A322106 Numerator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon.

2, 2, 50, 8, 10, 10, 1250, 29, 40, 40, 2738, 72, 82, 82, 176900, 17810, 1709690, 178, 11300, 260, 290, 290, 568690, 416, 2418050, 488, 3479450, 629, 2674061, 730

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

Table of n, a(n) for n=3..32.

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.
For comparison, the 12-gon abcdefghijkla with minimal area A070911(12) = 48 requires a larger enclosing circle with squared diameter A321693(12)/A322029(12) = 52/1, e.g. determined by the distance a - g, indicated by ***.

Crossrefs

Cf. A063984, A070911, A321693, A322029, A322107 (corresponding denominators).

Sequence in context: A345267 A316090 A345463 * A321693 A268536 A323872

Adjacent sequences: A322103 A322104 A322105 * A322107 A322108 A322109

Keyword

nonn,frac,more

Author

Hugo Pfoertner, Nov 26 2018

Extensions

a(27)-a(32) from Hugo Pfoertner, Dec 19 2018

Status

approved