The 14 reference contexts in paper B. Conquet, L. Zambrano F., N. Artyukhina K., R. Fiodоrtsev V., A. Silie R., Б. Конкет, Л. Самбрано Ф., Н. Артюхина К., Р. Фёдорцев В., А. Силие Р. (2018) “Алгоритм и математическая модель геометрического позиционирования асферического составного зеркала // Algorithm and mathematical model for geometric positioning of segments on aspherical composite mirror” / spz:neicon:pimi:y:2018:i:3:p:234-242

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    DOI: 10.21122/2220-9506-2018-9-3-234-242 235 Introduction In recent years, the largest terrestrial telescopes operating in a wide spectral range of wavelengths use the technology of segmented composite mirrors
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    . A typical representative of this class is the Keck Observatory of two telescopes (Keck Telescope) Mauna Kea, Hawaii (1996), in which two 10 meter mirrors consist of 36 segments.
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    A typical representative of this class is the Keck Observatory of two telescopes (Keck Telescope) Mauna Kea, Hawaii (1996), in which two 10 meter mirrors consist of 36 segments. The most well-known method of aligning segmented mirrors is described in the works of Mast and Nelson
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    and found practical application in the alignment of Keck telescopes. According to this method, each individual segment is described as a local curve of an aspherical surface, and all together they form a common curve of the aspherical surface.
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    Segments have different geometric dimensions and a distorted hexagonal shape, i. e. elongate in the radial direction to ensure minimization of the intersegment gap area. The exact mutual position of the mirror segments relative to the base surface is established in two steps
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    . The first stage involves geometrical positioning, during which the plane segments are displaced along three linear directions (along the coordinate axes OX, OY and OZ in the base coordinate system) and the maximum reduction of all optical rays in the central region close to the center of the curvature of the mirror is achieved.
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    respect to the top of the mirror segment (two slopes with respect to the optical axis and rotation around it) and minimization of the wave front difference (aberrations) at the working wavelength of the telescope (Figure 1). The error in positioning and relative positioning of individual segments should not exceed the dimensions of the working wavelength
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    . Figure 1 – Diagram of geometric and opto-technical positioning of mirror segments The comparison method involves the process relative to a common reference surface by means of actuators (eg piezo drives).
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    segment is performed under the condition that the normal vectors constructed from the center of the flat surface of each segment must intersect at one calculated point on the axis OZ coinciding with the center of curvature of the base spherical surface. The tangent plane in space is determined by three Cartesian or spherical coordinates (Figure 1)
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    . However, in practice it is impossible to realize completely identical mirror segments, and their normals do not converge at the point of the double focal length of the mirror spheroid, but in some region of this point.
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    The optimization problem usually reduces to minimizing this region of convergence, as well as to reducing the difference between the real and calculated wavefronts, both for the entire composite mirror, and for each segment separately
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    . In [6], the efficiency of using the method of geometric computer positioning of 20 controllable hexagonal mirror segments forming a 500 mm composite mirror for 1 hour with an error of not more than 0.01 mm is shown in [6].
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    The optimization problem usually reduces to minimizing this region of convergence, as well as to reducing the difference between the real and calculated wavefronts, both for the entire composite mirror, and for each segment separately [9]. In
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    , the efficiency of using the method of geometric computer positioning of 20 controllable hexagonal mirror segments forming a 500 mm composite mirror for 1 hour with an error of not more than 0.01 mm is shown in [6].
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    In [6], the efficiency of using the method of geometric computer positioning of 20 controllable hexagonal mirror segments forming a 500 mm composite mirror for 1 hour with an error of not more than 0.01 mm is shown in
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    . In the classical scheme of alignment of similar elements using an autocollimator, it takes about 20 hours. It should be noted that the complexity of the alignment increases exponentially, so it took 1 year to set up a composite of positioning each individual mirror segment 236 telescope mirror Gran Telescopio CANARIAS with Segmented mirrors with regular hexagon
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    In addition, when mounting segments on supports, convenient placement of actuators and sensors at the edge points is provided for their subsequent positioning and optimal position control
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    . 3) The projection of segments and the coordinate system OXYZ. The segmented mirror is described in such a way that when projecting onto the XOY plane, the individual segments are a circular array of regular and evenly spaced hexagons relative to the main optical axis collinear with the geometric axis OZ [11].
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    The segmented mirror is described in such a way that when projecting onto the XOY plane, the individual segments are a circular array of regular and evenly spaced hexagons relative to the main optical axis collinear with the geometric axis OZ
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    . In the plane of the optical axis, two parameters determine the geometry of the projection of segments: the length of the side of the segment and the intersegment gap. The length of the side of the segment is determined by the distance between two adjacent vertices of the projected segment, while the intersegment gap is determined by the distance b
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    Intersegment gaps allow avoiding contact between adjacent segments and are assigned taking into account processing tolerances, in addition they allow compensation of gravitational and thermal deformations of the mirror cell
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    . The geometric size of the gap should be as small and uniform as possible across the entire width [13]. 5) Segmentation approach. In order to minimize the distortion of segments caused by the curvature of the aspherical mirror surface, two possible approaches to segmentation are proposed: a) Consider the case of a mosaic with regular hexagons and equal
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    Intersegment gaps allow avoiding contact between adjacent segments and are assigned taking into account processing tolerances, in addition they allow compensation of gravitational and thermal deformations of the mirror cell [12]. The geometric size of the gap should be as small and uniform as possible across the entire width
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    . 5) Segmentation approach. In order to minimize the distortion of segments caused by the curvature of the aspherical mirror surface, two possible approaches to segmentation are proposed: a) Consider the case of a mosaic with regular hexagons and equal gaps across the surface.
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    In order to minimize the distortion of segments caused by the curvature of the aspherical mirror surface, two possible approaches to segmentation are proposed: a) Consider the case of a mosaic with regular hexagons and equal gaps across the surface. According to Dan Curley's method
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    , we make the assumption that all segments are identical flat hexagons, with the best relative position relative to the common aspherical surface. In the first approximation on a flat surface we create an array of identical regular hexagons separated by homogeneous gaps. b) An alternative technique was proposed by Mast and Nelson fo
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    In the first approximation on a flat surface we create an array of identical regular hexagons separated by homogeneous gaps. b) An alternative technique was proposed by Mast and Nelson for the TMT (Thirteen-meter telescope)
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    , according to which hexagonal segments are not separated by spaces. An array аb a diameter of 10.4 m (73 m2). The purpose of the research was to develop an algorithm for solving the problem of geometric positioning of hexagonal segments of a mirror telescope, constructing an optimal circuit for traversing elements when aligning to the nearest
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