Assignment 2

COMP9021, Trimester 3, 2023

1. General matter

   1. Aims. The purpose of the assignment is to:

      • design and implement an interface based on the desired behaviour of an application program;

      • practice the use of Python syntax;

      • develop problem solving skills.

   2. Submission. Your program will be stored in a file named polygons.py. After you have developed and tested your program, upload it using Ed (unless you worked directly in Ed). Assignments can be submitted more than once; the last version is marked. Your assignment is due by November 20, 10:00am.

   3. Assessment. The assignment is worth 13 marks. It is going to be tested against a number of input files. For each test, the automarking script will let your program run for 30 seconds.

      Assignments can be submitted up to 5 days after the deadline. The maximum mark obtainable reduces by 5% per full late day, for up to 5 days. Thus if students A and B hand in assignments worth 12 and 11, both two days late (that is, more than 24 hours late and no more than 48 hours late), then the maximum mark obtainable is 11.7, so A gets min(11.7, 11) = 11 and B gets min(11.7, 11) = 11. The outputs of your programs should be exactly as indicated.

      You will design and implement a program that will

      • extract and analyse the various characteristics of (simple) polygons, their contours being coded and stored in a file, and

      • – either display those characteristics: perimeter, area, convexity, number of rotations that keep the polygon invariant, and depth (the length of the longest chain of enclosing polygons)

– or output some Latex code, to be stored in a file, from which a pictorial representation of the polygons can be produced, coloured in a way which is proportional to their area.

Call encoding any 2-dimensional grid of size between between 2 × 2 and 50 × 50 (both dimensions can be different) all of whose elements are either 0 or 1.

Call neighbour of a member m of an encoding any of the at most eight members of the grid whose value is 1 and each of both indexes differs from m’s corresponding index by at most 1. Given a particular encoding, we inductively define for all natural numbers d the set of polygons of depth d (for this encoding) as follows. Let a natural number d be given, and suppose that for all dr < d, the set of polygons of depth dr has been defined. Change in the encoding all 1’s that determine those polygons to 0. Then the set of polygons of depth d is defined as the set of polygons which can be obtained from that encoding by connecting 1’s with some of their neighbours in such a way that we obtain a maximal polygon (that is, a polygon which is not included in any other polygon obtained from that encoding by connecting 1’s with some of their neighbours).

1

1. Examples

   1. First example. The file polys_1.txt has the following contents:

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      11111111111111111111111111111111111111111111111111

      Here is a possible interaction:

      $ python3

      …

      >>> from polygons import *

      >>> polys = Polygons(‘polys_1.txt’)

      >>> polys.analyse() Polygon 1:

      Perimeter: 78.4

      Area: 384.16

      Convex: yes

      Nb of invariant rotations: 4 Depth: 0

      Polygon 2:

      Perimeter: 75.2

      Area: 353.44

      Convex: yes

      Nb of invariant rotations: 4 Depth: 1

      Polygon 3:

      Perimeter: 72.0

      Area: 324.00

      Convex: yes

      Depth: 3

      Polygon 5:

      Perimeter: 65.6

      Area: 268.96

      Convex: yes

      Nb of invariant rotations: 4 Depth: 4

      Polygon 6:

      Perimeter: 62.4

      Area: 243.36

      Convex: yes

      Nb of invariant rotations: 4 Depth: 5

      Polygon 7:

      Perimeter: 59.2

      Area: 219.04

      Convex: yes

      Nb of invariant rotations: 4 Depth: 6

      Polygon 8:

      Perimeter: 56.0

      Area: 196.00

      Convex: yes

      Nb of invariant rotations: 4

      Depth: 7

      Polygon 9:

      Perimeter: 52.8

      Area: 174.24

      Convex: yes

      Nb of invariant rotations: 4 Depth: 8

      Polygon 10:

      Perimeter: 49.6

      Area: 153.76

      Convex: yes

      Nb of invariant rotations: 4 Depth: 9

      Polygon 11:

      Perimeter: 46.4

      Area: 134.56

      Convex: yes

      Nb of invariant rotations: 4 Depth: 10

      Polygon 12:

