Abstract: This machine comprises a carriage movable in translation and in oscillation in a direction parallel to and perpendicular to a shaft carrying trimming and bevelling grinding wheels, and further comprises a tracer (7) adapted to come in contact with the edge of a trimmed blank (3) of the glass in a direction perpendicular to this edge, and adapted to measure the distance L1 of the edge of the front side of the glass (3) relative to the edge of a planar glass in respect of which L1=0, and the value L2 which corresponds to the thickness of the edge of the glass. The tracer (7) has a curvature which is identical to and concentric with the curvature of the grinding wheel (4, 11) and formed by a plurality of parallel elements (9) acting on a sensor (10).