The present invention relates to a beverage preparation machine (1) with a machine body (2) and a machine dispensing head (3), characterized in that said dispensing head (3) and machine body (2) comprise cooperating means (12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ghd, gmb, gil, cmb) that allow movement of said head (3) relative to said body (2) along a non-linear path, and such that the trim of the dispensing head (3) is constant whatever its position.