      Perimeter: 43.2

      Area: 116.64

      Convex: yes

      Nb of invariant rotations: 4

      Depth: 12

      Polygon 14:

      Perimeter: 36.8

      Area: 84.64 Convex: yes

      Nb of invariant rotations: 4 Depth: 13

      Polygon 15:

      Perimeter: 33.6

      Area: 70.56 Convex: yes

      Nb of invariant rotations: 4 Depth: 14

      Polygon 16:

      Perimeter: 30.4

      Area: 57.76 Convex: yes

      Nb of invariant rotations: 4 Depth: 15

      Polygon 17:

      Perimeter: 27.2

      Area: 46.24 Convex: yes

      Nb of invariant rotations: 4

      Depth: 16

      Polygon 18:

      Perimeter: 24.0

      Area: 36.00 Convex: yes

      Nb of invariant rotations: 4 Depth: 17

      Polygon 19:

      Perimeter: 20.8

      Area: 27.04 Convex: yes

      Nb of invariant rotations: 4 Depth: 18

      Polygon 20:

      Perimeter: 17.6

      Area: 19.36 Convex: yes

      Nb of invariant rotations: 4 Depth: 19

      Polygon 21:

      Perimeter: 14.4

      Area: 12.96 Convex: yes

      Nb of invariant rotations: 4

      Depth: 21

      Polygon 23:

      Perimeter: 8.0

      Area: 4.00 Convex: yes

      Nb of invariant rotations: 4 Depth: 22

      Polygon 24:

      Perimeter: 4.8

      Area: 1.44 Convex: yes

      Nb of invariant rotations: 4 Depth: 23

      Polygon 25:

      Perimeter: 1.6

      Area: 0.16 Convex: yes

      Nb of invariant rotations: 4 Depth: 24

      >>> polys.display()

      The effect of executing polys.display() is to produce a file named polys_1.tex that can be given as argument to pdflatex to produce a file named polys_1.pdf that views as follows.

   2. Second example. The file polys_2.txt has the following contents:

      00000000000000000000000000000000000000000000000000

      01111111111111111111111111111111111111111111111110

      00111111111111111111111111111111111111111111111100

      00011111111111111111111111111111111111111111111000

      01001111111111111111111111111111111111111111110010

      01100111111111111111111111111111111111111111100110

      01110011111111111111111111111111111111111111001110

      01111001111111111111111111111111111111111110011110

      01111100111111111111111111111111111111111100111110

      01111110011111111111111111111111111111111001111110

      01111111001111111111111111111111111111110011111110

      01111111100111111111111111111111111111100111111110

      01111111110011111111111111111111111111001111111110

      01111111111001111111111111111111111110011111111110

      01111111111100111111111111111111111100111111111110

      01111111111110011111111111111111111001111111111110

      01111111111111001111111111111111110011111111111110

      01111111111111100111111111111111100111111111111110

      01111111111111110011111111111111001111111111111110

      01111111111111111001111111111110011111111111111110

      01111111111111111100111111111100111111111111111110

      01111111111111111110011111111001111111111111111110

      01111111111111111001111111111110011111111111111110

      01111111111111110011111111111111001111111111111110

      01111111111111100111111111111111100111111111111110

      01111111111111001111111111111111110011111111111110

      01111111111110011111111111111111111001111111111110

      01111111111100111111111111111111111100111111111110

      01111111111001111111111111111111111110011111111110

      01111111110011111111111111111111111111001111111110

      01111111100111111111111111111111111111100111111110

      01111111001111111111111111111111111111110011111110

      01111110011111111111111111111111111111111001111110

      01111100111111111111111111111111111111111100111110

      01111001111111111111111111111111111111111110011110

      01110011111111111111111111111111111111111111001110

      01100111111111111111111111111111111111111111100110

      01001111111111111111111111111111111111111111110010

      00011111111111111111111111111111111111111111111000

      00111111111111111111111111111111111111111111111100

      01111111111111111111111111111111111111111111111110

      00000000000000000000000000000000000000000000000000

      Here is a possible interaction:

