实现一个 Frieze
程序，能根据文本内容显示不同的图案。
General presentation
You will design and implement a program that will
- check whether some numbers, stored in a file, represent a particular coding of a frieze, and
- either display the period of the pattern of the frieze and the transformations that keep it invariant, based on a result that classifies friezes into 7 groups of symmetries,
- or output some Latex code in a file, from which a pictorial representation of the frieze can be produced.
The representation of a frieze is based on a coding with numbers in the range
0 15, each such number n being associated with a particular point p such that - if the rightmost digit of the representation of n in base 2 is equal to 1 then p is to be connected to its northern neighbour:
- if the second rightmost digit of the representation of n in base 2 is equal to 1 then p is to be connected to its north-eastern neighbour:
- if the third rightmost digit of the representation of n in base 2 is equal to 1 then p is to be connected to its eastern neighbour:
- if the fourth rightmost digit of the representation of n in base 2 is equal to 1 then p is to be connected to its south-eastern neighbour:
Examples
First example
The file frieze_1.txt has the following contents.
Here is a possible interaction:
>>> from frieze import *
>>> frieze = Frieze(‘frieze_1.txt’)
>>> frieze.analyse()
Pattern is a frieze of period 15 that is invariant under translation only.
>>> frieze.display()
—|—
The effect of executing frieze.display() is to produce a file named
frieze_1.tex that can be given as argument to pdflatex to produce a file named
frieze_1.pdf that views as follows.
Second example
The file frieze_2.txt has the following contents.
Here is a possible interaction:
>>> from frieze import *
>>> frieze = Frieze(‘frieze_2.txt’)
>>> frieze.analyse()
Pattern is a frieze of period 12 that is invariant under translation and vertical reflection only.
>>> frieze.display()
—|—
The effect of executing frieze.display() is to produce a file named
frieze_2.tex that can be given as argument to pdflatex to produce a file named
frieze_2.pdf that views as follows.
Third example
The file frieze_3.txt has the following contents.
Here is a possible interaction:
>>> from frieze import *
>>> frieze = Frieze(‘frieze_3.txt’)
>>> frieze.analyse()
Pattern is a frieze of period 3 that is invariant under translation and horizontal reflection only.
>>> frieze.display()
—|—
The effect of executing frieze.display() is to produce a file named
frieze_3.tex that can be given as argument to pdflatex to produce a file named
frieze_3.pdf that views as follows.
Fourth example
The file frieze_4.txt has the following contents.
Here is a possible interaction:
>>> from frieze import *
>>> frieze = Frieze(‘frieze_4.txt’)
>>> frieze.analyse()
Pattern is a frieze of period 6 that is invariant under translation and glided horizontal reflection only.
>>> frieze.display()
—|—
The effect of executing frieze.display() is to produce a file named
frieze_4.tex that can be given as argument to pdflatex to produce a file named
frieze_4.pdf that views as follows.
Fifth example
The file frieze_5.txt has the following contents.
Here is a possible interaction:
>>> from frieze import *
>>> frieze = Frieze(‘frieze_5.txt’)
>>> frieze.analyse()
Pattern is a frieze of period 8 that is invariant under translation and rotation only.
>>> frieze.display()
—|—
The effect of executing frieze.display() is to produce a file named
frieze_5.tex that can be given as argument to pdflatex to produce a file named
frieze_5.pdf that views as follows.
Sixth example
The file frieze_6.txt has the following contents.
Here is a possible interaction:
>>> from frieze import *
>>> frieze = Frieze(‘frieze_6.txt’)
>>> frieze.analyse()
Pattern is a frieze of period 4 that is invariant under translation, glided horizontal and vertical reflections, and rotation only.
>>> frieze.display()
—|—
The effect of executing frieze.display() is to produce a file named
frieze_6.tex that can be given as argument to pdflatex to produce a file named
frieze_6.pdf that views as follows.
Seventh example
The file frieze_7.txt has the following contents.
Here is a possible interaction:
>>> from frieze import *
>>> frieze = Frieze(‘frieze_7.txt’)
>>> frieze.analyse()
Pattern is a frieze of period 4 that is invariant under translation, horizontal and vertical reflections, and rotation only.
>>> frieze.display()
—|—
The effect of executing frieze.display() is to produce a file named
frieze_7.tex that can be given as argument to pdflatex to produce a file named
frieze_7.pdf that views as follows.
