Geodesics of hyperbolic paraboloid ( )

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SUMMARY

The discussion focuses on identifying the geodesics of the hyperbolic paraboloid defined by the equation z = xy, specifically those passing through the point (0,0,0). It is established that lines and normal sections are geodesics, with y = x and y = -x confirmed as two line geodesics. Additionally, the participant questions whether planes in the z-y and z-x axes through (0,0,0) qualify as geodesics, prompting a need for a clear definition of geodesics in this context.

PREREQUISITES
  • Understanding of hyperbolic paraboloids
  • Familiarity with the concept of geodesics in differential geometry
  • Knowledge of normal sections in 3D geometry
  • Basic skills in calculus and multivariable functions
NEXT STEPS
  • Research the definition and properties of geodesics in differential geometry
  • Explore the calculation of geodesics for surfaces using calculus of variations
  • Investigate the role of normal sections in determining geodesics
  • Examine examples of geodesics on other surfaces, such as ellipsoids and paraboloids
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Mathematicians, physics students, and anyone studying differential geometry or interested in the properties of surfaces and their geodesics.

akoska
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Geodesics of hyperbolic paraboloid (urgent!)

Help me find the geodesics of the hyperbolic paraboloid z=xy passing through (0,0,0).

I know that lines and normal sections are geodesics. Based on a picture, I think y=x and y=-x are 2 line geodesics. Then, maybe the planes in the z-y and z-x axis passing through (0,0,0) are geodesics? How can I check this?
 
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