## graphics – Project 3D points on a plane, then project again on 3D

I have a table in 3D space, represented by a plan.

I want to project arbitrary points representing an object (mug, toy, etc.) on this table, execute an analysis of 2D main components to obtain an oriented delimitation frame (https://www.sciencedirect.com/topics/computer- science / oriented- bounding box)

This gives me 2D oriented bounding boxes (defined by 8 corner points) that lie along the plane of the table. Now, I want to project these points in 3D so that I can get 3D bounding boxes aligned with the plane.

The projection from the 3D plane to the 2D plane is not invertible .. so how to do it?

## real analysis – The inverse of a Poincaré result on regular limit points

Let $$V$$ be a delimited open set $$mathbb {R} ^ n$$ with $$n> 1$$. According to a well-known result due to Poincaré, if $$x$$ is a point in the border $$partial V$$ and there is a ball $$B$$ such as $$x in partial B$$ and $$B cap V = emptyset$$, then $$x$$ is a regular point for the Dirichlet problem.

Is there a reciprocal for this result? More generally, on what condition (s) can we say that if $$x in partial V$$, there is a ball $$B$$ such as $$x in partial B$$ and
$$B cap V = emptyset$$?

## dg.differential geometry – Conjugated points and Jacobi matrices

Let $$(M, g)$$ be a compact and smooth Riemannian collector of dimension $$n geq 3$$ and leave $$gamma: (- 2.2) to M$$ be a geodesic that contains no conjugate point on $$(- 2.2)$$.

I have two questions, as follows.

(i) Is it possible to build transversely (at $$gamma$$) Jacobi fields $$J_1, ldots, J_ {n-1}$$ so that the determinant of the $$(n-1) times (n-1)$$ matrix $$X (t)$$ with columns $$J_1 (t), ldots, J_ {n-1} (t)$$ on the smallest interval $$(- 1.1)$$ only disappears to the point $$t = 0$$?

(ii) Is it possible to build $$gamma$$) Jacobi fields $$J_1, ldots, J_ {n-1}$$ so that the determinant of the $$(n-1) times (n-1)$$ matrix $$X (t)$$ with columns $$J_1 (t), ldots, J_ {n-1} (t)$$ on the smallest interval $$(- 1.1)$$ only disappears to the point $$t = 0$$, as well as the rank of $$X (0)$$ East $$n-3$$?

## tracing – Recovery of the limit points of the concave shell (or alpha form) in 3D

I am trying to find the limit points of a 3D concave region which is described by a list of points. If the region was convex, I would take the convex shell and simply extract the coordinates from it.

However, since the shape is concave, I have to use an alternative algorithm. For example, I can implement alpha forms as proposed in

DelaunayMesh in a specified closed region – creating a concave hull from a set of points

and

Find a concave shell.

I can successfully get a MeshRegion object that represents the shape I want. However, unlike the convex shell, required implementations such as Delauney triangulation or the code offered in https://mathematica.stackexchange.com/a/88769/45020 produces a MeshRegion which always contains all the points. It simply selects the triangles to include, but it retains the internal triangles. So, I cannot easily choose only the points which are at the border. I have tried to apply RegionBoundary which works for 2D shapes but when applied to 3D mesh it acts as an identity for some reason.

Examples of data:

``````pts = Join(RandomPoint(Cuboid({0, 0, 0}, {1, 1, 1}), 1000),
RandomPoint(Cuboid({1, 0, 0}, {2, 0.5, 0.5}), 1000));
``````

I want to get only those points at the limit of the form (reject the points inside).

## I live in Virginia and here you get your license suspended if you get 18 points within 2 years?

I have 15 points.

One of my tickets should fall this March 3rd. since I got it in 2018.

I would then "only" 11 points.

I just wonder, if I have a ticket now, I would get my license suspended for almost two weeks.

I have a day flower delivery job for Valentine's Day. I think, am I stupid for doing this. what if i get a ticket?

## graphics and networks – 2D cluster points in clusters of particular group size?

Imagine i have a set of 2D points `pts`, and I want to partition them into groups by spatial contiguity, but limit this partition to particular group sizes. I thought that `NearestNeighborGraph()` could be a starting point, but the problem is that percolation soon leads to a large connected component:

`pts = RandomReal(1, {1000, 2});`

`NeighborsToCluster = 4;`

`NearestNeighborGraph(pts, NeighborsToCluster)`

Given:

How could I limit clustering so that only `n` number of points belong to each group (but always use Euclidean distance as a base)?

