?	CACHE::MARSHALL	beware the fractal dragon	`Wed Sep 24 1986 15:26`	13
	By the way, how did you lay out this design on the table top? That is, a compass will draw a circle, two pins and a loop of string will draw an ellipse, what technique draws this shape (at least to a good approximation)? / ( ___ ) /// /
583.8	Nothing fancy	26205::YARBROUGH		`Wed Sep 24 1986 18:00`	2
	I calculated the data points for one octant and used a French curve to smooth it, then cut out a template from cardboard.
583.9	shucks	CACHE::MARSHALL	beware the fractal dragon	`Wed Sep 24 1986 19:32`	11
	re .8: I was afraid of that. I wonder if there is any 'neat' way of drawing this curve. / ( ___ ) /// /
583.10	does this work?	CACHE::MARSHALL	beware the fractal dragon	`Wed Sep 24 1986 19:54`	18
	extrapolating from the method used to draw ellipses: set up 4 pins arranged in a square place loop of string of length x over the pens use the string as your guide as you move the pencil. The problem is determining the ratio of the string length and the sides of the square. I've tried this and seems to draw something like a supercircle. Can anyone take it from here? / ( ___ ) /// /
583.11	try something else	26205::YARBROUGH		`Wed Sep 24 1986 20:33`	9
	What that draws is sections of ellipses. Since the portion of the string away from the moving pencil is constant = twice the distance between pins, the string is most of the time effectively rotating around two pins, which are the foci of the ellipse. One way of drawing it is to run a GKS version of the program described earlier and output to a D-size plotter (Yes, that's terribly expensive, unless you already have the plotter). One octant might fit on an LA210. It's close.
583.12	some more curves...	ENGINE::ROTH		`Thu Sep 25 1986 00:34`	18
	Here's another method, that you can generalize to generate Lissajous style patterns: Let x(t) = cos(t)^0.5, y(t) = sin(t)^0.5, t in [0,pi/2], the other quadrants can be generated by symmetry. It is interesting to put other exponents and frequencies as follows: x(t) = cos(mt)^e, y(t) = sin(nt)^e Other variations on polar patterns like epicycloids and epitrochoids can be interesting too, I once had a program that generated these for the PC/350. Unfortunately, I don't know a nice way of rectifying this curve for other than a circle... the number given before was from a Romberg integration program I had laying around. - Jim
583.13	Here's a recent reference on these	ENGINE::ROTH		`Tue Oct 14 1986 15:07`	13
	I came across an interesting reference on these curves: "Superquadraics and Angle Preserving Transformations" by Alan Barr, in the Jan 1981 issue of IEEE Computer Graphics and Applications. He shows how to generalize the usual quadratic curves using techniques similar to my idea of putting an exponent other than 2 in the equation of a circle [x = cos(t), y = sin(t)]... No closed form for rectifying the arc length tho, but a nice paper. - Jim
583.14	Scientific American cover story on Supercircles	NOBUGS::AMARTIN	Alan H. Martin	`Wed Nov 05 1986 18:23`	3
	I think there was a cover article on Supercircles in Scientific American once. Probably during the '70s. /AHM