What are the fundamental forms and curvatures of a helicoid in $\mathbb{R}^3$?

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In summary, a helicoid is a three-dimensional surface that resembles a spiral staircase and can be created by moving a straight line in a specific pattern. It differs from a helix in that it contains many helix curves and is two-dimensional. The fundamental forms of a helicoid are the first and second fundamental forms, which describe the local geometry of the surface at each point. The Gaussian curvature of a helicoid is constant and equal to 0, making it a unique object to study in mathematics. Helicoids have practical applications in architecture, antenna design, and the construction of certain objects, and can also be found in nature in the shells of certain sea creatures.
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Chris L T521
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Here's this week's problem.

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Problem: A helicoid in $\mathbb{R}^3$ is parameterized by $(s,t)\mapsto (s\cos t, s\sin t, t)$. Compute the helicoid's:

(a) first fundamental form
(b) second fundamental form
(c) Gaussian curvature
(d) mean curvature

as functions of $s$ and $t$.

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  • #2
No one answered this week's question. I don't have a solution ready at this time (too busy studying for preliminary exams this past week) -- please expect one sometime tomorrow.
 
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FAQ: What are the fundamental forms and curvatures of a helicoid in $\mathbb{R}^3$?

What is a helicoid in $\mathbb{R}^3$?

A helicoid is a three-dimensional surface that resembles a spiral staircase. It is a type of ruled surface, meaning that it can be created by moving a straight line in a specific pattern.

How is a helicoid different from a helix?

A helix is a curve that follows a spiral path, while a helicoid is a surface that contains many helix curves. In other words, a helix is one-dimensional, while a helicoid is two-dimensional.

What are the fundamental forms of a helicoid?

The fundamental forms of a helicoid are the first and second fundamental forms, which describe the local geometry of the surface at each point. The first fundamental form gives information about the lengths and angles of curves on the surface, while the second fundamental form describes how the surface curves in different directions.

What is the Gaussian curvature of a helicoid?

The Gaussian curvature of a helicoid is constant and equal to 0. This means that the surface is flat and has no curvature at any point. This is a unique property of helicoids and makes them interesting objects to study in mathematics.

How are helicoids used in real life?

Helicoids have many practical applications, such as in the design of architectural structures, such as spiral staircases and ramps. They are also used in the construction of helical antennas and in the design of certain types of gears and screws. In nature, helicoidal shapes can be found in the shells of certain sea creatures, such as the nautilus.

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