Abstract: A method and apparatus for conferring a cup shape to the terminal junction segment of pipes that are bi-axially oriented longitudinally and circumferentially and hence very sensitive to diameter and length reduction through heat. The apparatus includes a furnace which heats the segment to a differentiated temperature, increasing towards the end of the segment such that the inner diameter of the terminal segment progressively drops down to a controlled value as temperature increases (whilst length of the pipe is simultaneously reduced, with a corresponding increase in thickness of the wall of the terminal segment). Preferably then, in an appropriate station, an additional heating is executed to a plastic deformation temperature suited to obtain a correct preliminary dilation of the terminal segment, upon introducing a rigid element which acts as inner contrast, thereby inhibiting any retraction thereof.