UnivHypGeom10: Orthocenters exist!

In classical hyperbolic geometry, orthocenters of triangles do not in general exist. Here in universal hyperbolic geometry, they do. This is a crucial building block for triangle geometry in this subject. The dual of an orthocenter is called an ortholine---also not seen in classical hyperbolic geometry!

This lecture also introduces a number of basic important definitions: that of side, vertex, couple, triangle, trilateral. We also introduce Desargues theorem and use it to define the polar of a point with respect to a triangle. The lecture culminates in the definition of the orthic line, orthostar and ortho-axis of a triangle. The ortho-axis will prove to be the most important line in hyperbolic triangle geometry.

Category:

Education

Tags:

• HypGeom10
• universal
• hyperbolic
• geometry
• orthocenter
• duality
• altitudes
• education
• sides
• vertex
• triangle
• trilateral
• ortholine
• orthic
• axis
• orthostar
• ortho-axis
• Wildberger
• couple

[njwildberger]

Hi rationalinteger

That is not at all a silly question. First of all the geometry I am explaining here---hyperbolic geometry---has a direct generalization to the case of a more general conic. I will eventually explain this. However it is also possible to consider analogs say when one starts with a cubic curve. It turns out that we get a quite remarkable and new form of algebraic geometry! There is much more to be said about this... but I am still only learning...

Excellent question.

[rationalinteger]

I hope it's not a silly question but what I wonder if we could make geometries with curves other than the circle (conics) like the lemniscate or folium.

[njwildberger]

Hi Hardy,

Thanks for that--will try to keep that in mind.

[njwildberger]

Hi Simon

Those are both excellent questions, to which I currently do not have answers.

[peterhi503]

Excellent as always. I believe the preferred pronunciation of Giovanni Ceva's last name is "Chay-vah," and that of his last name's adjectival form is "Chay-vee-an". Hardy.

[simonsillitoe]

In the past you've illustrated the connection (equivalence?) between hyperbolic geometry and projective relativistic geometry. Given your assertion that the ortho-axis is the most important line in hyperbolic triangle geometry, is there a meaningful interpretation of it in the relativistic setting, and in particular is there a way to conceptualise what it represents in terms of the physical world around us?