In the special case of ideal triangles, where all the angles are zero, the tessellation corresponds to the Farey tessellation and the triangle function yields the modular lambda function. This is a special case of a general scheme of Henri Poincaré that associates automorphic forms with ordinary differential equations with regular singular points. ![]() On the orientation-preserving normal subgroup, this two-dimensional representation corresponds to the monodromy of the ordinary differential equation and induces a group of Möbius transformations on quotients of hypergeometric functions.Īs the inverse function of such a quotient, the triangle function is an automorphic function for this discrete group of Möbius transformations. By the Schwarz reflection principle, the reflection group induces an action on the two dimensional space of solutions. Through the theory of complex ordinary differential equations with regular singular points and the Schwarzian derivative, the triangle function can be expressed as the quotient of two solutions of a hypergeometric differential equation with real coefficients and singular points at 0, 1 and ∞. The corresponding inverse function, first defined by Schwarz, solved the problem of uniformization a hyperbolic triangle and is called the Schwarz triangle function. Each such tessellation yields a conformal mapping of the upper half plane onto the interior of the geodesic triangle, generalizing the Schwarz–Christoffel mapping, with the upper half plane replacing the complex plane. The hyperbolic refections generates a discrete group, with an orientation-preserving normal subgroup of index two. Through successive hyperbolic reflections in its sides, such a triangle generates a tessellation of the upper half plane (or the unit disk after composition with the Cayley transform). a geodesic triangle in the upper half plane with angles which are either 0 or of the form π over a positive integer greater than one. Schwarz as a way of tessellating the hyperbolic upper half plane by a Schwarz triangle, i.e. # Use the extent's spatial reference to project the outputĪ mathematics, the Schwarz triangle tessellation was introduced by H. # Should result in a 4x4 grid covering the extent. # Multiply the divided values together and specify an area unit from the linear # Divide the width and height value by three. # Find the width, height, and linear unit used by the input feature class' extent # Describe the input feature and extract the extent Output_feature = r"C:\data\project.gdb\sqtessellation" My_feature = r"C:\data\project.gdb\myfeature" # Purpose: Generate a grid of squares over the envelope of a provided feature To determine the area of a shape based on the length of a side, use one of the following formulas to calculate the value of the Size parameter: The tool generates shapes by areal units. To generate a grid that excludes tessellation features that do not intersect features in another dataset, use the Select Layer By Location tool to select output polygons that contain the source features, and use the Copy Features tool to make a permanent copy of the selected output features to a new feature class. ![]() For example, select all features in column A with GRID_ID like 'A-%', or select all features in row 1 with GRID_ID like '%-1'. ![]() ![]() This allows for easy selection of rows and columns using queries in the Select Layer By Attribute tool. The format for the IDs is A-1, A-2, B-1, B-2, and so on. The GRID_ID field provides a unique ID for each feature in the output feature class. The output features contain a GRID_ID field. This occurs because the edges of the tessellated grid will not always be straight lines, and gaps would be present if the grid was limited by the input extent. To ensure that the entire input extent is covered by the tessellated grid, the output features purposely extend beyond the input extent.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |