The TNBFrame(C, t, out) command computes the tangent, normal, binormal frame of the curve C.  The computed objects are returned in an expression sequence.

The curve can be specified as a free or position Vector or a procedure that returns a Vector. This determines the returned object type.

If t is not specified, the function tries to determine a suitable variable name by using the components of C.  To do this, it checks all of the indeterminates of type name in the components of C and removes the ones which are determined to be constants.

If the resulting set has a single entry, the single entry is the variable name.  If it has more than one entry, an error is raised.

The right side of the equation out, when specified, controls which of the Vectors in the tangent, normal, binormal frame return. If 'T' is specified, the tangent vector is returned.  If 'N' is specified, the normal vector is returned.  If 'B' is specified, the binormal vector is returned.  The default is $'\mathrm{output}'=\left['T'\,'N'\,'B'\right]$.

If a coordinate system attribute is specified on C, C is interpreted in that coordinate system.  Otherwise, the curve is interpreted as a curve in the current default coordinate system.  If the two are not compatible, an error is raised.

$\mathrm{with}\left(\mathrm{VectorCalculus}\right)\:$

$\mathrm{TNBFrame}\left(\mathrm{PositionVector}\left(\left[\cos \left(t\right)\,\sin \left(t\right)\,t\right]\right)\,t\right)$

$\left[\begin{array}{c}-\frac{\sqrt{2}\sin \left(t\right)}{2}\\ \frac{\sqrt{2}\cos \left(t\right)}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right],\left[\begin{array}{c}-\cos \left(t\right)\\ -\sin \left(t\right)\\ 0\end{array}\right],\left[\begin{array}{c}\frac{\sqrt{2}\sin \left(t\right)}{2}\\ -\frac{\sqrt{2}\cos \left(t\right)}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right]$

$\mathrm{TNBFrame}\left(⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\,'\mathrm{output}'=\left['T'\right]\right)$

$\left[\begin{array}{c}-\frac{\sqrt{2}\sin \left(t\right)}{2}\\ \frac{\sqrt{2}\cos \left(t\right)}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right]$

$\mathrm{TNBFrame}\left(⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\,'\mathrm{output}'=\left['N'\,'B'\right]\right)$

$\left[\begin{array}{c}-\cos \left(t\right)\\ -\sin \left(t\right)\\ 0\end{array}\right],\left[\begin{array}{c}\frac{\sqrt{2}\sin \left(t\right)}{2}\\ -\frac{\sqrt{2}\cos \left(t\right)}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right]$

$\mathrm{SetCoordinates}\left('\mathrm{cylindrical}'\right)$

$\mathrm{cylindrical}$

$\mathrm{TNBFrame}\left(⟨1\,t\,t⟩\right)$

$\left[\begin{array}{c}\frac{-\frac{\cos \left(t\right)\cos \left(\sin \left(t\right)\right)\sqrt{2}\sin \left(t\right)}{2}+\frac{{\cos \left(t\right)}^2\sin \left(\sin \left(t\right)\right)\sqrt{2}}{2}}{\sqrt{{\cos \left(t\right)}^2{\sin \left(\sin \left(t\right)\right)}^2+{\cos \left(t\right)}^2{\cos \left(\sin \left(t\right)\right)}^2}}\\ \frac{\frac{{\cos \left(t\right)}^2\cos \left(\sin \left(t\right)\right)\sqrt{2}}{2}+\frac{\cos \left(t\right)\sin \left(\sin \left(t\right)\right)\sqrt{2}\sin \left(t\right)}{2}}{\sqrt{{\cos \left(t\right)}^2{\sin \left(\sin \left(t\right)\right)}^2+{\cos \left(t\right)}^2{\cos \left(\sin \left(t\right)\right)}^2}}\\ \frac{\sqrt{2}}{2}\end{array}\right],\left[\begin{array}{c}\frac{-{\cos \left(t\right)}^2\cos \left(\sin \left(t\right)\right)-\cos \left(t\right)\sin \left(\sin \left(t\right)\right)\sin \left(t\right)}{\sqrt{{\cos \left(t\right)}^2{\sin \left(\sin \left(t\right)\right)}^2+{\cos \left(t\right)}^2{\cos \left(\sin \left(t\right)\right)}^2}}\\ \frac{-\cos \left(t\right)\cos \left(\sin \left(t\right)\right)\sin \left(t\right)+{\cos \left(t\right)}^2\sin \left(\sin \left(t\right)\right)}{\sqrt{{\cos \left(t\right)}^2{\sin \left(\sin \left(t\right)\right)}^2+{\cos \left(t\right)}^2{\cos \left(\sin \left(t\right)\right)}^2}}\\ 0\end{array}\right],\left[\begin{array}{c}\frac{\frac{\cos \left(t\right)\cos \left(\sin \left(t\right)\right)\sqrt{2}\sin \left(t\right)}{2}-\frac{{\cos \left(t\right)}^2\sin \left(\sin \left(t\right)\right)\sqrt{2}}{2}}{\sqrt{{\cos \left(t\right)}^2{\sin \left(\sin \left(t\right)\right)}^2+{\cos \left(t\right)}^2{\cos \left(\sin \left(t\right)\right)}^2}}\\ \frac{-\frac{{\cos \left(t\right)}^2\cos \left(\sin \left(t\right)\right)\sqrt{2}}{2}-\frac{\cos \left(t\right)\sin \left(\sin \left(t\right)\right)\sqrt{2}\sin \left(t\right)}{2}}{\sqrt{{\cos \left(t\right)}^2{\sin \left(\sin \left(t\right)\right)}^2+{\cos \left(t\right)}^2{\cos \left(\sin \left(t\right)\right)}^2}}\\ \frac{\sqrt{2}}{2}\end{array}\right]$