Detailed description
Input
The input is expected to consist of height + 1 lines of length + 1
numbers in {0, …, 15}, where length is at least equal to 4 and at most equal
to 50 and height is at least equal to 2 and at most equal to 16, with possibly
lines consisting of spaces only that will be ignored and with possibly spaces
anywhere on the lines with numbers. The xth number n of the y th line, with 0
x length and 0 y height,
- is to be associated with a point situated x ? 0.2 cm to the right and y? 0.2 cm below an origin,
- is to be connected to the point 0.2 cm above if the rightmost digit of n is 1,
- is to be connected to the point 0.2 cm above and 0.2 cm to the right if the second rightmost digit of n is 1,
- is to be connected to the point 0.2 cm to the right if the third rightmost digit of n is 1, and
- is to be connected to the point 0.2 cm to the right and 0.2 cm below if the fourth rightmost digit of n is 1.
To qualify as a frieze, the input is further constrained to fit in a rectangle
of length length ? 0.2 cm and of height heigth ? 0.2 cm, with horizontal lines
of length length at the top and at the bottom, identical vertical borders at
both ends, no crossing segments connecting pairs of neighbours inside the
rectangle, and a pattern of integral period at least equal to 2 that is fully
repeated at least twice in the horizontal dimension.
Output
Consider executing from the Python prompt the statement from frieze import * followed by the statement frieze = Frieze(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 numbers in {0, . . . , 15} besides spaces, or
in that it contains either too few or too many lines of numbers, or in that
some line of numbers contains too many or too few numbers, or in that two of
its lines of numbers do not contain the same number of numbers, then the
effect of executing frieze = Frieze(some\_filename) should be to generate
a FriezeError exception that reads
Traceback (most recent call last):
…
frieze.FriezeError: Incorrect input.
If the previous conditions hold but the further conditions spelled out above
for the input to qualify as a frieze do not hold, then the effect of executingfrieze = Frieze(some_filename) should be to generate a FriezeError
exception that reads
Traceback (most recent call last):
…
frieze.FriezeError: Input does not represent a frieze.
If the input is correct and represents a frieze, then executing frieze = Frieze(some_filename) followed by frieze.analyse() should have the
effect of outputting one or two lines that read
Pattern is a frieze of period N that is invariant under translation only. or
Pattern is a frieze of period N that is invariant under translation and
vertical reflection only. or
Pattern is a frieze of period N that is invariant under translation and
horizontal reflection only. or
Pattern is a frieze of period N that is invariant under translation and glided
horizontal reflection only. or
Pattern is a frieze of period N that is invariant under translation and
rotation only. or
Pattern is a frieze of period N that is invariant under translation, glided
horizontal and vertical reflections, and rotation only. or
Pattern is a frieze of period N that is invariant under translation,
horizontal and vertical reflections, and rotation only. with N an appropriate
integer at least equal to 2.
These 7 possible outputs are based on a mathematical result on the
classification of friezes that lists all possible complete lists of symmetries
that leave a frieze invariant under an isometry (that is, a transformation
that does not alter the distance between any two points). These possible lists
involve 5 symmetries.
- Translation by period; of course, any frieze is invariant under this symmetry.
- Vertical reflection about some vertical line; that line does not necessarily delimit the pattern nor does it necessarily go through its middle (these conditions are actually equivalent).
- Horizontal reflection about the line that goes through the middle of the frieze.
- Glided horizontal reflection, that is, horizontal reflection about the line that goes through the middle of the frieze and translation by half the period of the resulting lower half of the frieze.
- Rotation around some point situated on the horizontal line that goes through the middle of the frieze; this is equivalent to horizontal refection combined with vertical reflection.
Pay attention to the expected format, including spaces.
If the input is correct and represents a frieze, then executingfrieze = Frieze(some_filename)followed byfrieze.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. Segments are
drawn in purple with a single draw command for each longest segment, - starting with the vertical segments, from the topmost leftmost one to the bottommost rightmost one with the leftmost ones first,
- followed by the segments that go from north west to south east, from the topmost leftmost one to the bottommost rightmost one with the topmost ones first,
- followed by the segments that go from west to east, from the topmost leftmost one to the bottommost rightmost one with the topmost ones first,
- followed by the segments that go from the south west to the north east, from the topmost leftmost one to the bottommost rightmost one with the topmost ones first.
Pay attention to the expected format, including spaces and blank lines. Lines
that start with % are comments; there are 4 such lines, that have to be
present even when there is no item to be displayed of the kind described by
the comment. 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. When
testing locally, 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.
