[Classical Geometries] Does RP2 follow the three incidence axioms: (I1) Given any two discrete points there exists a unique line containing them (I2) Given any line there exists at least two distinct points lying on it (I3) There exists three non-collinear points We will use Hilbert’s axiom system that is constructed with the three primitive terms point, line and plane (lie on).
[Classical Geometries] Does RP2 follow the three incidence axioms: (I1) Given any two discrete points there exists a unique line containing them (I2) Given any line there exists at least two distinct points lying on it (I3) There exists three non-collinear points We will use Hilbert’s axiom system that is constructed with the three primitive terms point, line and plane (lie on).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Classical Geometries] Does RP2 follow the three incidence axioms:
(I1) Given any two discrete points there exists a unique line containing them
(I2) Given any line there exists at least two distinct points lying on it
(I3) There exists three non-collinear points
We will use Hilbert’s axiom system that is constructed with the three primitive terms point, line and plane (lie on).
![2. Rp²
lowing:
Consider the RP2 Geometry, whose interpretation is the fol-
point: equivalent classes of points on the sphere, obtained by identify
anti-polar pairs, i.e.,
RP² = {[x, y, z]|(x, y, z) = x² + y² + z² = 1,
(x, y, z) ~ (-x, -y, -z)}.
line The great circles of the unit sphere.
● lie on: A point p = [x, y, z] € RP² lies on a line 7 if (x, y, z) ≤ l.
E
Discuss if it is a model of geometry, by verifying the 3 axioms one by one.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff884895d-a63c-4839-b2c7-9f1431d12b88%2F304f458f-67b6-41f8-9930-6030d3938f33%2Fofztk9p_processed.png&w=3840&q=75)
Transcribed Image Text:2. Rp²
lowing:
Consider the RP2 Geometry, whose interpretation is the fol-
point: equivalent classes of points on the sphere, obtained by identify
anti-polar pairs, i.e.,
RP² = {[x, y, z]|(x, y, z) = x² + y² + z² = 1,
(x, y, z) ~ (-x, -y, -z)}.
line The great circles of the unit sphere.
● lie on: A point p = [x, y, z] € RP² lies on a line 7 if (x, y, z) ≤ l.
E
Discuss if it is a model of geometry, by verifying the 3 axioms one by one.
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