Offsets, Conchoids and Pedal Surfaces

We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces.

Estimation of satellite antenna phase center offsets for Galileo

Satellite antenna phase center offsets for the GalileoInOrbitValidation(IOV) and FullOperationalCapability (FOC) satellites are estimated by two different analysiscenters based on tracking data of a global GNSS network. The mean x- and y-offsets could be determined with a precision of a few centimeters. However, daily estimates of thex-offsets of the IOV satellites show pronounced systematic effects with a peak-to-peak amplitude of up to 70 cm that depend on the orbit model and the elevation of the Sun above the orbital plane. For the IOV y-offsets, no dependence on the orbit model exists but the scatter strongly depends on the elevation of the Sun above the orbital plane. In general, these systematic effects are significantly smaller for the FOC satellites. The z-offsets of the two analysis centers agree within the 10–15 cm level, and the time series do not show systematic effects. The application of an averaged Galileo satellite antenna model obtained from the two solutions results in a reduction of orbit day boundary discontinuities by up to one third—even if an independent software package is used.