I do not need an exact `n`, in fact (which will also be impossible in many cases), I would like a distribution around `n` (normal distribution, with an arbitrary average, std).

Any help with this goal is appreciated!

## Return ranking of user points track week, month, year from multiple tables with mySQL

Here is the structure of my tables:

User

``````|--------|------------|
| id     | name       |
|--------|------------|
| 1      | Name1      |
| 2      | Name2      |
| 3      | Name3      |
|--------|------------|
``````

Publish

``````|--------|------------|-------------|--------------------|
| id     | content    | user_id     |   created_at       |
|--------|------------|-------------|--------------------|
| 1      | Content1   |  1          |2020-01-17 14:03:31 |
| 2      | Content2   |  1          |2020-01-17 16:18:23 |
| 3      | Content3   |  2          |2020-01-17 16:29:13 |
|--------|------------|-------------|--------------------|
``````

Comment

``````|--------|------------|-------------|----------|---------------------|
| id     | comment    | user_id     | post_id  |   created_at        |
|--------|------------|-------------|----------|---------------------|
| 1      | Comment1   |  1          |   1      | 2020-01-20 18:29:19 |
| 2      | Comment2   |  1          |   1      | 2020-01-22 17:25:49 |
| 3      | Comment3   |  2          |   2      | 2020-01-28 11:37:59 |
|--------|------------|-------------|----------|---------------------|
``````

Vote

``````|--------|-------------|----------|-----------------------|
| id     |  user_id    | post_id  |    created_at         |
|--------|-------------|----------|-----------------------|
| 1      |   1         |   1      | 2020-01-20 15:08:55.0 |
| 2      |   1         |   2      | 2020-01-20 15:13:29   |
| 3      |   2         |   2      | 2020-01-20 15:13:32   |
|--------|-------------|----------|-----------------------|
``````

I want to find the best score of 10 users per week, month, year according to the formula:

A user creating 1 position will have 10 points.
A user creates 1 comment in a post that will have 5 points.
A user votes in a post that will have 2 points.

Thank you so much.

## Path for a tube between two points, including constraints

I have to create paths for tubes, pipes and conduits connecting two points in an automated way.

For example, I want to create a path for a tube between the two red endpoints in this screenshot (yes, it's a loo):

The path between these two points must satisfy some constraints, among which:

1. The distance between the path and any other object in any direction must be greater than a certain threshold

2. The path should not be unnecessarily long or be "too winding"

3. It will probably have to allow the incorporation of a "minimum radius of curvature".

I can think of two ways to approach this problem (but there are probably also other approaches):

Method A. Move an object from the start point to the end point using some kind of object avoidance, like raycasting

Method B. Use some sort of the least expensive path analysis. This would likely involve discretizing 3D space into quadrants and assigning a cost to each quadrant based on its distance from other objects.

Currently, I have implemented a version of Method A, as you can see in the gif below:

While this solution produces something usable, I'm not sure it's the best approach. At least in my implementation, the minimum distance to other objects is not guaranteed, because the object may, for example (not illustrated in this example) follow a path in a hole and not find any exit.

What other approaches are there? For example, is there a specific approach to method B mentioned above? How could we discretize space into separate quadrants?

## Masquerade vampire – How many blood points can a ghoul use during a round?

I recently started running a ghoul campaign, but no matter how hard I look, I can't find how many blood points a ghoul can use per round. I know that vampires are limited by their generation, lower generation vampires can use more blood points at the same time.

As such, my question is how many vitae can a ghoul use during a round?

Note: I am using the basic birthday edition book and the addiction book on fatal addiction.

## typography – Should vertical spacing ensure that each list points to a paragraph?

Very often, lists in continuous texts are composed in such a way that the vertical spacing between the points is greater than the regular spacing of lines in the same text. Well, like the lists here at SE:

• just a sample point,
• look at this spacing above!

Although the list items are placed on a new line each, they still form a sentence, so increasing the spacing between them breaks the "text – paragraph – sentence – word" logic. In addition, these lists are much less dense compared to the rest of the text.

What is the purpose of additional spacing in this case? Wouldn't it be better to densify the lists so that the text does not fall apart?