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co.combinatorics – reference for flat diagram results

A colleague and I write a paper in which we need to use some basic facts about planar graphs. If possible, I would prefer to just give references for the results, since the evidence is not even important for the (long) paper and I have always liked to write evidence about planar embedding and related things (so you save all evidence) -seid also readers). Unfortunately I could not find any references for these specific results and so I thought I would post them here and ask if anyone can provide references.

If you do not know any specific references, but you may know some short, preferably combinatorial, evidence for these results based on results for which you have references, this can also be helpful.

I assume that flat embeddings are polygonal, ie. H. Edges are embedded as finitely many straight line segments. I can not assume that my graphics are 3-connected. By face cycle I mean a cycle of the graph that is the boundary of a face in a planar embedding. Below are the results for which I am looking for references:

Sentence 1: Accept $ G $ is a plane graph with cycle $ C = u_1, ldots, u_k $, To let $ G & # 39; $ let the graph be obtained from $ G $ by adding a new vertex $ u $ next to everyone $ u_i $, Then $ C $ is a facial cycle of $ G $ then and only if $ G & # 39; $ is planar. In addition, any planar embedding of $ G $ in which $ C $ limited some face $ F $ can be extended to a planar embedding of $ G & # 39; $ Where $ u $ is embedded in $ F $ and then connected to the vertices $ u_1, ldots, u_k $,

Sentence 2: To let $ G $ be a graph and let it go $ S subseteq V (G) $, Then $ G $ has a planar embedding that has an area that is incident on each vertex of $ S $ if and only if the graphics $ G & # 39; $ received from $ G $ by adding a new vertex $ s $ adjacent to each element of $ S $ is planar. In addition, any planar embedding of $ G $ in which the vertices of $ S $ are incident on a common face $ F $ can be extended to a planar embedding of $ G & # 39; $ Where $ s $ is embedded in $ F $ and then connected to each vertex of $ S $,

Sentence 3: Suppose that $ G_1 $ and $ G_2 $ are plane graphs with facial cycles $ C_1 = u ^ 1_1, ldots, u ^ 1_ ell $ and $ C_2 = u ^ 2_1, ldots, u ^ 2_k $ respectively. To let $ G & # 39; $ let the graph be obtained from $ G_1 cup G_2 $ by adding edges $ u ^ 1_1u ^ 2_1, ldots, u ^ 1_m u ^ 1_m $ for some $ m le min { ell, k } $, Then $ G & # 39; $ is flat and $ C = = u ^ 1_m, u ^ 1_ {m + 1}, ldots, u ^ 1_ ell, u ^ 1_1, u ^ 2_1, u ^ 2_k, u ^ 2_ {k-1}, ldots, u ^ 2_m $ is a face cycle.

Flat shaded triangles with the ModelBuilder?

I decided to give it ModelBuilder a try, but I can not get rid of the smoothing of my terrain. So, what I have tried:

  • Create each triangle individually with its own vertex position.
  • Put each triangle in its own part,
  • Insert each triangle for yourself node,
  • After you set the vertex positions, set the indexes one at a time.
  • Used ensureTriangleIndices and ensureCapacity to set the vertex and the index number.

Executable test

import com.badlogic.gdx.ApplicationAdapter;
import com.badlogic.gdx.Gdx;
import com.badlogic.gdx.graphics.Color;
import com.badlogic.gdx.graphics.GL20;
import com.badlogic.gdx.graphics.PerspectiveCamera;
import com.badlogic.gdx.graphics.VertexAttributes.Usage;
import com.badlogic.gdx.graphics.g3d.Environment;
import com.badlogic.gdx.graphics.g3d.Material;
import com.badlogic.gdx.graphics.g3d.Model;
import com.badlogic.gdx.graphics.g3d.ModelBatch;
import com.badlogic.gdx.graphics.g3d.ModelInstance;
import com.badlogic.gdx.graphics.g3d.attributes.ColorAttribute;
import com.badlogic.gdx.graphics.g3d.environment.DirectionalLight;
import com.badlogic.gdx.graphics.g3d.model.Node;
import com.badlogic.gdx.graphics.g3d.utils.CameraInputController;
import com.badlogic.gdx.graphics.g3d.utils.MeshPartBuilder;
import com.badlogic.gdx.graphics.g3d.utils.ModelBuilder;
import com.badlogic.gdx.math.Vector3;

/**
 * libgdxtestenvironment (2019)
 * By Menno Gouw
 */
public class TerrainBuilderTest extends ApplicationAdapter {

