Definition and intuition of a tubular neighborhood of a submanifold

Let $dinmathbb N$, $kin{1,ldots,d}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $mathbb R^d$ with boundary$^1$.

Now let $$T_xM:=left{vinmathbb R^dmidexistsvarepsilon>0,gammain C^1((-varepsilon,varepsilon),M):gamma(0)=x,gamma'(0)=vright}$$ denote the tangent space of $M$ at $x$ and $N_xM:=(T_xM)^perp$ for $xin M$.

I’m trying to understand the definition of a tubular neighborhood of $M$. I was only able to find definitions of this notion in a way more general setting which built up on several concepts I’m not familiar with. So, I’d like to find a simplified, but equivalent, definition for my present setting.

What I’ve understood is that one considers the space $$N(M):={(x,v):xin Mtext{ and }vin N_xM}$$ and the map $$E(x,v):=x+v;;;text{for }(x,v)in N(M).$$

Now the usual definition of a tubular neighborhood $U$ of $M$ is that it is a (open?) neighborhood of $M$ in $mathbb R^d$ (really a neighborhood of all of $M$?) such that $U$ is the diffeomorphic image under $E$ of an open subset $V$ of $N(M)$ with $$V={(x,v)in N(M):left|vright|<delta(x)}tag1$$ for some continuous function $delta:Mto(0,infty)$.

I really got trouble to understand this. The vector $E(x,v)$ is simply a vector originating from $x$ and pointing in the direction $v$. If $M$ is a circle in $mathbb R^3$, I guess the intuition is that for each point $x$ on the circle all the $vin N_xM$ built a ring around $x$ by "rotating" the normals around $x$. Doing so for all $x$ on the circle, we obtain a torus. Is this correct so far? All the pictures I’ve found have confused me.

And how would a tubular neighborhood of a sphere in $mathbb R^3$ look like? For a sphere, the normal spaces are 1-dimensional …

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$^1$ i.e. each point of $M$ is locally $C^1$-diffeomorphic to $mathbb H^k:=mathbb R^{k-1}times[0,infty)$.

If $E_i$ is a $mathbb R$-Banach space and $B_isubseteq E_i$, then $f:B_1to E_2$ is called $C^1$-differentiable if $f=left.tilde fright|_{B_1}$ for some $E_1$-open neighborhood $Omega_1$ of $B_1$ and some $tilde fin C^1(Omega_1,E_2)$ and $g:B_1to B_2$ is called $C^1$-diffeomorphism if $g$ is a homeomorphism from $B_1$ onto $B_2$ and $g$ and $g^{-1}$ are $C^1$-differentiable.