Iterative constructions of central conic arcs using non-stationary IFS
Résumé
Several methods of subdivision exist to build parabola arcs or circle arcs in the usual Euclidean affine plane. Using a compass and a ruler, it is possible to construct, from three weighted points, circles arcs in the affine space without projective considerations. This construction is based on rational quadratic Bézier curve properties. However, when the conic is an ellipse or a hyperbola, the weight computation is relatively hard. As the equation of a conic is $\qaff(x,y)=1$, where $\qaff$ is a quadratic form, one can use the pseudo-metric associed to $\qaff$ in the affine plane and then, the conic geometry is also handled as an Euclidean circle. At each step of the iterative algorithm, the constructed point belongs to a principal perpendicular bissector of the control polyhedron and then, our construction is regular. Moreover, we can pass through the point at infinity when the bounds do not belong to the same branch of the hyperbola, using massic points defined by J.C. Fiorot: we compute two subdivisions with two collinear direction vectors of the same asymptote such that these two vectors have opposite senses. Moreover, at each step, we know the tangent lines to the conic at the built vertex. At each step, two weights are equal to $1$ or $0$, we just have to find an induction relation to compute the third weight and so, it is possible to run our algorithms using a non-stationary I.F.S.