The first step of the subdivision algorithm is to split input triangles into smaller ones, allowing to represent more geometry detail. In our implementation, we will focus only on individual triangles, and we will not try to adaptively subdivide the surface based on surface detail or curvature.

Each triangle will be subdivided into smaller triangles uniformly , and all new triangles will lie in the same plane as the original triangle.

## Application Of Remote Sensing In Metrology

To achieve this, we will make use of barycentric coodrinates , which provide a very simple way to parametrize a triangle:. Parameterising on u and v , we can then subdivide the original triangle into smaller triangles, each subdivision level adding 2 more triangles than the previous level i. The subdivided triangle can be rendered using a set of triangle strips, one for each subdivision level, saving on the output vertex count of our geometry shader.

The number of vertices for each subdivision level i. The total number of output vertices of the compute shader is computed via its sum :.

### Casting Rays into the scene

Given the usual vertex output limitation of vertices, the maximum subdivision level is This shader renders each triangle strip using a separate colour. With maximum subdivision level, this leads to a "stripy" render:. The equations describing the cubic patch is described in Section 3. In this section, we will not repeat its mathematical description, and focus only on implementation in GLSL. First, lets update the fragment shader to display per-polygon normals, which simplify the debugging of the geometry shader:.

We will store the cubic patch data in a structure Coeffs , which is computed in the body of the shader's main function.

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We then use this structure to emit individual vertices of each patch in the emitVertex function. Finally, a loop in main will generate separate triangles strips, based on the subdivision level. This leads to the following output notice the additional patches and the "bulging" of the tetrahedron's faces produced by the subdivision :. The final step is to implement the normals generation using a quadratic patch , as described in Section 3. Despite these checks, it is possible to obtain incorrect indexing solutions, especially if the geometry is not accurate or the peak-search parameters are not set appropriately.

Therefore, it is important to check the alignment of peaks visually, just as before when setting up the peak detection. Alternatively, the cxiview program from Cheetah can also be used. Some experience is needed to be able to judge whether or not the spot positions are correct.

The most important thing is that the patterns of reflections agree; for example, a line or arc of reflections should correspond to a line or arc of true spots, even if there might be an excessive number of reflections predicted either side of the true spots. If there are a large number of patterns where the calculated reflection patterns do not agree with the true spots, the earlier stages should be revisited. The prediction-refinement algorithm also refines the position of the central beam on the detector for each frame.

The resulting shifts are stored in the stream, and these should also be checked using a scatter plot. This can be performed using a script called detector-shift provided with CrystFEL :. The best outcome is for the shifts to be clustered tightly around the origin of the graph. Successful indexing despite an initially incorrect beam position will show as a cluster of points away from the origin for an example, see Fig. This script can also be used to correct the offset by running it again, this time telling it the name of the geometry file which should be updated:.

This command will result in my-predrefine. The detector-shift tool can only correct offsets of the entire detector in two directions perpendicular to the beam direction note that the detector panels need not necessarily be in this plane. A more advanced program in CrystFEL called geoptimiser can be used to refine the detector geometry down to the level of individual panels. The way that geoptimiser works has been described in detail in a separate paper Yefanov et al.

In particular, it allows the panels to be handled in groups which are either treated rigidly or used to calculate adjustments for panels for which there is not sufficient information for an individual refinement by using the average of the adjustments for the other panels in the group. A few thousand indexed patterns should be used for a good refinement, and a typical command line for geoptimser would be as follows:. Here, the stream my. The names asics and quadrants are defined in the geometry file.

For detectors which can be treated as single panels, which includes most detectors at synchrotron beamlines to a good approximation, it will not be necessary to use these panel-grouping options. Certain symmetry classes are subject to indexing ambiguities.

The situations in which this can happen, and algorithms to address the problem, have been amply discussed previously White et al. CrystFEL includes a simplified version of the Brehmâ€”Diederichs algorithm, the details of which have been given previously White, Mariani et al. If the space group of the structure under investigation is known in advance, it is easy to know whether an ambiguity is possible or not.

A table of symmetry classes point groups and space groups is included with CrystFEL which highlights the relevant groups. The mathematical basis of an indexing ambiguity, and the manifestation of an indexing ambiguity in the data from an SX experiment, is exactly the same as that of perfect twinning by merohedry or pseudomerohedry, in the case of approximate overlap of spot positions.

At a minimum, ambigator needs the stream file from the previous step which was called my. Typical use will add the further options --fg-graph , --lowres , --highres and -j , which are described below:. The -j option tells ambigator to perform calculations in parallel across multiple CPUs, as described earlier for indexamajig. The --fg-graph option tells ambigator to write some diagnostic information to a file called fg.

The notation used to specify point-group symbols is described in detail in Section 5. If the range is too narrow, there will not be enough reflections for an accurate calculation. The best resolution range depends on the data, so it needs to be adjusted based on the results.

### Introduction

Firstly, the re-indexing operation, calculated from the true and apparent point-group symmetries, is shown. The two values will be similar on the first iteration here, they are 0. If the resolution of the ambiguity is successful, the value of f should approximately double while the value of g should decrease approximately to zero.

Here, the final values are 0. If the final value of g is significantly above zero, the resolution range may be too large.

If the final value of f is not much larger than its starting value, a larger resolution range may be necessary, or the earlier data-processing steps should be revisited. Ambigator will create a new stream detwinned. If the ambiguity resolution was successful, this can be merged normally, as described in the next section.

The file fg. This can be used to plot graphs as shown in White, Mariani et al. For this purpose, a script called fg-graph is provided. The script must first be edited to set the filename usually fg. Then, it can be run very simply,. This will produced an image file correlation.

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In practice, about correlation coefficients are normally sufficient, depending on the quality and the resolution of the data. If the number of crystals is large, as is the case in the example above, with an average of This will make it finish sooner, usually without affecting the results. Before merging, it is wise to check for reflections that were too strong to be recorded by the detector. They can be removed while merging, but the highest reliable pixel value must be known.

Unfortunately, the maximum recordable value can vary as the calibration constants for the detector change over a period of days. This is a particular problem for detectors at XFEL facilities. The peakogram-stream script is provided to help to find the saturation value. It simply plots a point for every reflection in a stream, with the horizontal position being the resolution and the vertical position being the maximum intensity of all pixels in the integration region for that reflection. The plot is colour-coded according to the local density of spots.

It is simply run on the stream:.

## Geometry - Wikipedia

An example graph is shown in Fig. The maximum values for some reflections, particularly at low resolution, have been clipped to lower values. Normally, this would manifest itself as a sharp horizontal line in the graph.