Stitching B-spline Curves Symbolically

We present an algorithm for stitching B-spline curves, which is different from the generally used least square method. Our aim is to find a symbolic solution for unifying the control polygons of arcs separately described as 4th degree B-spline curves. We show the effect of interpolation conditions and fairing functions as well.

Key words: B-spline curve, B-spline surface, merging,
interpolation, fairing

Lamoenian Circles of the Collinear Arbelos

We give an infinite sets of circles which generate Archimedean circles of a collinear arbelos.

Key words: arbelos, collinear arbelos, radical circle, Lamoenian circle
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The Parabola in Universal Hyperbolic Geometry I

We introduce a novel definition of a parabola into the framework of universal hyperbolic geometry, show many analogs with the Euclidean theory, and also some remarkable new features. The main technique is to establish parabolic standard coordinates in which the parabola has the form xz = y². Highlights include the discovery of the twin parabola and the connection with sydpoints, many unexpected concurrences and collinearities, a construction for the evolute, and the determination of (up to) four
points on the parabola whose normals meet.

Key words: universal hyperbolic geometry, parabola

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Conchoids on the Sphere

The construction of planar conchoids can be carried over to the Euclidean unit sphere. We study the case of conchoids of (spherical) lines and circles. Some elementary constructions of tangents and osculating circles are stil valid on the sphere. Further, we aim at the illustration and a precise description of the algebraic properties of the principal views of spherical conchoids, i.e., the conchoid's images under orthogonal projections onto their symmetry planes.

Key words: spherical curves, conchoids, algebraic curves, tangent, osculating circle, singularities, orthogonal projection

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On the Isoptic Hypersurfaces in the
n-Dimensional Euclidean Space

The theory of the isoptic curves is widely studied in the Euclidean plane E² (see [1] and [13] and the references given there). The analogous question was investigated by the authors in the hyperbolic H ² and elliptic E² planes (see [3], [4]), but in the higher dimensional spaces there is no result according to this topic.
In this paper we give a natural extension of the notion of the isoptic curves to the n-dimensional Euclidean space Eⁿ (n≥ 3) which are called isoptic hypersurfaces. We develope an algorithm to determine the isoptic hypersurface H_D of an arbitrary (n−1) dimensional compact parametric domain D lying in a hyperplane in the Euclidean n-space. We will determine the equation of the isoptic hypersurfaces of rectangles D  E² and visualize them with Wolfram Mathematica. Moreover, we will show some possible applications of the isoptic hypersurfaces.

Key words: isoptic curves, hypersurfaces, differential ge-
ometry, elliptic geometry

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Introduction to the Planimetry of the Quasi-
Hyperbolic Plane

The quasi-hyperbolic plane is one of nine projective-metric
planes where the absolute figure is the ordered triple {j₁, j₂,F}, consisting of a pair of real lines j₁ and j₂ through the real point F. In this paper some basic geometric notions of the quasi-hyperbolic plane are introduced. Also the classification of qh-conics in the quasi-hyperbolic plane with respect to their position to the absolute figure is given. The notions concerning the qh-conic are introduced and some selected constructions for qh-conics are presented.

Key words: quasi-hyperbolic plane, perpendicular points, central line, qh-conics classification, osculating qh-circle

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Isometries in Escher's Work