In geometric optics the relationships are derived assuming that the light rays are paraxial; i.e. they lie close to the optical axis of the system and the angle between them and the axis is small. Strictly speaking the relationships derived in geometric optics are true in the limit as the deviations from the optical axis go to zero. Since this is clearly a severe restriction on the validity of optical theory it was a great accomplishment to find a relationship that did not presume paraxial conditions. Ernst Karl Abbe, a very able German mathematical physicist, found such a condition. He had been hired by Carl Zeiss to provide analyses useful for improving optical instruments. Abbe was working out the analysis of microscopes when he discovered the sine conditions.

Consider an image of an object formed by a spherical lens interface. Let y_O be the deviation of a point P_O in the object from the optic axis. The deviation of the corresponding point P_I in the image is denoted as y_I. Let C be the center of curvature of the spherical lens. The angle between the line connecting P_O and C and the optic axis is denoted as α_O. The angle between the line connecting C with P_I and the optic axis is denoted as α_I. If α_O is positive then α_I is negative so it is common practice to work with the absolute values of the angles. Let n_O and n_I be the indices of refraction on the object-side and image-side of the spherical interface.

The sine condition is then

y_On_Osin(|α_O|) = y_In_Isin(|α_I|)
 

If there were no distortion in the image the value of y_I would be proportional to y_O; i.e.,

y_I = ky_O
 

where k is a constant independent of y_O. Instead

y_I = (n_O/n_I)sin(|α_O|)/sin(|α_I|)y_O
 

where α_O depends upon y_O.

If the signs of the angles are taken into account Abbe's sine condition takes the form

y_On_Osin(α_O) + y_In_Isin(α_I) = 0