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Exact Methods for PDEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
A commonly used conformal map is the Schwartz–Christoffel transformation. This maps a closed polygonal figure (with n vertices) into a half plane. The mapping is given by the solution of dzdζ=C(ζ−ζ1)β1/π−1(ζ−ζ2)β2/π−1⋯(ζ−ζn)βn/π−1
Chapter 1
Published in Pearson Frederick, Map Projections:, 2018
A conformal projection is one in which the shape of a figure on the Earth is preserved in its transfer to a map. This concept is valid at a point, but must not be extended over large areas. The condition for a conformal map may be stated as, at a point, the scale is the same in all directions. Another basic feature is that at a point, angles are the same on the map as they are on the model of the Earth. Thus, locally, angular distortion is zero. This feature is purchased at the price of distortion in size. Figure 4 illustrates the result of this conformal transformation. The resulting quadrilaterals are in the same shape after transformation. However, the size has either increased or diminished. For the figure, it is evident that this projection is orthomorphic (same form), another term used for this projection.
Planar Shape Deformation
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
The domain Ω $ \Omega $ is equipped with a colored texture and we are interested in visualizing the map f $ \boldsymbol{f} $ by rendering a high quality image of the codomain. The algorithms that we consider for computing the underlying map f $ \boldsymbol{f} $ are indifferent to the choice of such texture. These algorithms are solely based on geometric considerations; however, for the result to be valuable from the graphics application standpoint, some requirements have to be added. A high quality result should preserve the fine details of the underlying texture. Hence, smoothness of the map is often required. Luckily, this is relatively easy to obtain when working with barycentric coordinates as it is implied by the smoothness of the coordinates themselves. In addition to smoothness, we often desire that the amount of metric distortion induced by the map is as low as possible. We consider both the conformal and the isometric distortion, where conformal distortion refers to the amount by which angles between arbitrary intersecting curves are altered under the action of the map, and isometric distortion refers to the amount of change in the length of arbitrary curves. A map is said to be conformal if it has zero conformal distortion everywhere. The Jacobian matrix Jf $ J_{\boldsymbol{f}} $ of a conformal map is a similarity transformation, allowing only rotations with isotropic scale, but it can have arbitrarily high isometric distortion. Ideally, a map should be isometric, with a Jacobian that is restricted to be a rotation (without scale). Unfortunately, unlike the space of conformal maps, the space of isometries in R2 $ \mathbb R ^2 $ is fairly restricted and contains only rigid motions (a global rotation and translation) and reflections. Hence, we can only hope for maps that are close to isometry, having a low amount of isometric and conformal distortion.
Evaluating the geometric aspects of integrating BIM data into city models
Published in Journal of Spatial Science, 2020
Jing Sun, Perola Olsson, Helen Eriksson, Lars Harrie
Because none of the BIMs are georeferenced (which is a common problem, see Ohori et al. (2017) it is impossible to compare absolute coordinates in the quantitative analysis. Therefore, a relative comparison of the models was performed and constructed in such a way that it would give a result similar to that of an absolute comparison if the BIM data were appropriately georeferenced. The BIMs have a Cartesian coordinate system (often called an engineering system), whereas the CityGML models have a geodetic reference system, most commonly based on a conformal map projection and an orthometric height system. The geodetic coordinates of the origin of the engineering system is defined in the IFC file by either IfcMapConversion (Easting, Northing, and OrthogonalHeight) or IfcSite (RefLatitude, RefLongitude, and RefElevation; BuildingSMART 2018), where the latter is mainly used for an approximate location. If this information is stored in the IFC model, it is said to be georeferenced. However, the georeferencing is still not straightforward because there are geometric differences between the engineering and geodetic systems, especially if the latter is based on a map projection. First, the map projection introduces scale distortions, and second, the height axis (for the orthometric heights) is not perfectly orthogonal to the map projection plane. However, if an appropriate map projection is chosen and the extent of the BIM is small, the errors introduced are negligible. In our study, the footprint and laser scanning data are stored in local projections (with small scale distortions) and our BIMs are small (single buildings) and we can therefore safely disregard the problems arising from the different coordinate systems (see Uggla and Horemuz (2018) for techniques to handle coordinate transformations for larger-scale BIM data).