    PerspectiveCamera cam;
    CameraInputController camController;
    ModelBatch modelBatch;
    Model model;
    ModelInstance instance;
    Environment environment;

    //CreateTerrain createTerrain;
    // Hardcoded heightmap for testing
    private final int()() heightMap = {
            {0, 1, 3, 1, 0},
            {1, 2, 1, 3, 1},
            {2, 3, 2, 2, 1},
            {2, 4, 2, 1, 2},
            {1, 3, 2, 1, 2},
    };

    @Override
    public void create () {
        modelBatch = new ModelBatch();
        setEnvironment();

        setCamera();

        //createTerrain = new CreateTerrain(16, 16);
        instance = new ModelInstance(myModelBuilder(heightMap));

        Gdx.input.setInputProcessor(camController = new CameraInputController(cam));
    }

    private void setCamera() {
        cam = new PerspectiveCamera(67, Gdx.graphics.getWidth(), Gdx.graphics.getHeight());
        cam.position.set(0f, 7f, 10f);
        cam.lookAt(0, 0, 0);
        cam.near = 1f;
        cam.far = 50f;
        cam.update();
    }

    private void setEnvironment() {
        environment = new Environment();
        environment.set(new ColorAttribute(ColorAttribute.AmbientLight, .4f, .4f, .4f, 1f));
        environment.add(new DirectionalLight().set(1, 1, 1, -.2f, -.7f, -.1f));
    }

    @Override
    public void render () {
        camController.update();

        Gdx.gl.glViewport(0, 0, Gdx.graphics.getBackBufferWidth(), Gdx.graphics.getBackBufferHeight());
        Gdx.gl.glClearColor(0, 0, 0, 1);
        Gdx.gl.glClear(GL20.GL_COLOR_BUFFER_BIT | GL20.GL_DEPTH_BUFFER_BIT);

        modelBatch.begin(cam);
        modelBatch.render(instance, environment);
        modelBatch.end();
    }

    @Override
    public void dispose () {
        modelBatch.dispose();
        model.dispose();
    }

    private Model myModelBuilder(int()() heightMap){
        ModelBuilder modelBuilder = new ModelBuilder();

        modelBuilder.begin();
        MeshPartBuilder mpb = null;

        // Build the terrain, I tried with one part/node and with one for each triangle.
        modelBuilder.node();
        for (int y = 0; y < heightMap(0).length - 1; y++){
            for (int x = 0; x < heightMap.length - 1; x++) {
                mpb = modelBuilder.part("quad(" + x + "," + y + ")" + "tri-1", GL20.GL_TRIANGLES, Usage.Position | Usage.Normal | Usage.ColorUnpacked,
                        new Material(ColorAttribute.createDiffuse(Color.WHITE)));
                mpb.triangle(
                        new Vector3(x, heightMap(x)(y) / 2f, y),
                        new Vector3(x, heightMap(x)(y + 1) / 2f, y + 1),
                        new Vector3(x + 1, heightMap(x + 1)(y) / 2f, y)
                );

                modelBuilder.node();
                mpb = modelBuilder.part("quad(" + x + "," + y + ")" + "tri-2", GL20.GL_TRIANGLES, Usage.Position | Usage.Normal | Usage.ColorUnpacked,
                        new Material(ColorAttribute.createDiffuse(Color.WHITE)));
                mpb.triangle(
                        new Vector3(x + 1, heightMap(x + 1)(y) / 2f, y),
                        new Vector3(x, heightMap(x)(y + 1) / 2f, y + 1),
                        new Vector3(x + 1, heightMap(x + 1)(y + 1) / 2f, y + 1)
                );
                modelBuilder.node();
            }
        }

        // Try setting indices manually
        for (int i = 0; i < (heightMap(0).length - 1) * (heightMap.length - 1); i += 3){
            mpb.triangle((short)i, (short)(i + 1),(short)(i + 2));
        }

        // Try setting indice numbers
        mpb.ensureTriangleIndices((heightMap(0).length - 1) * (heightMap.length - 1) * 2);

        mpb.ensureCapacity(
                (heightMap(0).length - 1) * (heightMap.length - 1) * 6,
                (heightMap(0).length - 1) * (heightMap.length - 1) * 6
        );

        return modelBuilder.end();
    }
}

Agal Algebraic Geometry – Defeat a brewer's class with a flat, projective morphism

To let $ X $ be a noetherian scheme and $ beta in H ^ 2 (X, mathbb {G} _m) $ that is the picture of some $ alpha in H ^ 1 (X, mathrm {PGL} _ {n + 1}) $ for some $ n ge 0 $, in order to $ beta $ is a class in the brewers group of $ X $, There seems to be a projective, flat, surjective schema morphism $ pi colon Y rightarrow X $ With $ pi ^ * ( beta) = 0 $ in the $ H ^ 2 (Y, mathbb {G} _m) $, This is stated in Theorem 3.6 by Edidin, Hassett, Kresch, Vistoli ("Brewer Groups and Quotient Stacks," https://arxiv.org/abs/math/9905049) (which seems to be important to the overall goals of this article).

