Understanding the Basics of Extended Plane and Space

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SUMMARY

The discussion clarifies the concepts of extended plane and extended space in geometry. It establishes that while every plane in regular space corresponds to a unique extended plane in extended space, there is only one ideal line for each plane when two distinct planes intersect. The distinction between extended plane and extended space is emphasized, with the latter encompassing multiple extended planes. The conversation highlights the need for clearer educational resources on these topics.

PREREQUISITES
  • Understanding of basic geometric concepts, including planes and lines.
  • Familiarity with the definition of ideal points in geometry.
  • Knowledge of the properties of parallel lines in regular space.
  • Basic comprehension of dimensionality in geometric contexts.
NEXT STEPS
  • Research the concept of ideal points and lines in projective geometry.
  • Study the differences between Euclidean and non-Euclidean geometries.
  • Explore resources on projective planes and their properties.
  • Learn about the implications of dimensionality in extended geometric spaces.
USEFUL FOR

Students of geometry, educators seeking clearer explanations of advanced concepts, and anyone interested in the foundational principles of extended geometric spaces.

*best&sweetest*
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Could anybody please explain the concept of extended plane and extended space? I know the definition of an ideal point and that's the only part that I really understand. If each two distinct planes (in extended space) meet in just one line, does it mean that there's only one ideal line for each plane?
What's the difference between extended plane and extended space? There's only one extended plane in extended space, right?
I know that for some of you these questions will sound very silly, but the book I have has a very poor explanation. If you know some web page where I can learn the basics, please tell me. Thank you!
 
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I don't believe the term "extended space" is standard. There are many ways to "extend space".

If each two distinct planes (in extended space) meet in just one line, does it mean that there's only one ideal line for each plane?
Yes, that's true. In particular planes that are "parallel" in regular space meet in that one "ideal" line (and a pair of lines, one on each plane, meet in an "ideal" point on that "ideal" line.

What's the difference between extended plane and extended space? There's only one extended plane in extended space, right?
No, every plane in regular space corresponds to a different extended plane in extended space.
 

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