      $ python3

      …

      >>> from polygons import *

      >>> polys = Polygons(‘polys_2.txt’)

      >>> polys.analyse() Polygon 1:

      Perimeter: 37.6 + 92*sqrt(.32) Area: 176.64

      Convex: no

      Nb of invariant rotations: 2 Depth: 0

      Polygon 2:

      Perimeter: 17.6 + 42*sqrt(.32) Area: 73.92

      Convex: yes

      Nb of invariant rotations: 1 Depth: 1

      Polygon 3:

      Perimeter: 16.0 + 38*sqrt(.32) Area: 60.80

      Convex: yes

      Depth: 0

      Polygon 5:

      Perimeter: 14.4 + 34*sqrt(.32) Area: 48.96

      Convex: yes

      Nb of invariant rotations: 1 Depth: 3

      Polygon 6:

      Perimeter: 16.0 + 40*sqrt(.32) Area: 64.00

      Convex: yes

      Nb of invariant rotations: 1 Depth: 0

      Polygon 7:

      Perimeter: 12.8 + 30*sqrt(.32) Area: 38.40

      Convex: yes

      Nb of invariant rotations: 1 Depth: 4

      Polygon 8:

      Perimeter: 14.4 + 36*sqrt(.32) Area: 51.84

      Convex: yes

      Nb of invariant rotations: 1

      Depth: 1

      Polygon 9:

      Perimeter: 11.2 + 26*sqrt(.32) Area: 29.12

      Convex: yes

      Nb of invariant rotations: 1 Depth: 5

      Polygon 10:

      Perimeter: 14.4 + 36*sqrt(.32) Area: 51.84

      Convex: yes

      Nb of invariant rotations: 1 Depth: 1

      Polygon 11:

      Perimeter: 9.6 + 22*sqrt(.32) Area: 21.12

      Convex: yes

      Nb of invariant rotations: 1 Depth: 6

      Polygon 12:

      Perimeter: 12.8 + 32*sqrt(.32) Area: 40.96

      Convex: yes

      Nb of invariant rotations: 1

      Depth: 7

      Polygon 14:

      Perimeter: 12.8 + 32*sqrt(.32) Area: 40.96

      Convex: yes

      Nb of invariant rotations: 1 Depth: 2

      Polygon 15:

      Perimeter: 6.4 + 14*sqrt(.32) Area: 8.96

      Convex: yes

      Nb of invariant rotations: 1 Depth: 8

      Polygon 16:

      Perimeter: 11.2 + 28*sqrt(.32) Area: 31.36

      Convex: yes

      Nb of invariant rotations: 1 Depth: 3

      Polygon 17:

      Perimeter: 4.8 + 10*sqrt(.32) Area: 4.80

      Convex: yes

      Nb of invariant rotations: 1

      Depth: 9

      Polygon 18:

      Perimeter: 11.2 + 28*sqrt(.32) Area: 31.36

      Convex: yes

      Nb of invariant rotations: 1 Depth: 3

      Polygon 19:

      Perimeter: 3.2 + 6*sqrt(.32) Area: 1.92

      Convex: yes

      Nb of invariant rotations: 1 Depth: 10

      Polygon 20:

      Perimeter: 9.6 + 24*sqrt(.32) Area: 23.04

      Convex: yes

      Nb of invariant rotations: 1 Depth: 4

      Polygon 21:

      Perimeter: 1.6 + 2*sqrt(.32) Area: 0.32

      Convex: yes

      Nb of invariant rotations: 1

      Depth: 4

      Polygon 23:

      Perimeter: 8.0 + 20*sqrt(.32) Area: 16.00

      Convex: yes

      Nb of invariant rotations: 1 Depth: 5

      Polygon 24:

      Perimeter: 8.0 + 20*sqrt(.32) Area: 16.00

      Convex: yes

      Nb of invariant rotations: 1 Depth: 5

      Polygon 25:

      Perimeter: 6.4 + 16*sqrt(.32) Area: 10.24

      Convex: yes

      Nb of invariant rotations: 1 Depth: 6

      Polygon 26:

