• General presentation

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 22 and 5050 (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 ms 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 d⁰ < d, the set of polygons of depth d⁰ has been defined. Change in the encoding all 1s 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 1s 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 1s with some of their neighbours).

• Assessment

The assignment is worth 10 marks. the automarking script will allocate 30 seconds to each run of your program.

Late assignments will be penalised: the mark for a late submission will be the minimum of the awarded mark and 10 minus the number of full and partial days that have elapsed from the due date.

The outputs of your programs should be exactly as indicated.

• Reminder on plagiarism policy

You are permitted, indeed encouraged, to discuss ways to solve the assignment with other people. Such discussions must be in terms of algorithms, not code. But you must implement the solution on your own. Submissions are routinely scanned for similarities that occur when students copy and modify other peoples work, or work very closely together on a single implementation. Severe penalties apply.

• Examples
  • First example

Given a file named polys_1.txt whose contents is

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

11111111111111111111111111111111111111111111111111

11111111111111111111111111111111111111111111111111

11111111111111111111111111111111111111111111111111

11111111111111111111111111111111111111111111111111

11111111111111111111111111111111111111111111111111

11111111111111111111111111111111111111111111111111

11111111111111111111111111111111111111111111111111

your program when run as python3 polygons.py file polys_1.txt should output

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

Nb of invariant rotations: 4

Depth: 2 Polygon 4:

Perimeter: 68.8

Area: 295.84 Convex: yes

Nb of invariant rotations: 4

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: 11

Polygon 13:

Perimeter: 40.0

Area: 100.00

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: 20

Polygon 22:

Perimeter: 11.2

Area: 7.84

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

and when run as python3 polygons.py -print file polys_1.txt should produce some output saved in a file named polys_1.tex, which can be given as argument to pdflatex to produce a file named polys_1.pdf that views as follows.

2.2 Second example

Given a file named polys_2.txt whose contents is

00000000000000000000000000000000000000000000000000

01111111111111111111111111111111111111111111111110

00111111111111111111111111111111111111111111111100

00011111111111111111111111111111111111111111111000

01001111111111111111111111111111111111111111110010

01100111111111111111111111111111111111111111100110

01110011111111111111111111111111111111111111001110

01111001111111111111111111111111111111111110011110

01111100111111111111111111111111111111111100111110

01111110011111111111111111111111111111111001111110

01111111001111111111111111111111111111110011111110

01111111100111111111111111111111111111100111111110

01111111110011111111111111111111111111001111111110

01111111111001111111111111111111111110011111111110

01111111111100111111111111111111111100111111111110

01111111111110011111111111111111111001111111111110

01111111111111001111111111111111110011111111111110

01111111111111100111111111111111100111111111111110

01111111111111110011111111111111001111111111111110

01111111111111111001111111111110011111111111111110

01111111111111111100111111111100111111111111111110

01111111111111111110011111111001111111111111111110

01111111111111111111001111110011111111111111111110

01111111111111111111100111100111111111111111111110

01111111111011111111110011001111111111011111111110

01111111111111111111100111100111111111111111111110

01111111111111111111001111110011111111111111111110

01111111111111111110011111111001111111111111111110

01111111111111111100111111111100111111111111111110

01111111111111111001111111111110011111111111111110

01111111111111110011111111111111001111111111111110

01111111111111100111111111111111100111111111111110

01111111111111001111111111111111110011111111111110

01111111111110011111111111111111111001111111111110

01111111111100111111111111111111111100111111111110

01111111111001111111111111111111111110011111111110

01111111110011111111111111111111111111001111111110

01111111100111111111111111111111111111100111111110

01111111001111111111111111111111111111110011111110

01111110011111111111111111111111111111111001111110

01111100111111111111111111111111111111111100111110

01111001111111111111111111111111111111111110011110

01110011111111111111111111111111111111111111001110

01100111111111111111111111111111111111111111100110

01001111111111111111111111111111111111111111110010

00011111111111111111111111111111111111111111111000

00111111111111111111111111111111111111111111111100

01111111111111111111111111111111111111111111111110

00000000000000000000000000000000000000000000000000

your program when run as python3 polygons.py file polys_2.txt should output

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

Nb of invariant rotations: 1

Depth: 2

Polygon 4: Perimeter: 16.0 + 40*sqrt(.32)

Area: 64.00

Convex: yes

Nb of invariant rotations: 1

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: 2

Polygon 13:

Perimeter: 8.0 + 18*sqrt(.32)

Area: 14.40

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: 11

Polygon 22:

Perimeter: 9.6 + 24*sqrt(.32)

Area: 23.04

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: 8

Polygon 31:

Perimeter: 1.6 + 4*sqrt(.32)

Area: 0.64

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:

Nb of invariant rotations: 1

Depth: 4

Polygon 37:

Perimeter: 11.2 + 26*sqrt(.32)Area: 29 12

Perimeter: 12.8 + 30*sqrt(.32)

Area: 38.40

Convex: yes

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: 7

Polygon 40:

Perimeter: 6.4 + 14*sqrt(.32)

Area: 8.96

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

and when run as python3 polygons.py -print file polys_2.txt should produce some output saved in a file named polys_2.tex, which can be given as argument to pdflatex to produce a file named polys_2.pdf that views as follows.

