In this paper, the drawing of the intersection between a straight line and an ellipsoid, using auxiliary graphic constructions, is presented. This kind of intersection demands many preparatory steps and auxiliary graphic drawings, as the finding of the tangent planes. The draft has been obtained using both the means offered by the descriptive geometry and the AutoCAD program. So, a very precise construction of the intersection points coordinates is guaranteed. Ellipsoidal shells are used in the chemical and petrochemical industry, as ellipsoidal heads for: pressure vessels, heat exchangers, posts etc. On these heads, some fittings can be installed and their axes can have various angles to the horizontal or vertical axis.