Frenet Trihedron

Given a parametrized curve of the form r(t) = f(t)i + g(t)j + h(t)k, also denoted (f(t),g(t),h(t)), we can find three mutually perpendicular vectors, commonly called the Frenet Trihedron. We use (cos(t),sin(t),t/2) for t=0..10 to illustrate this.
View Curve.

The unit tangent vector u(t) is f'(t)i + g'(t)j + h'(t)k normalized so it is length 1. In our case u(t) = 2/sqrt(5)*(-sin(t),cos(t),1/2).
View Unit Tangent Vector.

The unit principal normal vector is the normalized derivative of the unit tangent vector p(t) = u'(t)/|u'(t)|. Note that if u'(t) is zero, this is undefined. In our example p(t) = (-cos(t),-sin(t),0).
View Unit Principal Normal Vector.

The unit binormal vector is the cross product of the unit tangent vector and the unit principal normal vector b(t) = u(t) X p(t). For us b(t) = 2/sqrt(5)*(sin(t)/2,-cos(t)/2,1).
View Unit Binormal Vector.

The Frenet Trihedron is the vectors consisting of the unit tangent vector, unit principal normal vector, and unit binormal vector.
View Frenet Trihedron.