Answer
a) See the explanation.
b) See the explanation.
Work Step by Step
a)
A formula for unit normal vector, $N(t)$ (that is, perpendicular to the tangent vector of the curve) of a smooth space curve $r(t)$ is defined as:
$N(t)=\dfrac{T'(t)}{|T'(t)|}$.
A formula for binormal vector, $B(t)$ (that is, perpendicular to both $T(t)$ and $N(t)$ of the curve) of a smooth space curve $r(t)$ is defined as:
$B(t)=T(t) \times N(t)$
b)
(i) Consider a curve $C$ whose normal plane is determined by the normal and binormal vectors, that is $N$ and $B$ respectively, at a point $A$ on $C$.
(ii) Consider a curve $C$ whose osculating plane is determined by the unit normal, that is $T$ and $N$ respectively at a point $A$ on $C$ and a curve intersects the osculating plane at the point of contact when it comes closet to it.
(iii) Osculating circle of a curve (let us say $C$ at a point $A$) is defined as the circle that has the same tangent as C at a point $A$ and the radius $ \rho$ is the reciprocal of the curvature, that is, $ \rho=\frac{1}{\kappa}$.