My question is: how do you build one? $ Y $? For clarifying comments on this I would be very grateful.

In fact, in Note 3.9, the authors hint at the evidence for their claim: $ Y $ is just the famous brewer Severi scheme of $ alpha $, the turn of $ mathbb {P} ^ n_X $ determined by $ alpha $ with the isomorphism $ mathrm {Aut} ( mathbb {P} ^ n) = mathrm {PGL} _ {n + 1} $, I can not understand why that is $ Y $ works, and let me explain my doubts with the following example (which a priori is not a counterexample to their claim because the authors want to kill the brewer class, not a $ mathrm {PGL} _ {n + 1} $-torsor).

Suppose that $ X = mathrm {Spec} (k) $ for a field $ k $ and let it go $ Z rightarrow mathrm {Spec} (k) $ be that $ mathrm {PGL} _ {n + 1} $-torsor accordingly $ alpha $, I claim that the base is changing $ Z_Y $ is not a trivial torso, though $ alpha $ is not trivial: indeed everyone $ k $-Morphismus $ Y rightarrow Z $ it has to be constant $ Y $ is projective and $ Z $ is affine. What happens in this example is that Severi brewery schemes across a field are trivial once they have a section, but this no longer applies to any bases like $ Y $ in the present example so base change on $ Y $ does not trivialize the Severi Brewers scheme $ Y $ (nor the torso $ Z $).

magento2 – Delete or save category works in the bootstrap file, but not in the API controller when flat categories are enabled

I try to create and delete categories in Magento 2, but encounter some problems Flat Categories Option is activated.

The strange thing is that my code works well when called by a bootstrap file like this one.

getObjectManager();
$state = $obj->get('MagentoFrameworkAppState');
$state->setAreaCode(MagentoFrameworkAppArea::AREA_FRONTEND);
$registry = $obj->get('MagentoFrameworkRegistry');
$registry->register("isSecureArea", true);



$category = $obj->create('MagentoCatalogModelCategoryFactory')->create();
$data = ('name'=>'Test', 'url_key'=> 'test');
$parentId = 52;
$parent = $categoryFactory->create()->load($parentId);
$category = $categoryFactory->create();
$category->setData($data);
$category->addData((
    'parent_id' => $parentId,
    'path' => $parent->getPath(),
    'default_sort_by' => 'position',
    'display_mode' => MagentoCatalogModelCategory::DM_PRODUCT,
    'include_in_menu' => 0,
    'is_anchor' => 1
));
$category->setStoreId(0)->save();

die('
done');

However, when I run the same code from a custom API controller, the following error occurs when invoked $category->save() if storeId is set to 0.

SQLSTATE(42S02): Base table or view not found: 1146 Table 'm23.catalog_category_flat' doesn't exist, query was: SELECT `catalog_category_flat`.* FROM `catalog_category_flat` WHERE (`catalog_category_flat`.`entity_id`='52')

If I set storeId to 1, this error will be displayed

Uncaught Error: Call to undefined method Magento\Catalog\Model\ResourceModel\Category\Flat::getEntityTable()

I have solved a similar problem in Magento 1.9 with it

$category = Mage::getModel('catalog/category', array('disable_flat' => true));

Any idea how to solve this in Magento 2? : /

Thank you very much

Composition – How can I display a gradient? Even mountain roads appear flat on my photos

You must shoot from an angle – if all the trees / characters are in front of you, they will be displayed vertically in the image, regardless of the slope.

In fact, this is a familiar illusion that leads to a "gravitational hill," a road that leads downhill but is viewed from the front without visual cues. The brain interprets them as flat / sloping, which causes objects to roll uphill!


(Source: flickr.com)

Photo by Fluxn

If you shoot from the side or from an angle, there is a marked difference in the line of the trees and the road that gives the viewer the information that the road is falling off.

Here is an extreme example:

The slope is clearly visible when looking at the trees. Note that the photo was taken by a friend with my camera since I am in it!