[Classical Geometries] Is this a model of geometry? Verify by showing that they 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] Is this a model of geometry? Verify by showing that they 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).
Computer Networking: A Top-Down Approach (7th Edition)
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[Classical Geometries] Is this a model of geometry? Verify by showing that they 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).
![3. H
• point: Point on the upper sheet of the hyperboloid.
H² = {(x, y, z) = R³|z² = x² + y² + 1, z>0}
● line: great hyperbola 2 on H².
lie on: A point p
(x, y, z) = 1.
Discuss if it is a model of geometry, by verifying the 3 axioms one by one.
=
(x, y, z) = H² lies on a great hyperbola 1 if](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff884895d-a63c-4839-b2c7-9f1431d12b88%2F0ef96256-7710-4f2a-9b85-e87d7bfcc541%2F8o8mqz_processed.png&w=3840&q=75)
Transcribed Image Text:3. H
• point: Point on the upper sheet of the hyperboloid.
H² = {(x, y, z) = R³|z² = x² + y² + 1, z>0}
● line: great hyperbola 2 on H².
lie on: A point p
(x, y, z) = 1.
Discuss if it is a model of geometry, by verifying the 3 axioms one by one.
=
(x, y, z) = H² lies on a great hyperbola 1 if
![2A great hyperbola is the intersection of H2 with any plane through the origin of R³.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff884895d-a63c-4839-b2c7-9f1431d12b88%2F0ef96256-7710-4f2a-9b85-e87d7bfcc541%2Futb2v3g_processed.png&w=3840&q=75)
Transcribed Image Text:2A great hyperbola is the intersection of H2 with any plane through the origin of R³.
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