UnivHypGeom4: First steps in hyperbolic geometry

[xuanji07]

Hi Prof Norman, I was wondering what is the difference between classical hyperbolic geometry and your "universal hyperbolic geometry"? From earlier videos I thought that they are the same theory and your development of it is just a different exposition, but if orthocenters don't necessarily exist in classical hyperbolic geometry then it seems that the two theories are different.

[njwildberger]

@xuanji07 There are many differences. For one thing, universal hyperbolic geometry is a logically correct theory, while classical hyperbolic geometry is not. For another, UHG works over a general field, classical HG only over the `real numbers'. For another the formulas in UHG work also for points and lines outside the usual Beltrami Poincare disk. However the mathematical reality that both theories try to capture is essentially the same. UHG is just a lot more successful at capturing that.

[brangelito]

I appreciated this lecture. Thank you!

[Junkbox09]

hi norman. just asking a question about the slide on the altitudes meeting in a point. is the choice of points of the triangle arbitrary? if yes, would that mean that there would be many orthocentres depending on the choice of points?

[njwildberger]

Hi Junkbox09 The three points forming the triangle are general. ALmost any three points work---there are a few exceptions. But for any general three points the three altitudes meet in exactly one point, so each triangle has exactly one orthocenter.

[grichard24]

On page 3 (definition of perpendicular) if points a and b are collinear with the center of the circle, wouldn't lines A and B be non-intersecting and if so, would they still be described as perpendicular?

[njwildberger]

Hi grichard24,

Yes A and B in that case would still be perpendicular lines. At this point I have only referred indirectly to `points at infinity', but shortly we will see that they are part of the story of hyperbolic geometry. This is what is meant by a `projective' view. So then the lines A and B would meet (in fact any two lines meet).

[rationalinteger]

I am enjoying this series a lot!

I was wondering if spread can sometimes be infinity then what number system is being used?

[njwildberger]

Hi rationalinteger, Let's just say at this point that if one of a or b is a null point, then the quadrance q(a,b) is undefined: the ratio involved has a denominator 0. However at some later point we will consider the possibility that we can extend our number system to include a value "infinity" (still in a completely rigorous way) so that this kind of quadrance still has a meaning.