In general for each point
System Message: WARNING/2 ((x_{0}, y_{0}))
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
, we can define the family of lines that goes through that point as:
System Message: WARNING/2 (r_{\theta} = x_{0} \cdot \cos \theta + y_{0} \cdot \sin \theta)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
Meaning that each pair
System Message: WARNING/2 ((r_{\theta},\theta))
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
represents each line that passes by
System Message: WARNING/2 ((x_{0}, y_{0}))
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
.
If for a given
System Message: WARNING/2 ((x_{0}, y_{0}))
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
we plot the family of lines that goes through it, we get a sinusoid. For instance, for
System Message: WARNING/2 (x_{0} = 8)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
and
System Message: WARNING/2 (y_{0} = 6)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
we get the following plot (in a plane
System Message: WARNING/2 (\theta)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
-
System Message: WARNING/2 (r)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
):
We consider only points such that
System Message: WARNING/2 (r > 0)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
and
System Message: WARNING/2 (0< \theta < 2 \pi)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
.
We can do the same operation above for all the points in an image. If the curves of two different points intersect in the plane
System Message: WARNING/2 (\theta)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
-
System Message: WARNING/2 (r)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
, that means that both points belong to a same line. For instance, following with the example above and drawing the plot for two more points:
System Message: WARNING/2 (x_{1} = 9)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
,
System Message: WARNING/2 (y_{1} = 4)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
and
System Message: WARNING/2 (x_{2} = 12)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
,
System Message: WARNING/2 (y_{2} = 3)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
, we get:
The three plots intersect in one single point
System Message: WARNING/2 ((0.925, 9.6))
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
, these coordinates are the parameters (
System Message: WARNING/2 (\theta, r)
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
) or the line in which
System Message: WARNING/2 ((x_{0}, y_{0}))
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
,
System Message: WARNING/2 ((x_{1}, y_{1}))
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
and
System Message: WARNING/2 ((x_{2}, y_{2}))
latex exited with error
[stdout]
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2014/05/01>
Babel <3.9k> and hyphenation patterns for 78 languages loaded.
(/usr/local/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/local/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.165 \endinput
^^M
No pages of output.
Transcript written on math.log.
lay.