Rational quartic spectrahedra

Rational quartic spectrahedra in $3$-space are semialgebraic convex subsets in $\mathbb{R} ^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in $\mathbb{C}\mathbb{P} ^3$ of the boundary of a rational spectrahedron is a rational complex symmetroid. We give necessary conditions on the configurations of singularities of the corresponding real symmetroids in $\mathbb{R} \mathbb{P} ^3$ of rational quartic spectrahedra. We provide an almost exhaustive list of examples realizing the configurations, and conjecture that the missing example does not occur.

Blending Cylinders and Cones using Canal Surfaces

There is reviewed the construction of a rational blending surface between cylinders and cones in some interlocation cases. This surface is constructed as a patch of rolling ball envelope, i.e. as a patch of tangent canal surface of rational-variable radius. This construction defines rational parameterization of a blending surface. The constructed surface is Laguerre invariant.

Specified–Precision Computation of Curve/Curve Bisectors

The bisector of two plane curve segments (other than lines and circles) has, in general, no simple — i.e., rational — parameterization, and must therefore be approximated by the interpolation of discrete data. A procedure for computing ordered sequences of point/tangent/curvature data along the bisectors of polynomial or rational plane curves is described, with special emphasis on (i) the identification of singularities (tangent–discontinuities) of the bisector; (ii) capturing the exact rational form of those portions of the bisector with a terminal footpoint on one curve; and (iii) geometrical criteria the characterize extrema of the distance error for interpolants to the discretely–sample data. G1 piecewise– parabolic and G2 piecewise–cubic approximations (with O(h4) and O(h6) convergence) are described which, used in adaptive schemes governed by the exact error measure, can be made to satisfy any prescribed geometrical tolerance.