We prove that if annth degree rational Bézier curve has a singular point, then it belongs to the two (n − 1)th degree rational Bézier curves defined in the (n − 1)th step of the de Casteljau algorithm. Moreover, both curves are tangent at the singular point. A procedure to construct Bézier curves with singularities of any order is given. 2001 Elsevier Science B.V. All rights reserved.