Symmetry of linear aggregates of second-order curves (conoprimas)
Abstract
It is clear that the complete aggregate, that is, the quint of conoprimas, possesses the highest possible symmetry, that is, circular symmetry. The symmetry of quarts is completely determined by the symmetry of one conoprima, because the symmetry is derived from it is completely and unambiguously. Therefore, in the general case, such a aggregate has a twofoldaxis of symmetry and two perpendicular planes of symmetry (orthorhombic type of symmetry in the plane). In the particular case of the parabola, only one plane of symmetry remains (the hemiorthorhombic type of symmetry). The circle possesses absolutely exceptional symmetry, and therefore there exist linear quarts that exhibit circular symmetry. From this we conclude that if one takes an arbitrary conoprima and a pentad axis of symmetry, from which five equal elements are derived to define a linear quart, the resulting a quart will possess circular symmetry. All curves contained in it, are in every orientation, arranged in continuous circles of equal elements.
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