Question

define new cartesian sign convention

Answer 1

Cartesian Sign Convention. 1) Light initially propagates from left to right. 2) The origin of the Cartesian coordinate system is at the centre of the optical component. 3) Distances measured normal to the optic axis are positive above and negative below.

Answer 2

sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. ... Sometimes, the term "sign convention" is used more broadly to include factors of i and 2π, rather than just choices of sign.
Cartesian Sign Convention: In the case of spherical mirror all signs are taken from Pole of the spherical mirror, which is often called origin or origin point. ... Sign is taken as – (negative) from pole of a spherical mirror towards object along the principal axis.