5 deleted 1 character in body edited Apr 11 at 22:11 Cinaed Simson 19211 silver badge88 bronze badges Frenet Frame See [Kinematics of the helical motion of organisms with up to six degrees of freedom] (link.springer.com/article/10.1007%2FBF02460302) referenced by the OP which uses the Frenet frame. The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c}^2 sin\frac{s}{c},0)$$$$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c^2}sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{c},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. Frenet Frame See [Kinematics of the helical motion of organisms with up to six degrees of freedom] (link.springer.com/article/10.1007%2FBF02460302) referenced by the OP which uses the Frenet frame. The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c}^2 sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{c},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. Frenet Frame See [Kinematics of the helical motion of organisms with up to six degrees of freedom] (link.springer.com/article/10.1007%2FBF02460302) referenced by the OP which uses the Frenet frame. The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c^2}sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{c},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. 4 added 140 characters in body edited Apr 11 at 22:03 Cinaed Simson 19211 silver badge88 bronze badges Frenet Frame See [Kinematics of the helical motion of organisms with up to six degrees of freedom] (link.springer.com/article/10.1007%2FBF02460302) referenced by the OP which uses the Frenet frame. The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c}^2 sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{c},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. Frenet Frame The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c}^2 sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{c},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. Frenet Frame See [Kinematics of the helical motion of organisms with up to six degrees of freedom] (link.springer.com/article/10.1007%2FBF02460302) referenced by the OP which uses the Frenet frame. The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c}^2 sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{c},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. 3 edited body edited Apr 11 at 20:54 Cinaed Simson 19211 silver badge88 bronze badges Frenet Frame The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c}^2 sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{s},-sin\frac{s}{c},0)$$$$N(s)=(-cos\frac{s}{c},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. Frenet Frame The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c}^2 sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{s},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. Frenet Frame The Frenet frame or trihedron is used for describing the turning and twisting of curves in $$R^3$$. There are 6 degree of freedom, namely, 3 degrees of freedom to fix a point in $$R^3$$, and 3 degrees of freedom to fix the $$T$$ tangent vector or a velocity vector to the point. From these 6 degrees of freedom, all else follows, namely, the Frenet frame or trihedron. The Frenet frame consists of 3 vector, namely, $$T{'},N{'},B{'}$$ which are calculated from $$T$$. The vector $$T{'}$$ measures the curvature of the curve, and the vector $$B{'}$$ measures the torsion of the curve. The parameter $$s$$ generates the trajectory. Let $$\beta:I\rightarrow R^3$$ be the mapping of cylindrical helix curve in $$R^3$$ with $$s$$ in the open interval $$0\lt s \lt1$$ in $$I$$ $$\beta(s)=(a_{ } cos\frac{s}{c},a_{ }sin\frac{s} {c},\frac {bs}{c})$$ where $$c=(a^2+b^2)^{1/2}$$ and $$a>0$$. The unit tangent vector to the curve $$T$$ is $$T(s)=\beta{'}(s)=(-\frac {a}{c}sin\frac{s}{c},\frac {a}{c}cos\frac{s}{c},\frac {b}{c})$$ where $$\beta{'}(s)$$ is short hand for $$\frac {d\beta(s)}{ds}$$. Then $$T^{'}(s)=\beta{''}(s)=(-\frac {a}{c^2}cos\frac{s} {c},-\frac {a}{c}^2 sin\frac{s}{c},0)$$ and the curvature is $$\kappa(s)=||T^{'}(s)||=\frac {a}{c^2}=\frac{a}{a^2+b^2} > 0$$ Since $$T^{'}=\kappa N$$, where $$N$$ is the normal vector $$N(s)=(-cos\frac{s}{c},-sin\frac{s}{c},0)$$ which always points toward the axis of the cylinder on which $$\beta(s)$$ lies regardless of values of $$a$$ and $$b$$. Using the cross product, $$B=T\times N$$ gives $$B(s)=(\frac {b}{c}sin\frac{s}{c},-\frac {b}{c} cos\frac{s}{c},\frac {a}{c})$$ To compute the torsion, by definition, $$B{'}=-\tau N$$ $$B^{`}(s)=(\frac {b}{c^2}cos\frac{s}{c},\frac {b}{c^2} sin\frac{s}{c},0)$$ which implies $$\tau=\frac {b}{c^2}=\frac {b}{a^2+b^2}$$ In summary, the Frenet frame consists of the 3 orthonormal vectors $$T{'}=\kappa N$$ $$N{'}=-\kappa T +\tau B$$ $$B{'}=-\tau N,$$ The first equation is the definition of curvature and last equation is the definition of torsion - both equations were calculated directly from the $$T$$ vector. The second equation has been written as orthonormal expansion of the first and last equations. 2 deleted 13 characters in body edited Apr 11 at 13:21 Cinaed Simson 19211 silver badge88 bronze badges 1 answered Apr 11 at 13:13 Cinaed Simson 19211 silver badge88 bronze badges