This thesis presents an exact and complete approach for visualization of segments and points of real plane algebraic curves given in implicit form f(x, y) = 0. A curve segment is a distinct curve branch consisting of regular points only. Visualization of algebraic curves having self-intersection and isolated points constitutes the main challenge. Visualization of curve segments involves even mo...

Subdivision schemes with matrix masks are a natural extension of the well studied case of subdivision schemes with scalar masks Such schemes arise in the analysis of multivariate scalar schemes in subdivision processes corresponding to shift invariant spaces generated by more than one function in geometric modeling where each component of the curve surface is designed by a di erent linear combi...

If (Q,A) is a marked polygon with one interior point, then a general polynomial f ∈ K[x, y] with support A defines an elliptic curve Cf on the toric surface XA. If K has a non-archimedean valuation into R we can tropicalize Cf to get a tropical curve Trop(Cf ). If the Newton subdivision induced by f is a triangulation, then Trop(Cf ) will be a graph of genus one and we show that the lattice len...

Subdivision provides a powerful scheme for building smooth and complex surfaces. But the number of faces in the uniformly refined meshes increases exponentially with respect to subdivision depth. Adaptive rendering reduces the number of faces needed to yield a smooth approximation to the limit surface and, consequently, makes the rendering process more efficient. In this paper, we present a new...

Given a segment of a conic section in the form of a rational Bézier curve, a quadratic spline approximation is constructed and an explicit error bound is derived. The convergence order of the error bound is shown to be O(h) which is optimal, and the spline curve is both C and G. The approximation method is very efficient as it is based on local Hermite interpolation and subdivision. The approxi...

This article describes an image segmentation procedure applied over magnetic resonance (MR) data-images. The applied technique is based on a contour propagation process deened by an energy minimization analysis. The nal contour of each volume of the heart is deened by a propagation process (spline deformable curve) added to a re-sampling procedure which increases the number of control points. T...

A new parametrization technique and its applications for general Catmull-Clark subdivision surfaces are presented. The new technique extends J. Stam’s work by redefining all the eigen basis functions in the parametric representation for general Catmull-Clark subdivision surfaces and giving each of them an explicit form. Therefore, the new representation can be used not only for evaluation purpo...

Implicit curves are widely used in computer graphics because of their powerful features for modeling and their ability for general function description. The most popular rasterization techniques for implicit curves are space subdivision and curve tracking. In this paper we are introducing an efficient curve tracking algorithm that is also more robust then existing methods. We employ the Predict...

Given a Cylindrical Algebraic Decomposition of an implicit algebraic curve, visualizing distinct curve arcs is not as easy as it stands because, despite the absence of singularities in the interior, the arcs can pass arbitrary close to each other. We present an algorithm to visualize distinct connected arcs of an algebraic curve efficiently and precise (at a given resolution), irrespective of h...

A sufficient condition for C 1-continuity of subdivision surfaces was proposed by Reif [17] and extended to a more general setting in [22]. In both cases, the analysis of C 1-continuity is reduced to establishing injectivity and regularity of a characteristic map. In all known proofs of C 1-continuity, explicit representation of the limit surface on an annular region was used to establish regul...