Let $\alpha$ be a unit speed curve. Its tangent indicatrix $\sigma$ is defined by $\sigma(t)=T(t)$. Find torsion and curvature of $\sigma$ with respect to the torsion and curvature of $\alpha$.

There is an answer on my book. But I don't understand (*) part

$$ \kappa_\sigma=\frac{||\alpha'\times \alpha''||}{||\alpha'||^3}$$

$$ \tau_\sigma=\frac{\left<\alpha'\times\alpha'',\alpha'''\right>}{||\alpha'\times\alpha''||^2} $$

$$\sigma'(t)=T'(t) \Rightarrow \sigma'=T'=\kappa N$$


we find $\sigma'\times\sigma''=\kappa^3B+\kappa^2\tau T$

$$||\sigma'||=\sqrt{\left<\sigma',\sigma'\right>}=\sqrt{\left<\kappa N,\kappa N\right>}={\kappa^2\left<N,N\right>}=|\kappa|=||\alpha''||$$

$$||\sigma'\times\sigma''||=\sqrt{(\kappa^2\tau)^2+(\kappa^3)^2}=\kappa^2\sqrt{\tau^2+\kappa^2} \tag{*}$$



I couldn't understand the equation (*). Can you explain it please? (expression "tangents indicator" may not true. I translated into English and $\kappa$ is the curvature and $\tau$ , $T$ tangent vector, and $B=T\times N$

  • 1
    $\begingroup$ $B$ and $T$ are orthogonal unit vectors. You know that $\|a+b\|^2 = \|a\|^2 + \|b\|^2$ if $a$ and $b$ are orthogonal, right? Apply it to $\|\sigma'\times\sigma''\| = \|\kappa^3B+\kappa^2\tau T\|$. $\endgroup$ – Rahul Jan 4 '14 at 14:39
  • $\begingroup$ In English this is called the tangent indicatrix. Your $x$s need to be written as '\times' :) $\endgroup$ – Ted Shifrin Jan 4 '14 at 14:39
  • $\begingroup$ @RahulNarain thanks. appreciated. $\endgroup$ – lyme Jan 4 '14 at 15:15
  • $\begingroup$ @TedShifrin I see. Could you suggest me a book or an article about this subject? $\endgroup$ – lyme Jan 4 '14 at 15:15
  • $\begingroup$ You just have to use the formulas for curvature/torsion of non-arclength parametrized curves. The tangent indicatrix appears in curve theory, but only as a tool. What book are you studying from? $\endgroup$ – Ted Shifrin Jan 4 '14 at 16:04

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