Search results for "Definite symmetric bilinear form"
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Iterative construction of Dupin cyclides characteristic circles using non-stationary Iterated Function Systems (IFS)
2012
International audience; A Dupin cyclide can be defined, in two different ways, as the envelope of an one-parameter family of oriented spheres. Each family of spheres can be seen as a conic in the space of spheres. In this paper, we propose an algorithm to compute a characteristic circle of a Dupin cyclide from a point and the tangent at this point in the space of spheres. Then, we propose iterative algorithms (in the space of spheres) to compute (in 3D space) some characteristic circles of a Dupin cyclide which blends two particular canal surfaces. As a singular point of a Dupin cyclide is a point at infinity in the space of spheres, we use the massic points defined by J.C. Fiorot. As we su…
Iterative constructions of central conic arcs using non-stationary IFS
2012
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, t…