Steiner Conics and Quadratic Bézier Curves with Control Mass Points

The authors propose a new approach of the Steiner problem. The Steiner problem consists in constructing a conic tangent to three sides of a triangle—possibly extended sides-. The method in use is based on Bézier curves with control mass points. Bézier curves with control mass points generalize classical rational Bézier curves by introducing control vectors in addition to control points. Here, any quadratic rational Bézier curves with three control mass points model conic arcs, semi-ellipses, and hyperbola branches while preserving rational parameterizations with any homographic parameter change. This method simplifies the construction and avoids implicit conic equations, solving systems of five linear equations in five unknowns, or reducing the implicit form of a conic. All conic features can be obtained by reparametrizations.