2.3 Third example

Given a file named polys_3.txt whose contents is

• 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
• 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 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 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

1 1 1 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 1 1 1

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

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

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

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

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

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

• 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 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 10 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

your program when run as python3 polygons.py file polys_3.txt should output

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

Nb of invariant rotations: 1 Depth: 0

Polygon 4:

Perimeter: 17.6 + 40*sqrt(.32)

Area: 59.04

Convex: no

Nb of invariant rotations: 2

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 Depth: 0

and when run as python3 polygons.py -print file polys_3.txt should produce some output saved in a file named polys_3.tex, which can be given as argument to pdflatex to produce a file named polys_3.pdf that views as follows.

2.4 Fourth example

Given a file named polys_4.txt whose contents is

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

0100010001000100010000100010100010100011

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

11101 1101110 1 1 1 0111011101100000001111000 000000000000000000000001100000011000100 0

1 111001100111111100000000111111000 010000

110 01 0 1 1 0 1011111100011111000000000001000

001 1000011 10 000000000 11111111111111111 00

your program when run as python3 polygons.py file polys_4.txt should output

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: 0.8 + 8*sqrt(.32)

Area: 1.92

Convex: yes

Nb of invariant rotations: 2

Depth: 0 Polygon 4:

Perimeter: 3.2 + 1*sqrt(.32)

Area: 0.88

Convex: yes

Nb of invariant rotations: 1

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 13:

Perimeter: 1.6

Area: 0.16

Convex: yes

Nb of invariant rotations: 4

Depth: 0 Polygon 14:

Perimeter: 5.6 + 3*sqrt(.32)

Area: 1.36

Convex: no

Nb of invariant rotations: 1 Depth: 0

and when run as python3 polygons.py -print file polys_4.txt should produce some output saved in a file named polys_4.tex, which can be given as argument to pdflatex to produce a file named polys_4.pdf that views as follows.

• Detailed description
  • Input

The input is expected to consist of y_dim lines of x_dim 0s and 1s, where x_dim and y_dim 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 x^th digit of the y^th line with digits, with 0 x < x_dim and 0 y < y_dim, 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.

• Output

The program should be run as either

python3 polygons.py filefilename.txt

or as

python3 polygons.py -print filefilename.txt

(where filename.txt is the name of a file that stores the input). You can study the program ascii_art.py from Lecture 7 to find out how this can be done.

If the input is incorrect, that is, does not satisfy the conditions spelled out in the previous section, then the program should print out a single line that reads

Incorrect input. and immediately exit.

3.2.1 When the program is run without -print as command-line argument

If the input is correct, then the program should output a first line that reads one of

Cannot get polygons as expected.

in case it is not possible to use all 1s in the input and make them the contours of polygons of depth d, for any natural number d, as defined in the general presentation. Otherwise, the program should output a first line that reads

Polygon N:

with N an appropriate integer at least equal to 1 to refer to the Nth 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. Note that your program should output no blank line. For a given test, the output of your program will be compared with the expected output; your program will pass the test if and only if both outputs are absolutely identical, character for character, including spaces. For the provided examples, the expected outputs are available in files that end in _output.txt. To check that the output of your program on those examples is correct, you can redirect it to a file and compare the contents of that file with the contents of the appropriate _output.txt file using the diff command. If diff silently exits then your program passes the test; otherwise it fails it. For instance, run

python3 polygons.py file polys_1.txt >polys_1_my_output.txt

and then

diff polys_1_my_output.txt polys_1_output.txt

to check whether your program succeeds on the first provided example.

3.2.2 When the program is run with -print as command-line argument

If the input is correct, then the program should output some lines saved in a file named filename.tex, that can be given as an argument to pdflatex to produce a file named filename.pdf that depicts the maze. The provided examples will show you what 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 polygonss 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 5 rounded up to z + 1.

Pay attention to the expected format, including spaces and blank lines. Lines that start with % are comments. The contents of the file output by your program will be compared with the expected output (saved in a file of a different name of course) using the diff command. For your program to pass the associated test, diff should silently exit, which requires that the contents of both files be absolutely identical, character for character, including spaces and blank lines. Check your program on the provided examples using the associated .tex files. For instance, rename the provided file polys_1.tex to polys_1_expected.tex, and then run

python3 polygons.py -print file polys_1.txt

and then

diff polys_1.tex polys_1_expected.tex

to check whether your program succeeds on the first provided example.