Quasi-Circles
In the previous pages, we have seen a number of times, that "circles" appeared, where only straight lines were drawn. Let's turn it around. Is it possible to draw (that is approximate) circles by straight lines? Yes, of course, it can! It means drawing polygons, with all angular points at – or close to – a circle. But the limiting factor is our checkered paper: the lines should begin and end on a raster-point. Now it becomes more interesting. Over the years, I experimented a lot with all sorts of circle approximating polygons on checkered paper. The resulting drawings were so many, that I feel I need to categorize them, especially so, now I have started the Gallery.
So, let's examine the possibilities to approximate circles with not too long, raster limited, straight lines. Or, which of these quasi-circles pass through or almost through (more than four) grid-points? That means, that the radius R of such a circle must satisfy the Pythagorean equation R² = x² + y², where x and y are integer numbers. In the following table, the value of R² is denoted for each pair of integer values x and y up to 10.


Table 1.   R² for each pair of integer values x and y
x 012345678910
0149162536496481100
y
0001491625364964 81100
1125101726375065 82101
248132029405368 85104
39182534455873 90109
4163241526580 97116
52550617489 106125
6367285100117 136
74998113130 149
864 128145164
981162181
10100200

Now we have to find radii, that are equal or almost equal. It is easiest to put all values of R² in ascending order, so that equal and almost equal values become adjacent. Table 2 illustrates this. Following the R² column, the appropriate x and y values are denoted, followed by a "circle number" with the relevant characteristics (if applicable – "no suitable circle" means, that the sides of a possible polygon are too far off the circumference of a circle), and an example of how it looks like on checkered paper. Sometimes, an alternative is possible with the same characteristics.
Some of the quasi-circles are "exact" in the way, that the angular points lie exactly on a circle; these are displayed with a dotted circle. Mostly, values of R² are combined, that do not differ more than 2, but quasi-circles with a difference in R² of more than 2 also exist in the Gallery, as do polygons with more than 2 different radii – especially some greater ones. On checkered paper, it is not absolutely mandatory to stick to the raster-points. There are a number of quasi-circles, that use halfway values on the raster. Therefore, some polygons in table 2 have half-sized counterparts shown additionally.
In the Gallery, the various quasi-circles will be put together within their own "family".


Table 2.   All suitable quasi-circles for the values of R²
  x y  available quasi-circle  example quasi-circle

110 (no suitable circle)
211 (no suitable circle)
420 (no suitable circle)
521 "Circle 1", radius 2x1 Circle 1
822 "Circle 2", radii 2x2 and 3x0 Circle 2
930
1031 "Circle 3", radius 3x1 Circle 3
1332 (no suitable circle)
1640 "Circle 4", radii 4x0 and 3x3 Circle 4
1833
1741 "Circle 5", radii 4x1 and 3x3 Circle 5
1833
Alternative
1741 "Circles 6", radii 4x1 and 3x3 Circles 6
1833
2042 "Circle 7", double of "Circle 1" Circle 7
2543 "Circle 8", radii 4x3 and 5x0 Circle 8
2550
Alternative
2543 "Circles 9", radii 4x3 and 5x0 Circles 9
2550
2543 "Circle 10", radii 4x3 and 5x1 Circle 10
2651
Alternative
2543 "Circles 11", radii 4x3 and 5x1 Circles 11
2651
2952 "Circle 12", radius 5x2 Circle 12
3244 "Circle 13", radii 4x4 and 6x0 Circle 13
3660
3453 "Circle 14", radii 5x3 and 6x0 Circle 14
3660
3453 "Circle 15", radii 5x3 and 6x1 Circle 15
3761
4062 "Circle 16", radii 6x2 and 5x4 Circle 16
4154
4154 "Circle 17", radii 5x4 and 7x0 Circle 17
4970
4563 "Circle 18", three times "Circle 1" Circle 18
4970 "Circle 19", radii 7x0 and 5x5 Circle 19
5055
5055 "Circle 20", radii 5x5 and 7x1 Circle 20
5071
Half of "Circle 20", radii 2.5x2.5 and 3.5x0.5 Half of Circle 20
Alternative
5055 "Circles 21", radii 5x5 and 7x1 Circles 21
5071
5071 "Circle 22", radii 7x1 and 6x4 Circle 22
5264
5264 "Circle 23", radii 6x4 and 7x2 Circle 23
5372
5873 "Circle 24", radius 7x3 Circle 24
Half of "Circle 24", radius 3.5x1.5 Half of Circle 24
6165 (no suitable circle)
6480 "Circle 25", radii 8x0 and 7x4 Circle 25
6574
6574 "Circle 26", radii 7x4 and 8x1 Circle 26
6581
Alternative 1
6574 "Circles 27", radii 7x4 and 8x1 Circles 27
6581
Alternative 2
6574 "Circles 28", radii 7x4 and 8x1 Circles 28
6581
6882 "Circle 29", radii 8x2 and 6x6 Circle 29
7266
7266 "Circle 30", radii 6x6 and 8x3 Circle 30
7383
7383 "Circle 31", radii 8x3 and 7x5 Circle 31
7475
8084 "Circle 32", radii 8x4 and 9x0 Circle 32
8190
8291 (no suitable circle)
8576 "Circle 33", radii 7x6 and 9x2 Circle 33
8592
Alternative
8576 "Circles 34", radii 7x6 and 9x2 Circles 34
8592
8985 "Circle 35", radii 8x5 and 9x3 Circle 35
9093
9794 "Circle 36", radii 9x4 and 7x7 Circle 36
9877
9794 "Circle 37", radii 9x4, 7x7 and 10x0 Circle 37
9877
100100
10086 "Circle 38", double of "Circle 7" Circle 38
100100
10086 "Circle 39", radii 8x6 and 10x1 Circle 39
101101
104102 "Circle 40", radii 10x2, 9x5 and 8x7 Circle 40
10695
11387
109103 (no suitable circle)
116104 (no suitable circle)
11796 (no suitable circle)
125105 "Circle 41", radii 10x5, 11x2 and 8x8 Circle 41
125112
12888
13097 "Circle 42", radii 9x7 and 11x3 Circle 42
130113
Half of "Circle 42", radii 4.5x3.5 and 5.5x1.5 Half of Circle 42
136106 (no suitable circle)
14598 (no suitable circle)
149107 (no suitable circle)
16299 (no suitable circle)
164108 (no suitable circle)
181109 (no suitable circle)
2001010 (no suitable circle)
234153 "Circle 43", radii 15x3 and 13x9 Circle 43
250139
Half of "Circle 43", radii 7.5x1.5 and 6.5x4.5 Half of Circle 43

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All drawings and figures, copyright © 1978-2009, Zef Damen, The Netherlands.
Personal use only, commercial use prohibited.

Nederlandse versie Nederlandse versie Last updated: 12-December-2009