      Perimeter: 6.4 + 16*sqrt(.32) Area: 10.24

      Convex: yes

      Nb of invariant rotations: 1

      Depth: 6

      Polygon 27:

      Perimeter: 4.8 + 12*sqrt(.32) Area: 5.76

      Convex: yes

      Nb of invariant rotations: 1 Depth: 7

      Polygon 28:

      Perimeter: 4.8 + 12*sqrt(.32) Area: 5.76

      Convex: yes

      Nb of invariant rotations: 1 Depth: 7

      Polygon 29:

      Perimeter: 3.2 + 8*sqrt(.32) Area: 2.56

      Convex: yes

      Nb of invariant rotations: 1 Depth: 8

      Polygon 30:

      Perimeter: 3.2 + 8*sqrt(.32) Area: 2.56

      Convex: yes

      Nb of invariant rotations: 1

      Depth: 9

      Polygon 32:

      Perimeter: 1.6 + 4*sqrt(.32) Area: 0.64

      Convex: yes

      Nb of invariant rotations: 1 Depth: 9

      Polygon 33:

      Perimeter: 17.6 + 42*sqrt(.32) Area: 73.92

      Convex: yes

      Nb of invariant rotations: 1 Depth: 1

      Polygon 34:

      Perimeter: 16.0 + 38*sqrt(.32) Area: 60.80

      Convex: yes

      Nb of invariant rotations: 1 Depth: 2

      Polygon 35:

      Perimeter: 14.4 + 34*sqrt(.32) Area: 48.96

      Convex: yes

      Nb of invariant rotations: 1

      Depth: 3

      Polygon 36:

      Perimeter: 12.8 + 30*sqrt(.32) Area: 38.40

      Convex: yes

      Nb of invariant rotations: 1 Depth: 4

      Polygon 37:

      Perimeter: 11.2 + 26*sqrt(.32) Area: 29.12

      Convex: yes

      Nb of invariant rotations: 1 Depth: 5

      Polygon 38:

      Perimeter: 9.6 + 22*sqrt(.32) Area: 21.12

      Convex: yes

      Nb of invariant rotations: 1 Depth: 6

      Polygon 39:

      Perimeter: 8.0 + 18*sqrt(.32) Area: 14.40

      Convex: yes

      Nb of invariant rotations: 1

      Depth: 8

      Polygon 41:

      Perimeter: 4.8 + 10*sqrt(.32) Area: 4.80

      Convex: yes

      Nb of invariant rotations: 1 Depth: 9

      Polygon 42:

      Perimeter: 3.2 + 6*sqrt(.32) Area: 1.92

      Convex: yes

      Nb of invariant rotations: 1 Depth: 10

      Polygon 43:

      Perimeter: 1.6 + 2*sqrt(.32) Area: 0.32

      Convex: yes

      Nb of invariant rotations: 1 Depth: 11

      >>> polys.display()

      The effect of executing polys.display() is to produce a file named polys_2.tex that can be given as argument to pdflatex to produce a file named polys_2.pdf that views as follows.

   3. Third example. The file polys_3.txt has the following contents:

      0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0

      1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1

      0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 0

      0 1 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0

      0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0

      0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0

      0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0

      0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0

      0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0

      0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0

      1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1

      1 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 1 1

      1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 1 1 1

      1 1 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1

      1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1

      1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1

      1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1

      0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0

      0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0

      0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0

      0 1 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0

      0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 0

      1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1

      0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0

      Here is a possible interaction:

      $ python3

      …

      >>> from polygons import *

      >>> polys = Polygons(‘polys_3.txt’)

      >>> polys.analyse() Polygon 1:

      Perimeter: 2.4 + 9*sqrt(.32) Area: 2.80

      Convex: no

      Nb of invariant rotations: 1 Depth: 0

      Polygon 2:

      Perimeter: 51.2 + 4*sqrt(.32) Area: 117.28

      Convex: no

      Nb of invariant rotations: 2 Depth: 0

      Polygon 3:

