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'line in space'
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| Title of object: |
line in space |
| Canonical Name: |
LineInSpace |
| Type: |
Topic |
| Created on: |
2007-08-16 12:43:42 |
| Modified on: |
2007-08-16 12:43:42 |
| Classification: |
msc:51N20, msc:53A04 |
| Synonyms: |
line in space=line in $\mathbb{R}^3$
line in space=space line |
Revision comment (for changes between this and next version):
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{xypic}
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\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
A line $l$ in the space $\mathbb{R}^3$ can be thought as the intersection of two planes
and therefore be represented by two simultaneous equations of planes in the form
\begin{align}
\begin{cases}
A_1x+B_1y+C_1z+D_1 = 0\\
A_2x+B_2y+C_2z+D_2 = 0,
\end{cases}
\end{align}
where $A_i$, $B_i$, $C_i$, $D_i$ are real \PMlinkescapetext{constants} and\, $A_i^2+B_i^2+C_i^2 > 0$.
Eliminating one of the variables $x$, $y$, $z$ from the pair of equations one gets the projection of the line on the corresponding coordinate plane. If one e.g. first elininates $x$ and secondly $y$, one gets the equations of the form
\begin{align}
y = mz+p, \quad x = nz+q.
\end{align}
The value triples \,$(x,\,y,\,z)$\, satisfying (1) satisfy also (2), and hence (2) represents the same line as well. Separately, both of the equations (2) \PMlinkescapetext{mean} planes, of which the former is parallel to the $x$-axis and the latter to the $y$-axis. The line itself is the intersection of these planes, and can be projected along the planes on the $yz$-plane and the $zx$-plane, respectively.
In a narrower meaning, the equation\, $y = mz+p$\, represents the projection line of $l$ on the $yz$-plane and the equation\, $x = nz+q$\, the projection line of $l$ on the $zx$-plane.\\
Not ready...
\begin{thebibliography}{8}
\bibitem{LL}{\sc L. Lindel\"of}: {\em Analyyttisen geometrian oppikirja}.\, Kolmas painos.\, Suomalaisen Kirjallisuuden Seura, Helsinki (1924).
\end{thebibliography} |
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