Dorsa R.
asked • 04/13/20find the curve r = r(t):
torsion = 1
curvature = 1
r(0) = unit tangent vector(0) =i
unit normal(0) = j
unit binormal (0) = k
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![{\displaystyle {\begin{aligned}&\color {black}{\text{wyzant.com/resources/answers/752487}}\\[3ex]&{\underline {\text{Problem:}}}\\&{\text{Find the curve }}r=r(t){\text{ with curvature }}\kappa (t)\equiv 1{\text{ and torsion }}\tau (t)\equiv 1\\&{\text{whose initial unit tangent, normal and binormal vectors are }}T(0)={\hat {\mathbf {i} }},\\&N(0)={\hat {\mathbf {j} }},{\text{and }}B(0)={\hat {\mathbf {k} }}.\\[2ex]&{\underline {\text{Solution:}}}\\&{\text{The Frenet-Serret equations (with arc length as parameter):}}\\&\qquad \left\{{\begin{aligned}T'(s)&={\phantom {-T(s)+{}}}N(s)\\N'(s)&=-T(s){\phantom {{}+N(s)}}+B(s)\\B'(s)&={\phantom {-T(s)}}-N(s)\end{aligned}}\right.\\&{\text{They yield }}\\&\qquad T'''=N''=-T'+B'=-T'-N=-2T'\,.\\&T'(s)=N(s){\text{ satisfies the ODE}}\\&\qquad N''+2N=0\qquad \left[N(0)={\hat {\mathbf {j} }},\ N'(0)=-{\hat {\mathbf {i} }}+{\hat {\mathbf {k} }}\right]\,.\\&{\text{Therefore, }}N(s)=\cos(s{\sqrt {2}}){\hat {\mathbf {j} }}+{\tfrac {\sin(s{\sqrt {2}})}{\sqrt {2}}}{\bigl (}-{\hat {\mathbf {i} }}+{\hat {\mathbf {k} }}{\bigr )}{\text{ or}}\\&\qquad {\begin{aligned}N(s)&={\frac {1}{\sqrt {2}}}{\bigl \langle }-\sin(s{\sqrt {2}}),{\sqrt {2}}\cos(s{\sqrt {2}}),\sin(s{\sqrt {2}}){\bigr \rangle }\\T(s)&={\frac {1}{2}}{\bigl \langle }1+\cos(s{\sqrt {2}}),{\sqrt {2}}\sin(s{\sqrt {2}}),1-\cos(s{\sqrt {2}}){\bigr \rangle }&&{\bigl [}T(0)=\langle 1,0,0\rangle {\bigr ]}\\B(s)&={\frac {1}{2}}{\bigl \langle }1-\cos(s{\sqrt {2}}),-{\sqrt {2}}\sin(s{\sqrt {2}}),1+\cos(s{\sqrt {2}}){\bigr \rangle }\,.\end{aligned}}\\&{\text{The binormal followed from }}(T+B)'=0,{\text{ i.e., }}T(s)+B(s)\equiv \langle 1,0,1\rangle .\\&{\text{Lastly, }}r'(s)=T(s){\text{, and with }}r(0)=0,\\&\qquad r(s)={\frac {1}{2{\sqrt {2}}}}{\bigl \langle }s{\sqrt {2}}+\sin(s{\sqrt {2}}),{\sqrt {2}}-{\sqrt {2}}\cos(s{\sqrt {2}}),s{\sqrt {2}}-\sin(s{\sqrt {2}}){\bigr \rangle }\,.\\&{\text{The (optional) reparametrization }}t=s{\sqrt {2}}{\text{ yields}}\\&\qquad r(t)={\frac {1}{2{\sqrt {2}}}}{\bigl \langle }t+\sin t,{\sqrt {2}}-{\sqrt {2}}\cos t,t-\sin t{\bigr \rangle }\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee87fdd0170ebba47df3575d98c636e68b45dadf)
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