Equation of a Curve in R3 with Constant Inclination: Need Help!

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The discussion focuses on deriving the equation of a curve in R3 characterized by constant inclination, defined by the relationship k=t=a/(s^2+b), where k is curvature, t is torsion, and a and b are constants in R. The user expresses frustration over encountering complex integrals while attempting to solve this problem. The concept of constant inclination is emphasized, where the ratio t/k equals 1, simplifying the integration process. Additionally, the user mentions the availability of free software that can visualize curves based on specified curvature and torsion values.

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  • Understanding of curvature and torsion in differential geometry
  • Familiarity with the concepts of constant inclination curves
  • Knowledge of integral calculus and its applications
  • Experience with mathematical software for curve visualization
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  • Explore techniques for solving complex integrals in differential geometry
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Mathematicians, physics students, and engineers interested in the geometric properties of curves in three-dimensional space, particularly those working with curvature and torsion concepts.

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I'm trying to find the equation of a curve in R3 where k=t=a/(s^2+b) where k is the curvature, t is the torsion and a,b are constants contained in R. I've spent weeks on this problem and at the moment it's driving me nuts since I always seem to end up with an impossible integral. Any help or hints would be much appreciated. Thanks.
 
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This is called a curve of 'constant inclination', as t/k=1=cotan(a),
where a is a constant angle formed
by the tangent to the curve and a specific fixed direction. This will make integration easier.


Ps. There is (free) software on the internet, which will draw a curve on the input of its k and t.
 

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