Graphics – Project 3D points onto a plane and then project them back into 3D

I have a table in 3D space represented by a layer.

I want to project arbitrary points representing an object (cup, toy, etc.) onto this table and do a 2D principal component analysis to get an oriented bounding box (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 table level. Now I want to somehow project these points back in 3D so that I can get 3D bounding boxes aligned with the plane.

The projection from the 3D to the 2D plane cannot be inverted. How do I do that?

convex polytopes – number of regions formed by $ n $ points in the general position

given $ n $ Points in $ mathbb {R} ^ d $ in general position where $ n geq d + 1 $, For each $ d $ Points form the hyperplane defined by this $ d $ Points. Intersect these hyperplanes $ mathbb {R} ^ d $ in several regions. My questions are:
(1) Is there a formula regarding $ d $ and $ n $ that describes the number of regions?
(2) the same question for the number of limited regions?

I tried many keywords on Google but found nothing useful. Any references or ideas are appreciated. Thank you very much

Conic sections – Tangents are drawn to the circle $ x ^ 2 + y ^ 2 = 4 $ from points on the line $ 3x-4y + 12 = 0 $.

Then the contact chord passes through a fixed point. Find the chord slope of the circle with this fixed point as the center.

Let the tangents be drawn from the point (h, k)

$$ 3h-4k + 12 = 0 $$

The chord of the contact is
$$ hx + ky-4 = 0 $$

Now$$ h = frac {4k-12} {3} $$

Then $$ frac {4k-12} {3} x + ky-4 = 0 $$
$$ 4kx-12x + 3ky-12 = 0 $$
$$ k (4x + 3k) -12x-12 = 0 $$
Solve the given line family
$$ x = -1 $$ and $$ y = frac 43 $$

The fixed point is $ (- 1, frac 43) $

Slope of the line connection This point is to the center of the circle $ – frac $ 43

Then the chord slope will be $ frac $ 34

But the answer is $ frac $ 43 , What's going wrong

Differential Geometry – Conjugate points and Jacobi matrices

To let $ (M, g) $ be a smooth compact Riemannian manifold $ n geq 3 $ and let $ gamma: (- 2.2) to M $ be a geodesist with no conjugate points $ (- 2.2) $,

I have two questions like this.

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

(ii) Is it possible to construct transversal (to $ gamma $) Jacobi fields $ J_1, ldots, J_ {n-1} $ so the determinant of $ (n-1) times (n-1) $ matrix $ X (t) $ with columns $ J_1 (t), ldots, J_ {n-1} (t) $ on the smaller interval $ (- 1.1) $ just disappears at the point $ t = 0 $and additionally rank of $ X (0) $ is $ n-3 $?

Integration – Is there a way to find "polynomrational functions" with turning points in their graphs?

For example, if the derivative of $ f (x) $ is$$ f & (39) (x) = frac {(x-1) ^ 2 (x-3)} {(x-2)} $$
then $ f (x) $ has a turning point at $ x = 1 $,

But $ f (x) $ is not a polynomial rational!

Is there a way to determine which antiderivatives are polynomial relations?

Graphics and networks – cluster 2D points in certain clusters with group size?

Imagine that I have a series of 2D points ptsand I want to divide them into groups by spatial proximity, but limit this division to certain group sizes. I thought that NearestNeighborGraph() might be a starting point, but the problem is that the infiltration will soon result in a large connected component:

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

NeighborsToCluster = 4;

NearestNeighborGraph(pts, NeighborsToCluster)

Gives:

Enter image description here

How could I limit clustering so that only n Number of points belongs to each group (but still use Euclidean distance as a basis)?

I don't need an exact one nIn fact (which will be impossible in many cases) I want some distribution n (Normal distribution with any mean, standard).

Any help towards this goal is appreciated!

real analysis – a question about attracting fixed points

I am trying to prove that the function $ varphi: A rightarrow A $ where A is the set $ (- frac { pi} {2}, frac { pi} {2}) $ and $ varphi (x) = arctgx $ has an attractive fixed point.

