Geodesics of hyperbolic paraboloid ( )

In summary, geodesics of hyperbolic paraboloid are lines on the surface of a hyperbolic paraboloid that have the shortest distance between any two points on the surface. They are unique because they are not straight lines, but instead curved lines that follow the shape of the surface. These geodesics have practical applications in architecture and engineering, as well as in the study of curved surfaces and geometry. They can be calculated using differential geometry and the concept of geodesic curvature. Additionally, they can be observed in nature, such as in the shape of sea shells, plants, and animal shells.
  • #1
akoska
22
0
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?
 
Physics news on Phys.org
  • #2
What is your definition of a geodesic?
 

1. What are geodesics of hyperbolic paraboloid?

Geodesics of hyperbolic paraboloid are the lines that lie on the surface of a hyperbolic paraboloid and have the shortest distance between any two points on the surface.

2. How are geodesics of hyperbolic paraboloid different from geodesics of other surfaces?

Geodesics of hyperbolic paraboloid are unique because they are not straight lines, unlike geodesics of other surfaces. They are instead curved lines that follow the shape of the surface.

3. What is the significance of geodesics of hyperbolic paraboloid?

Geodesics of hyperbolic paraboloid have practical applications in architecture and engineering, as they can be used to design structures with minimal material while still maintaining strength and stability. They also have important implications in the study of curved surfaces and geometry.

4. How are geodesics of hyperbolic paraboloid calculated?

Geodesics of hyperbolic paraboloid can be calculated using differential geometry and the concept of geodesic curvature. This involves finding the tangent vectors to the surface and using the geodesic equation to determine the path of the geodesic.

5. Can geodesics of hyperbolic paraboloid be observed in nature?

Yes, geodesics of hyperbolic paraboloid can be observed in nature. One example is the shape of sea shells, which often follow the pattern of a hyperbolic paraboloid. They can also be seen in the structure of some plants and animal shells.

Similar threads

MHB How is z=2xy a Hyperbolic Paraboloid in the rotated 45° in the xy-plane?
    • Calculus
Replies
7
Views
2K
Shortest path between two points on a paraboloid
    • Calculus and Beyond Homework Help
Replies
11
Views
2K
Classification of a second order partial differential equation
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Reduced equation of quadratic forms
    • Calculus and Beyond Homework Help
Replies
1
Views
988
How to model a function of a box's volume using Lagrange multiplier methods
    • Calculus and Beyond Homework Help
Replies
1
Views
804
A Schwarzschild metric in Fermi normal coordinates about a radially infalling timelike geodesic
    • Special and General Relativity
Replies
13
Views
615
Determine the line of force to F and if it's conservative.
    • Calculus and Beyond Homework Help
Replies
4
Views
866
Finding Volume of Solid Bounded by Paraboloid and Planes
    • Calculus and Beyond Homework Help
Replies
2
Views
932
B Volume of paraboloid in a cylinder
    • Calculus
Replies
2
Views
1K
Explain why all geodesics on a sphere are arcs of great circ
    • Calculus and Beyond Homework Help
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top