      Perimeter: 2.4 + 9*sqrt(.32) Area: 2.80

      Convex: no

      Depth: 1

      Polygon 5:

      Perimeter: 3.2 + 28*sqrt(.32) Area: 9.76

      Convex: no

      Nb of invariant rotations: 1 Depth: 2

      Polygon 6:

      Perimeter: 27.2 + 6*sqrt(.32) Area: 5.76

      Convex: no

      Nb of invariant rotations: 1 Depth: 2

      Polygon 7:

      Perimeter: 4.8 + 14*sqrt(.32) Area: 6.72

      Convex: no

      Nb of invariant rotations: 1 Depth: 1

      Polygon 8:

      Perimeter: 4.8 + 14*sqrt(.32) Area: 6.72

      Convex: no

      Nb of invariant rotations: 1

      Depth: 1

      Polygon 9:

      Perimeter: 3.2 + 2*sqrt(.32) Area: 1.12

      Convex: yes

      Nb of invariant rotations: 1 Depth: 2

      Polygon 10:

      Perimeter: 3.2 + 2*sqrt(.32) Area: 1.12

      Convex: yes

      Nb of invariant rotations: 1 Depth: 2

      Polygon 11:

      Perimeter: 2.4 + 9*sqrt(.32) Area: 2.80

      Convex: no

      Nb of invariant rotations: 1 Depth: 0

      Polygon 12:

      Perimeter: 2.4 + 9*sqrt(.32) Area: 2.80

      Convex: no

      Nb of invariant rotations: 1

   4. Fourth example. The file polys_4.txt has the following contents:

      1 1 101 11 0 1 1 1 0 1 1 1011 10 1 1 1 0 000 1 1 1 0 00 1 001 11 1

      01 01000100010001000100100 110010010101001

      100 0010 0 0 1 00 0 1 0 00 100 01000 100 0 1 01 0001011 1

      1000101010101010101000100101010100010000

      0100010001000100010000100010100011100011

      100 1 0 0 0 10 0 0 1 00 0 1 00 01 010 000 0000 0 0 0 0 00 01 11

      11101 1101110 1	1	1	0111011101100000001111000
      000000000000000000000001100000011000100			0
      1		111001100111111100000000111111000	010000
      110 01	0 1 1 0		1011111100011111000000000001000

      Here is a possible interaction:

      $ python3

      …

      >>> from polygons import *

      >>> polys = Polygons(‘polys_4.txt’)

      >>> polys.analyse() Polygon 1:

      Perimeter: 11.2 + 28*sqrt(.32) Area: 18.88

      Convex: no

      Nb of invariant rotations: 2 Depth: 0

      Polygon 2:

      Perimeter: 3.2 + 5*sqrt(.32) Area: 2.00

      Convex: no

      Nb of invariant rotations: 1 Depth: 0

      Polygon 3:

      Perimeter: 1.6 + 6*sqrt(.32) Area: 1.76

      Convex: yes

      Depth: 0

      Polygon 5:

      Perimeter: 4*sqrt(.32) Area: 0.32

      Convex: yes

      Nb of invariant rotations: 4 Depth: 1

      Polygon 6:

      Perimeter: 4*sqrt(.32) Area: 0.32

      Convex: yes

      Nb of invariant rotations: 4 Depth: 1

      Polygon 7:

      Perimeter: 4*sqrt(.32) Area: 0.32

      Convex: yes

      Nb of invariant rotations: 4 Depth: 1

      Polygon 8:

      Perimeter: 4*sqrt(.32) Area: 0.32

      Convex: yes

      Nb of invariant rotations: 4

      Depth: 1

      Polygon 9:

      Perimeter: 1.6 + 1*sqrt(.32) Area: 0.24

      Convex: yes

      Nb of invariant rotations: 1 Depth: 0

      Polygon 10:

      Perimeter: 0.8 + 2*sqrt(.32) Area: 0.16

      Convex: yes

      Nb of invariant rotations: 2 Depth: 0

      Polygon 11:

