Python代写：COMP9021Polygons

代写一个绘图程序，根据文本文件中的内容，根据算法生成对应的图形。

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 dierent) 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 diers 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 d0 < d , the set of polygons of depth d0 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).

Submission

Your programs will be stored in a file named polygons.py. After you have
developed and tested your program, upload your files using Ed. Assignments can
be submitted more than once: the last version is marked.

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.

Examples

First example

Given a file named polys_1.txt whose contents is
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.

Second example

Given a file named polys_2.txt whose contents is
your program when run as python3 polygons.py –file polys_2.txt should output
Polygon 1:
Perimeter: 37.6 + 92sqrt(.32)
Area: 176.64
Convex: no
Nb of invariant rotations: 2
Depth: 0
Polygon 2:
Perimeter: 17.6 + 42sqrt(.32)
Area: 73.92
Convex: yes
Nb of invariant rotations: 1
Depth: 1
Polygon 3:
Perimeter: 16.0 + 38sqrt(.32)
Area: 60.80
Convex: yes
Nb of invariant rotations: 1
Depth: 2
Polygon 4:
Perimeter: 16.0 + 40sqrt(.32)
Area: 64.00
Convex: yes
Nb of invariant rotations: 1
Depth: 0
Polygon 5:
Perimeter: 14.4 + 34sqrt(.32)
Area: 48.96
Convex: yes
Nb of invariant rotations: 1
Depth: 3
Polygon 6:
Perimeter: 16.0 + 40sqrt(.32)
Area: 64.00
Convex: yes
Nb of invariant rotations: 1
Depth: 0
Polygon 7:
Perimeter: 12.8 + 30sqrt(.32)
Area: 38.40
Convex: yes
Nb of invariant rotations: 1
Depth: 4
Polygon 8:
Perimeter: 14.4 + 36sqrt(.32)
Area: 51.84
Convex: yes
Nb of invariant rotations: 1
Depth: 1
Polygon 9:
Perimeter: 11.2 + 26sqrt(.32)
Area: 29.12
Convex: yes
Nb of invariant rotations: 1
Depth: 5
Polygon 10:
Perimeter: 14.4 + 36sqrt(.32)
Area: 51.84
Convex: yes
Nb of invariant rotations: 1
Depth: 1
Polygon 11:
Perimeter: 9.6 + 22sqrt(.32)
Area: 21.12
Convex: yes
Nb of invariant rotations: 1
Depth: 6
Polygon 12:
Perimeter: 12.8 + 32sqrt(.32)
Area: 40.96
Convex: yes
Nb of invariant rotations: 1
Depth: 2
Polygon 13:
Perimeter: 8.0 + 18sqrt(.32)
Area: 14.40
Convex: yes
Nb of invariant rotations: 1
Depth: 7
Polygon 14:
Perimeter: 12.8 + 32sqrt(.32)
Area: 40.96
Convex: yes
Nb of invariant rotations: 1
Depth: 2
Polygon 15:
Perimeter: 6.4 + 14sqrt(.32)
Area: 8.96
Convex: yes
Nb of invariant rotations: 1
Depth: 8
Polygon 16:
Perimeter: 11.2 + 28sqrt(.32)
Area: 31.36
Convex: yes
Nb of invariant rotations: 1
Depth: 3
Polygon 17:
Perimeter: 4.8 + 10sqrt(.32)
Area: 4.80
Convex: yes
Nb of invariant rotations: 1
Depth: 9
Polygon 18:
Perimeter: 11.2 + 28sqrt(.32)
Area: 31.36
Convex: yes
Nb of invariant rotations: 1
Depth: 3
Polygon 19:
Perimeter: 3.2 + 6sqrt(.32)
Area: 1.92
Convex: yes
Nb of invariant rotations: 1
Depth: 10
Polygon 20:
Perimeter: 9.6 + 24sqrt(.32)
Area: 23.04
Convex: yes
Nb of invariant rotations: 1
Depth: 4
Polygon 21:
Perimeter: 1.6 + 2sqrt(.32)
Area: 0.32
Convex: yes
Nb of invariant rotations: 1
Depth: 11
Polygon 22:
Perimeter: 9.6 + 24sqrt(.32)
Area: 23.04
Convex: yes
Nb of invariant rotations: 1
Depth: 4
Polygon 23:
Perimeter: 8.0 + 20sqrt(.32)
Area: 16.00
Convex: yes
Nb of invariant rotations: 1
Depth: 5
Polygon 24:
Perimeter: 8.0 + 20sqrt(.32)
Area: 16.00
Convex: yes
Nb of invariant rotations: 1
Depth: 5
Polygon 25:
Perimeter: 6.4 + 16sqrt(.32)
Area: 10.24
Convex: yes
Nb of invariant rotations: 1
Depth: 6
Polygon 26:
Perimeter: 6.4 + 16sqrt(.32)
Area: 10.24
Convex: yes
Nb of invariant rotations: 1
Depth: 6
Polygon 27:
Perimeter: 4.8 + 12sqrt(.32)
Area: 5.76
Convex: yes
Nb of invariant rotations: 1
Depth: 7
Polygon 28:
Perimeter: 4.8 + 12sqrt(.32)
Area: 5.76
Convex: yes
Nb of invariant rotations: 1
Depth: 7
Polygon 29:
Perimeter: 3.2 + 8sqrt(.32)
Area: 2.56
Convex: yes
Nb of invariant rotations: 1
Depth: 8
Polygon 30:
Perimeter: 3.2 + 8sqrt(.32)
Area: 2.56
Convex: yes
Nb of invariant rotations: 1
Depth: 8
Polygon 31:
Perimeter: 1.6 + 4sqrt(.32)
Area: 0.64
Convex: yes
Nb of invariant rotations: 1
Depth: 9
Polygon 32:
Perimeter: 1.6 + 4sqrt(.32)
Area: 0.64
Convex: yes
Nb of invariant rotations: 1
Depth: 9
Polygon 33:
Perimeter: 17.6 + 42sqrt(.32)
Area: 73.92
Convex: yes
Nb of invariant rotations: 1
Depth: 1
Polygon 34:
Perimeter: 16.0 + 38sqrt(.32)
Area: 60.80
Convex: yes
Nb of invariant rotations: 1
Depth: 2
Polygon 35:
Perimeter: 14.4 + 34sqrt(.32)
Area: 48.96
Convex: yes
Nb of invariant rotations: 1
Depth: 3
Polygon 36:
Perimeter: 12.8 + 30sqrt(.32)
Area: 38.40
Convex: yes
Nb of invariant rotations: 1
Depth: 4
Polygon 37:
Perimeter: 11.2 + 26sqrt(.32)
Area: 29.12
Convex: yes
Nb of invariant rotations: 1
Depth: 5
Polygon 38:
Perimeter: 9.6 + 22sqrt(.32)
Area: 21.12
Convex: yes
Nb of invariant rotations: 1
Depth: 6
Polygon 39:
Perimeter: 8.0 + 18sqrt(.32)
Area: 14.40
Convex: yes
Nb of invariant rotations: 1
Depth: 7
Polygon 40:
Perimeter: 6.4 + 14sqrt(.32)
Area: 8.96
Convex: yes
Nb of invariant rotations: 1
Depth: 8
Polygon 41:
Perimeter: 4.8 + 10sqrt(.32)
Area: 4.80
Convex: yes
Nb of invariant rotations: 1
Depth: 9
Polygon 42:
Perimeter: 3.2 + 6sqrt(.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.

