I was going through Hassani's Mathematical methods for physics, and in the part where he discusses the definition of an angle(to generalise to solid angles), I have a confusion-
There is an arbitrary curve. A segment $C_1C_2$ subtends an angle $d\theta$ at a point P. This is calculated by constructing an arc $C_1C_2$ of a circle centered at P. Then, as Hassani shows, the projection of the curve on this circle is- $ds=dl (\hat{e_R}.{\hat e_n})$ , where $\hat e_R,\hat e_n$ are the position vector (relative to P) and the normal vector to the segment of curve, respectively, and $dl$ is the length of that curve. Then, you have $d\theta=ds/R$, and all that is good.
Now, when we say that $ds$ is the projection of $\vec{dl}$ on the 'bigger' circle, $C_1C_2$, we mean $ds=|\vec{dl} . \hat{C_1C_2}|=dl(\hat{e_t}.\hat{C_1C_2})$, where $\hat{e_t}$ is the tangent vector along $dl$. Now, it is easy to see that the dot product of 2 unit vectors is the same as that of the dot product of 2 unit vectors normal to them(barring a negative sign), so $|(\hat{e_t}.\hat{C_1C_2})|=|\hat{e_n}.\hat{e_R}|$, as $\hat{e_R}$ is normal to the arc $C_1C_2$, and so $ds$ is indeed the projection of $\vec{dl}$ on the bigger circle.
Now, the problem is, we can apply the same argument for the smaller circle, because $\hat{e_R}$ is the same for both $C_1C_2$ and $A_1A_2$, so this would mean that the above mentioned dot product is also the projection of the segment $\vec{dl}$ on $A_1A_2$. But clearly, this is wrong- because $C_1C_2>A_1A2$! ( in fact, since they subtend the same angle, we have $d\theta=\frac{C_1C_2}{R}=\frac{A_1A_2}{a}$, so $A_1A_2=\frac{a}{R} C_1C_2=a\{\frac{dl(\hat{e_R}.{\hat e_n})}{R}$).
I don't understand where my mistake lies. Clearly, $ds=dl(\hat{e_R}.{\hat e_n})$ is the projection of the segment ('radially', so to speak) on an auxiliary circle passing through $\vec R$, but why is ds not the projection on the circle passing through $\vec a$?(I know it is supposed to be $a(ds/R)$) The unit vectors are the same in both cases, so the dot product should be the same, right? I'm confused!
Any help will be appreciated.