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.
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.
The Frenet Trihedron is explained in the book Advanced Engineering Mathematics 8th Edition Section 8.7 pg 440-441.