Aplanatic point of refraction of single sphere _ Popular Science

Aplanatic point of single spherical refraction

Recently, looking at the "Modern Optics Foundation" edited by Mr. Zhong Xihua, the first chapter introduces the aplanatic point and gives an example, which is the aplanatic point of the single spherical refraction that is to be discussed today. The position of the aplanatic point is given in the book, but the derivation process is not given. I tried to deduce it and put it here to make a note.
Ziming (aplanatic points) is also known as the Equal optical path point or no Halo point. Simply put, the light emitted here can be accurately assembled at a point with no ball aberration, coma aberration, and astigmatic.
The following figure:

Q0 Q_0 Point is a matter point, q′0 Q ' _0 point is like a dot. The light from the object point to the right will converge to the image point. The refractive index in the sphere is n N, the outer is n′n ', and the N>n′n>n '. According to the triangle sine theorem, the following two groups of relations are known:
Sin (u) r=sin (i) ssin (u′) r=sin (i′) s′\frac{sin (u)}{r} = \frac{sin (i)}{s}\\ \frac{sin (U ')}{r} = \frac{sin (i ')}{s '}
By the law of refraction we also know:
Nsin (i) =n′sin (i′) n sin (i) = n ' sin (i ')
The first two equations can be shaped, while using the refraction law:
S=sin (i) Sin (u) r=n′nsin (i′) sin (u) rs′=sin (i′