# COMP9021代做、代写Python程序语言

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COMP9021, Trimester 3, 2023

1. General matter

1.1. 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.

1.2. 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.

1.3. 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.

1.4. 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 people’s work, or work very closely together on a single implementation. Severe penalties

apply.

2. 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 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 d

0 < d, the set of polygons of depth d

0 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

2

3. Examples

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

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

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

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

5

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

>>> polys.display()

6

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.

7

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

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

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

9

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

10

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

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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

12

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: 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

>>> polys.display()

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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.

14

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

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

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

16

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

>>> polys.display()

The effect of executing polys.display() is to produce a file named polys_3.tex that can be given as

argument to pdflatex to produce a file named polys_3.pdf that views as follows.

17

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

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

Nb of invariant rotations: 1

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

19

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

>>> 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.

20

4. Detailed description

4.1. 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 x

th digit of the y

th 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.

4.2. 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

Traceback (most recent call last):

...

polygons.PolygonsError: Cannot get polygons as expected.

If the input is correct and it is 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 executing the statement

polys = Polygons(some_filename) followed by polys.analyse() should have the effect of outputting a first

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

21

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 polygons of depth d, for any natural number d, as defined in the general presentation, then executing the statement polys = Polygons(some_filename) followed by polys.display() should have the effect of producing 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

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,

renaming them as they have the names of the files expected to be generated by your program.

COMP9021, Trimester 3, 2023

1. General matter

1.1. 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.

1.2. 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.

1.3. 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.

1.4. 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 people’s work, or work very closely together on a single implementation. Severe penalties

apply.

2. 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 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 d

0 < d, the set of polygons of depth d

0 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

2

3. Examples

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

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

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

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

5

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

>>> polys.display()

6

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.

7

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

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

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

9

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

10

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

11

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

12

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: 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

>>> polys.display()

13

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.

14

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

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

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

16

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

>>> polys.display()

The effect of executing polys.display() is to produce a file named polys_3.tex that can be given as

argument to pdflatex to produce a file named polys_3.pdf that views as follows.

17

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

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

Nb of invariant rotations: 1

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

19

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

>>> 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.

20

4. Detailed description

4.1. 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 x

th digit of the y

th 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.

4.2. 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

Traceback (most recent call last):

...

polygons.PolygonsError: Cannot get polygons as expected.

If the input is correct and it is 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 executing the statement

polys = Polygons(some_filename) followed by polys.analyse() should have the effect of outputting a first

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

21

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 polygons of depth d, for any natural number d, as defined in the general presentation, then executing the statement polys = Polygons(some_filename) followed by polys.display() should have the effect of producing 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

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,

renaming them as they have the names of the files expected to be generated by your program.