      Perimeter: 12.0 + 7*sqrt(.32) Area: 5.68

      Convex: no

      Nb of invariant rotations: 1 Depth: 0

      Polygon 12:

      Perimeter: 2.4 + 3*sqrt(.32) Area: 0.88

      Convex: no

      Nb of invariant rotations: 1

      Depth: 0

      Polygon 14:

      Perimeter: 5.6 + 3*sqrt(.32) Area: 1.36

      Convex: no

      Nb of invariant rotations: 1 Depth: 0

      >>> polys.display()

      The effect of executing polys.display() is to produce a file named polys_4.tex that can be given as argument to pdflatex to produce a file named polys_4.pdf that views as follows.

      

2. Detailed description

   1. Input. The input is expected to consist of ydim lines of xdim 0’s and 1’s, where xdim and ydim are at least equal to 2 and at most equal to 50, with possibly lines consisting of spaces only that will be ignored and with possibly spaces anywhere on the lines with digits. If n is the xth digit of the yth line with digits, with 0 ≤ x < xdim and 0 ≤ y < ydim , then n is to be associated with a point situated x × 0.4 cm to the right and y × 0.4 cm below an origin.

   2. Output. Consider executing from the Python prompt the statement from polygons import * followed by the statement polys = Polygons(some_filename). In case some_filename does not exist in the working directory, then Python will raise a FileNotFoundError exception, that does not need to be caught. Assume that some_filename does exist (in the working directory). If the input is incorrect in that it does not contain only 0’s and 1’a besides spaces, or in that it contains either too few or too many lines of digits, or in that some line of digits contains too many or too few digits, or in that two of its lines of digits do not contain the same number of digits, then the effect of executing polys = Polygons(some_filename) should be to generate a PolygonsError exception that reads

Traceback (most recent call last):

…

polygons.PolygonsError: Incorrect input.

If the previous conditions hold but it is not possible to use all 1’s in the input and make them the contours of polygons of depth d, for any natural number d, as defined in the general presentation, then the effect of executing polys = Polygons(some_filename) should be to generate a PolygonsError exception that reads

line that reads

Polygon N:

with N an appropriate integer at least equal to 1 to refer to the N’th polygon listed in the order of polygons with highest point from smallest value of y to largest value of y, and for a given value of y, from smallest value of x to largest value of x, a second line that reads one of

Perimeter: a + b*sqrt(.32) Perimeter: a

Perimeter: b*sqrt(.32)

with a an appropriate strictly positive floating point number with 1 digit after the decimal point and b an appropriate strictly positive integer, a third line that reads

Area: a

with a an appropriate floating point number with 2 digits after the decimal point, a fourth line that reads one of

Convex: yes Convex: no

a fifth line that reads

Nb of invariant rotations: N

with N an appropriate integer at least equal to 1, and a sixth line that reads

Depth: N

with N an appropriate positive integer (possibly 0).

Pay attention to the expected format, including spaces.

If the input is correct and it is possible to use all 1’s in the input and make them the contours of poly- gons of depth d, for any natural number d, as defined in the general presentation, then executing the state- ment polys = Polygons(some_filename) followed by polys.display() should have the effect of produc- ing a file named some_filename.tex that can be given as argument to pdflatex to generate a file named some_filename.pdf. The provided examples will show you what some_filename.tex should contain.

• Polygons are drawn from lowest to highest depth, and for a given depth, the same ordering as previously described is used.

• The point that determines the polygon index is used as a starting point in drawing the line segments that make up the polygon, in a clockwise manner.

• A polygons’s colour is determined by its area. The largest polygons are yellow. The smallest polygons are orange. Polygons in-between mix orange and yellow in proportion of their area. For instance, a polygon whose size is 25% the difference of the size between the largest and the smallest polygon will receive 25% of orange (and 75% of yellow). That proportion is computed as an integer. When the value is not an integer, it is rounded to the closest integer, with values of the form z.5 rounded up to z + 1.

Pay attention to the expected format, including spaces and blank lines. Lines that start with % are comments. The output of your program redirected to a file will be compared with the expected output saved in a file (of a