Abstract: In order to represent conic sections with cubic weighted Lupaş q-Bézier curves, the necessary and sufficient con-ditions and classification for representing conics by cubic weighted Lupaş q-Bézier curves are investigated. Firstly, the necessary and sufficient conditions for a cubic weighted Lupaş q-Bézier curves degenerating into a quadratic weighted Lupaş q-Bézier curve are obtained. Then, by adopting Wachspress coordinates, several necessary and sufficient conditions for representing conics by cubic weighted Lupaş q-Bézier curves are approached. Moreover, the shape-invariant factor and classification for representing conics with cubic weighted Lupaş q-Bézier curves are obtained. Particularly, some essential geometrical conditions are gained by introducing Wachspress coordi-nates. Numerical experiments show that conic sections represented by cubic weighted Lupaş q-Bézier curves can adjust the curves’s type and shape flexibily by choosing different shape parameters.