What Is the Parallel of Least Radius on a Hyperboloid of Revolution?

In summary, the "parallel of least radius" on the hyperboloid of revolution x^2+y^2-z^2=1 is the shortest line, parallel to an axis, connecting two points on the hyperboloid. A "line of striction" is a curve that lies on the surface and is perpendicular to the tangent vector at each point. It can be calculated using the formula u=-\frac{<a',w'>}{<w',w>} and the points on a line of striction are referred to as "central points".
  • #1
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Homework Statement


Show that on of the hyperboloid of revolution x^2+y^2-z^2=1, the parallel of least radius is the line of striction, ...


What's the parallel of least radius?
 
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  • #2
The "parallel of least radius" is the line, parallel to an axis, that is shortest from one point on the hyperboloid to another.

Now, clear up my confusion: what is a "line of striction"?
 
  • #3
Given a ruled surface x(t,v)=a(t)+vw(t), a line of striction is a curve b(t) such that <b'(t),w'(t)>=0 for all t and b lies on the trace of x, ie b(t)=a(t)+u(t)w(t) for some real valued function u(t). It be can then shown that u(t) is given by

[tex]u=-\frac{<a',w'>}{<w',w>}[/tex].

The points of a line of striction are the "central points" of the ruled surface.
 
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FAQ: What Is the Parallel of Least Radius on a Hyperboloid of Revolution?

What is the Parallel of Least Radius?

The Parallel of Least Radius is a term used in mathematical analysis and optimization to refer to a line or curve that is tangent to a given function at its point of least curvature.

How is the Parallel of Least Radius used in scientific research?

The Parallel of Least Radius is used in various fields of science, such as physics, engineering, and economics, to optimize and model real-world systems. It helps to identify the most efficient or optimal solution for a given problem.

What is the difference between the Parallel of Least Radius and the Parallel of Maximum Radius?

The Parallel of Least Radius is the line or curve that is tangent to a function at its point of least curvature, while the Parallel of Maximum Radius is the line or curve that is tangent to a function at its point of maximum curvature. In other words, the Parallel of Least Radius minimizes the curvature of a function, while the Parallel of Maximum Radius maximizes it.

Can the Parallel of Least Radius be used in non-mathematical settings?

Yes, the concept of the Parallel of Least Radius can be applied in various non-mathematical settings, such as in design and architecture, to create the most efficient and aesthetically pleasing structures. It can also be used in decision-making processes to find the most optimal solution.

How is the Parallel of Least Radius calculated?

The Parallel of Least Radius is calculated using mathematical techniques such as calculus and optimization algorithms. It involves finding the point of minimum curvature on a given function and then constructing the tangent line or curve at that point.

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