Asymptotes of hyperbolic sections of a given cone

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SUMMARY

The discussion centers on the geometric properties of hyperbolic sections of a cone, specifically regarding the angles formed by their asymptotes. It is established that while all hyperbolic sections of the same cone share a common angle between their asymptotes, this angle can vary based on the orientation of the intersecting plane. The book "Companion to Concrete Math Vol. I" by Melzak highlights that only hyperbolas with asymptotes at sufficiently small angles can occur as plane sections of a fixed cone. The participants clarify that tilting the intersecting plane alters the angle of the asymptotes.

PREREQUISITES
  • Understanding of conic sections, specifically hyperbolas and ellipses.
  • Familiarity with the geometry of cones and their sections.
  • Knowledge of angles and their measurement in geometric contexts.
  • Basic principles of plane geometry and intersection theory.
NEXT STEPS
  • Study the properties of conic sections in detail, focusing on hyperbolas.
  • Explore the geometric implications of intersecting planes with cones.
  • Learn about the mathematical definitions and properties of asymptotes.
  • Investigate the relationship between angles formed by conic sections and their geometric representations.
USEFUL FOR

Mathematicians, geometry enthusiasts, educators teaching conic sections, and students studying advanced geometry concepts will benefit from this discussion.

imurme8
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A book I'm reading (Companion to Concrete Math Vol. I by Melzak) mentions, "...any ellipse occurs as a plane section of any given cone. This is not the case with hyperbolas: for a fixed cone only those hyperbolas whose asymptotes make a sufficiently small angle occur as plane sections."

It seems to me that all hyperbolic sections of the same cone must have asymptotes that make exactly the same angle with each other (the angle formed by two antipodal generators of the cone). Is this incorrect? The wording in the book suggests their angles fall a range of values.
 
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hi imurme8! :smile:
imurme8 said:
It seems to me that all hyperbolic sections of the same cone must have asymptotes that make exactly the same angle with each other (the angle formed by two antipodal generators of the cone).

no, the asymptotes will be parallel to the intersection of the cone with the parallel plane through the vertex …

tilt the plane away from the "vertical", and you reduce the angle :wink:
 
Thank you, I see it now. :)
 

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