# Why is there a diference in sign convention for the radius of curvature of spherical mirrors compared to curved refracting surfaces?

In the context of spherical mirrors we have

• $R > 0$ for concave mirrors
• $R < 0$ for convex mirrors

But for curved refracting surfaces we have

• $R < 0$ for concave mirrors
• $R > 0$ for convex mirrors

Why is that so?

• this is the convention that makes most sense in my opinion but quite many people use the opposite convention ($R<0$ for concave mirrors ans $R>0$ for convex mirrors). One should bear that in mind and always double check, and best is to always clarify (e.g. using vex and cav as abbreviations) – riddleculous May 11 '17 at 9:55

The general sign rule for the radius of curvature of a spherical surface is the following

When the center of curvature $C$ is on the same side as the outgoing light, $R> 0$ (the radius of curvature is positive), otherwise $R < 0$

Now in spherical mirrors we have

• For concave mirrors, $C$ is always on the same side as the outgoing light, hence $R > 0$
• For convex mirrors, $C$ is never on the same side as the outgoing light, hence $R < 0$

But in general curved refracting surfaces (i.e. not reflecting like mirrors) we have

• For concave mirrors, $C$ is never on the same side as the outgoing light, hence $R < 0$
• For convex mirrors, $C$ is always on the same side as the outgoing light, hence $R > 0$