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Rational patches on Darboux and isotropic cyclides and their modeling applications - Part I Krasauskas, Rimvydas


Darboux and isotropic cyclides in \(\mathbb{R} P^3\) are projections of intersections of certain pairs of quadrics in \(\mathbb{R} P^4\). Hence they are particular cases of real Del Pezzo surfaces (of degree 4 or less), that are are known to be rational. Low-degree rational patches on these cyclides described in a form convenient for shape manipulations are the most interesting for modeling applications. Let \(\mathcal{A} P^1\) be a projective line over two cases of Clifford algebras \(\mathcal{A} = Cl(\mathbb{R}^3), Cl(\mathbb{R}^{2,0,1})\), generated by euclidean space \(\mathbb{R}^3\) and pseudo-euclidean space \(\mathbb{R}^{2,0,1}\) with signature \((++0)\). Our approach is to treat \(\mathcal{A} P^1\) as an ambient space and to consider toric Bezier patches in the corresponding homogeneous coordinates. It is proved that such patches of formal degree 2 with standard and non-standard real structures cover almost all cases of real Darboux and isotropic cyclides. The corresponding implicitization and parametrization algorithms are studied. The applications are related to families of circles on surfaces (including generation of 3-webs of circles) in case of Darboux cyclides and blending of Pythagorean-normal surfaces in case of isotropic cyclides.

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