Pitch for a 3d hyperbolic spiral

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SUMMARY

The discussion centers on the mathematical formulation of a hyperbolic spiral space curve, specifically the formula r(t)=1/t * cos(alpha) + 1/t * sin(alpha) + t. The user has successfully derived the tangent vector T, normal vector N, and curvature (kappa) but seeks clarification on calculating the pitch of the spiral. The pitch is defined as pitch=arctan(axial speed/tangential speed), with questions raised about the definitions of axial speed and tangential speed in this context.

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  • Understanding of hyperbolic spiral equations
  • Familiarity with vector calculus concepts, including tangent and normal vectors
  • Knowledge of curvature in three-dimensional space
  • Basic trigonometry, particularly the arctangent function
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  • Explore vector calculus applications in physics and engineering
  • Investigate the relationship between axial and tangential speeds in curvilinear motion
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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,
 
Physics news on Phys.org
What "alpha" is ?
What "X" is ?
 
Alpha is the angle typically in the range from 0 (zero) to 2 phi.
X is a multiplication symbol
 

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