A closed, planar curve $C$ is said to have constant breadth $μ$ if the distance between parallel tangent lines to $C$ is always $μ.$ Assume for the rest of this problem that the curve is $C^2$ and $κ \neq 0.$
Prove that if $C$ has constant breadth $μ,$ then the chord joining opposite points is normal to the curve at both points.
My attempt:- Author gave us hint that If $β(s)$ is opposite $α(s)$, then $β(s) = α(s) + λ(s)T(s) + μN(s)$. I really don't understand how did the author write like this. Please help me. After that I can do the problem using the hint in the given answer A closed regular planar curve of constant width - finding an expression for the opposite point to $\alpha(s)$.
The vectors $T(s)$ and $N(s)$ form a basis of the plane. The vector $\beta(s)-\alpha(s)$ is a linear combination: $$ \beta(s)-\alpha(s)=\lambda(s)T(s)+\mu(s)N(s) $$ But you know that the distance between the tangents, which is the normal component of the above vector, is fixed, $\mu(s)=\mu$.