Third example

Given a file named polys_3.txt whose contents is
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
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
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
your program when run as python3 polygons.py –file polys_3.txt should output
Polygon 1:
Perimeter: 2.4 + 9sqrt(.32)
Area: 2.80
Convex: no
Nb of invariant rotations: 1
Depth: 0
Polygon 2:
Perimeter: 51.2 + 4sqrt(.32)
Area: 117.28
Convex: no
Nb of invariant rotations: 2
Depth: 0
Polygon 3:
Perimeter: 2.4 + 9sqrt(.32)
Area: 2.80
Convex: no
Nb of invariant rotations: 1
Depth: 0
Polygon 4:
Perimeter: 17.6 + 40sqrt(.32)
Area: 59.04
Convex: no
Nb of invariant rotations: 2
Depth: 1
Polygon 5:
Perimeter: 3.2 + 28sqrt(.32)
Area: 9.76
Convex: no
Nb of invariant rotations: 1
Depth: 2
Polygon 6:
Perimeter: 27.2 + 6sqrt(.32)
Area: 5.76
Convex: no
Nb of invariant rotations: 1
Depth: 2
Polygon 7:
Perimeter: 4.8 + 14sqrt(.32)
Area: 6.72
Convex: no
Nb of invariant rotations: 1
Depth: 1
Polygon 8:
Perimeter: 4.8 + 14sqrt(.32)
Area: 6.72
Convex: no
Nb of invariant rotations: 1
Depth: 1
Polygon 9:
Perimeter: 3.2 + 2sqrt(.32)
Area: 1.12
Convex: yes
Nb of invariant rotations: 1
Depth: 2
Polygon 10:
Perimeter: 3.2 + 2sqrt(.32)
Area: 1.12
Convex: yes
Nb of invariant rotations: 1
Depth: 2
Polygon 11:
Perimeter: 2.4 + 9sqrt(.32)
Area: 2.80
Convex: no
Nb of invariant rotations: 1
Depth: 0
Polygon 12:
Perimeter: 2.4 + 9sqrt(.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.

Detailed description

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.

Output

The program should be run as either
python3 polygons.py –file filename.txt
or as
python3 polygons.py -print –file filename.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.

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 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.
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 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 + bsqrt(.32)
Perimeter: a
Perimeter: bsqrt(.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.

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 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 dierence 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 contents of the file output by your
  program will be compared with the expected output (saved in a file of a
  dierent 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.