OFFSET
3,1
COMMENTS
If the smallest possible enclosing circle is essentially determined by 3 vertices of the polygon, the squared diameter may be rational and thus A322107(n) > 1.
The first difference of the sequences A321693(n) / A322029(n) from a(n) / A322107(n) occurs for n = 12.
The ratio (A321693(n)/A322029(n)) / (a(n)/A322107(n)) will grow for larger n due to the tendency of the minimum area polygons to approach elliptical shapes with increasing aspect ratio, whereas the polygons leading to small enclosing circles will approach circular shape.
For n>=19, polygons with different areas may fit into the enclosing circle of minimal diameter. See examples in pdf at Pfoertner link.
REFERENCES
See A063984.
LINKS
Hugo Pfoertner, Illustration of convex n-gons fitting into smallest circle, (2018).
Hugo Pfoertner, Illustration of convex n-gons fitting into smallest circle, n = 27..32, (2018).
EXAMPLE
By n-gon a convex lattice n-gon is meant, area is understood omitting the factor 1/2. The following picture shows a comparison between the minimum area polygon and the polygon fitting in the smallest possible enclosing circle for n=12:
.
0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
6 H ##### Gxh +++++ g
| # + # * +
| # + # +
| # + * # +
5 I i F f
| # + * # +
| # + # +
| # + * # +
4 J j # e
| # @+ * # +
| # + @ #+
| # + @ * +#
3 K + @ + E
| # + * @ + #
| # @ + #
| + # * +@ #
2 k # d D
| + # * + #
| + # + #
| + # * + #
1 l L c C
| + # * + #
| + # + #
| + * # + #
0 a ++++ Axb ##### B
0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
.
The 12-gon ABCDEFGHIJKLA with area 52 fits into a circle of squared diameter 40, e.g. determined by the distance D - J, indicated by @@@. No convex 12-gon with a smaller enclosing circle exists. Therefore a(n) = 40 and A322107(12) = 1.
CROSSREFS
KEYWORD
nonn,frac,more
AUTHOR
Hugo Pfoertner, Nov 26 2018
EXTENSIONS
a(27)-a(32) from Hugo Pfoertner, Dec 19 2018
STATUS
approved