Pitch for a 3d hyperbolic spiral

In summary, the conversation discusses the formula for a hyperbolic spiral space curve and obtaining the tangent vector, normal vector, and curvature. However, the individual is having trouble finding the formula for the pitch at any point on the spiral. They suggest using the formula pitch=arctan(axial speed/tangential speed) but have questions about what the axial and tangential speeds refer to, as well as the meaning of "alpha" and "X" in the formula.
  • #1
trustthrust
5
0
Hi, have worked out the formula for a hyperbolic spiral space curve to be r(t)=1/t X cos(alpha)+1/t X sine(alpha) + t and obtained the tangent vector T, normal vector N, and curvature (kappa) with the z axis being the central axis of the spiral. Am having trouble finding the formula for the pitch at any point of this spiral. What I have so far is pitch=arctan(axial speed/tangential speed).

Questions:
1. Is the axial speed the normal vector? Seems to me should be the z component of the normal vector. Is this correct?
2. Is the tangential speed in the above pitch formula the tangent vector T?

Thanks in advance for any help,
 
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  • #2
What "alpha" is ?
What "X" is ?
 
  • #3
Alpha is the angle typically in the range from 0 (zero) to 2 phi.
X is a multiplication symbol
 

1. What is a 3d hyperbolic spiral?

A 3d hyperbolic spiral is a three-dimensional curve that follows the path of a hyperbolic function. It can be defined mathematically as a spiral that increases in size and distance from the center as it wraps around a central axis.

2. What is the purpose of a pitch for a 3d hyperbolic spiral?

A pitch for a 3d hyperbolic spiral is typically used to describe and visualize the shape and characteristics of the spiral. It can also be used to explain the theoretical or practical applications of the spiral in various fields such as mathematics, physics, and engineering.

3. How is a 3d hyperbolic spiral different from a 2d spiral?

A 3d hyperbolic spiral is different from a 2d spiral in that it exists in three dimensions instead of two. This means that the spiral not only curves around a central axis, but also extends outwards in a circular motion. Additionally, the shape and characteristics of the spiral are different due to the inclusion of a hyperbolic function.

4. What are some real-life examples of 3d hyperbolic spirals?

One of the most well-known real-life examples of a 3d hyperbolic spiral is the shape of a nautilus shell. Other examples include the structure of some plants, such as pinecones and sunflower seeds, and the shape of certain galaxies in space.

5. What are the potential applications of a 3d hyperbolic spiral?

A 3d hyperbolic spiral has many potential applications in various fields. In mathematics, it can be used to model complex systems and phenomena. In physics, it can be used to describe the behavior of particles and waves. In engineering, it can be used in the design of structures and machinery. It also has potential applications in architecture, art, and design.

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