fringent needles. The same treatment does not affect the cobalt crystals noticeably: Boiling a quinoline mount of the cobalt-nickel-quinoline mixed crystals left two species, the colorless birefringent needles and blue-violet crystals.Quinoline has been used as a reagent
The identification of isometric displacements of studied objects with utilization of the vector product is the aim of the analysis conducted in this paper. Isometric transformations involve translation and rotation. The behaviour of distances between check points on the object in the first and second measurements is a necessary condition for the determination of such displacements. For every three check points about the measured coordinate, one can determine the vector orthogonal to the two neighbouring sides of the triangle that are treated as vectors, using the definition of the vector product in three-dimensional space. If vectors for these points in the first and second measurements are parallel to the studied object has not changed its position or experienced translation. If the termini of vectors formed from vector products treated as the vectors are orthogonal to certain axis, then the object has experienced rotation. The determination of planes symmetric to these vectors allows the axis of rotation of the object and the angle of rotation to be found. The changes of the value of the angle between the normal vectors obtained from the first and second measurements, by exclusion of the isometric transformation, are connected to the size of the changes of the coordinates of check points, that is, deformation of the object. This paper focuses mainly on the description of the procedure for determining the translation and rotation. The main attention was paid to the rotation, due to the new and unusual way in which it is determined. Mean errors of the determined parameters are often treated briefly, and this subject requires separate consideration.
Various curves are used for shaping the geometry of roads and a curve applied the most often is the circular arc. Clothoid, spiral, ellipse, sets of curves shaped with polynomial functions and other similar curves are also worth mentioning. The road axis is usually set during practical realization of the curves in the terrain. The external edges are determined by measuring the width of the carriageway along the normal line to this axis. That is how two offset curves, which usually are of different type, are created. If the road axis is a circular arc with a radius R, then by displacing the width of the carriageway s along the radius of the circle, the circular arcs with radii R-s and R+s are also created, assuming that s is constant. While shaping turbo-roundabouts different curves are suggested, e.g. an ellipse. In case of its application, by displacing the width of the carriageway from its axis, the curves that fulfill the conditions of ellipses' equations are not created, but only the curves similar to them. The question is how large are the deviations? Are they important in practical realization? The analyses included in this article give answers to these questions.
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