In an earlier paper we introduced a new quadric metric for simplifying triangle meshes using the edge collapse operation. The quadric measures both the geometric accuracy of the simplified mesh surface and the fidelity of appearance fields defined on the surface (such as normals or colors). The minimization of this quadric metric involves solving a linear system of size (3+m)×(3+m), where m is ...

We classify mutual position of a quadratic Bézier curve and a regular quadric in three dimensional Euclidean space. For given first and last control point, we find the set of all quadratic Bézier curves having no common point with a regular quadric. This system of such quadratic Bézier curves is represented by the set of their admissible middle control points. The spatial problem is reduced to ...

Many algorithms for reducing the number of triangles in a surface model have been proposed, but to date there has been little theoretical analysis of the approximations they produce. Previously we described an algorithm that simplifies polygonal models using a quadric error metric. This method is fast and produces high quality approximations in practice. Here we provide some theory to explain w...

For q odd, it was shown in [3] that the elliptic quadric Q−(7, q) possesses a unique 1-system, the so-called classical 1-system. Here, the same result will be obtained for even q. 1 Basic properties of 1-systems of Q−(7, q) A 1-system M of the elliptic quadric Q−(7, q) is a set {L0, L1, . . . , Lq4} of q4+1 lines of Q−(7, q) with the property that every plane of Q−(7, q) containing a line Li of...

Up to now, the shortest invariant representation of a quadric has 138 summands and there has been no invariant representation of a twisted cubic in 3D projective space, which limit to some extent the applications of invariants in 3D space. In this paper, we give a very short invariant representation of a quadric cone, a special quadric, which has only two summands similar to the invariant repre...

We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration generalizing the Kapranov’s description of...

A scheme is described for incorporation of scene constraints into the structure from motion problem. Specifically, the absolute quadric is recovered with constraints imposed by orthogonal scene planes. The scheme involves a number of steps. A projective reconstruction is first obtained, followed by a linear technique to form an initial estimate of the absolute quadric. A nonlinear iteration the...

The design-for-assembly technique requires realistic physically based simulation algorithms and in particular efficient geometric collision detection routines. Instead of approximating mechanical parts by large polygonal models, we work with the much smaller original CAD-data directly, thus avoiding precision and tolerance problems. We present a generic algorithm, which can decide whether two s...

We present a method for simulating slot-scanning X-ray imaging using the general-purpose Monte Carlo simulation package PENELOPE and penEasy Imaging. Different phantoms can be defined with the PENGEOM package, which defines bodies as combinations of volumes limited by quadric surfaces. The source-detector geometry, the position of the object, the collimator, the X-ray tube properties, the detec...

Interpolating trajectories of points and geometric entities is an important problem for kinematics. To describe these trajectories, several algorithms have been proposed using matrices, quaternions, dual-quaternions, the Study quadric; last one allows embedding motors as 8D vectors into projective space P7, where interpolation rotations translations becomes a linear problem. Furthermore, confor...