Now in my book I found this definition:
If $ exists epsilon ge0 $ | $ forall x in $(Domain)$ cap (p- epsilon, p + epsilon) $ and $ lim_ {n rightarrow inf} varphi ^ n (x) $ as p is an attractive fixed point.

I just found out that 0 is a fixed point and I tried to apply this definition to my case, but I am unable to understand how $ varphi ( varphi (… varphi (x))) = varphi ^ n (x) = arctan (arctan (arctan dots arctan (x))) $ can converge to 0.

Vampire the Masquerade – How many blood points can a ghoul use in a round?

I recently started a ghoul campaign, but no matter how precise I look, I can't figure out how many blood points a ghoul can use in one round. I know that vampires are limited by their generation, with lower generation vampires being able to consume more blood points in one round.

So I wonder how many vitae points a ghoul can use in a round.

Note: I use the land register for the anniversary edition and the supplementary book for the fatal addiction.

Unit – Generate a sprite with a list of points

I'm trying to dynamically generate a sprite in Unity to visualize a list of points.

So far I've created a small Texture2D object and then tried to overwrite the geometry:

static SpriteRenderer SpriteRendererFromPoints(SpriteRenderer spriteRenderer, List points) { 
    var texture = Texture2D.whiteTexture;
    spriteRenderer.sprite = Sprite.Create(texture, new Rect(0.0f, 0.0f, texture.width, texture.height), new Vector2(0.5f, 0.5f));
    Sprite sprite = spriteRenderer.sprite;
    sprite.OverrideGeometry(points.ToArray(), sprite.triangles);
    return spriteRenderer;
}

I add this SpriteRenderer and a PolygonCollider2D to my game object and instantiate it as follows:

    GameObject meshObject = new GameObject();

    PolygonCollider2D polygonCollider = meshObject.AddComponent();
    PolygonCollider2DFromPoints(polygonCollider, points);

    SpriteRenderer spriteRenderer = meshObject.AddComponent();
    SpriteRendererFromPoints(spriteRenderer, (from point in points select new Vector2(point.x, point.y)).ToList());
    Instantiate(meshObject);

Everything is going well, but with this version the sprite is nowhere to be seen:
unity_points

Any help what I miss here or alternative ways to achieve this would be greatly appreciated.

Numenera – Life points on NPCs are higher than expected

I want to understand the health inflation that is often printed in Cypher System material (often in module adventures or in the small sidebars when describing the hiring of NPCs).

In Numenera Health (HP) is usually determined by the standard destination number

Numenera – Discovery, p. 222 (also the same in the 1st edition)

Health: A creature's target number is usually its health. This is the amount of damage she can take before she is dead or unable to act. For easy reference, the entries always list a creature's health, even if that's the normal amount for a creature of its level.

Which is 3 x the level of difficulty, just for reference.

The designers evade a limitation that monsters sometimes just break the usual defined health for a much higher number. I remember referring to it in the first part of Ed Numenera to provide more challenging battles for higher level characters.

A quick tour of Discovery / Destiny I picked out a few examples:

  • Discovery p 367 – teratoma – level: 3 HP: 12
  • Discovery p 381 – Octopus- Level: 3 HP: 15
  • Discovery p. 369 – Teratoma (M) – Level: 4 HP: 15
  • Destiny p 371 – Assassin – Level: 4 HP: 20
  • Discovery p 375 – Weymel – Level: 5 HP: 20
  • Discovery p 385 – Latos – Level: 5 HP: 25
  • Destiny p 389 – Halcus – Level: 5 HP: 20
  • Destiny p 389 – Drayva – Level: 5 HP: 20
  • Destiny p 362 – Khagun Semper – Level: 5 HP: 26
  • Destiny p 373 – Soludi – Level: 6 HP: 24
  • Destiny p 398 – Heri – Level: 6 HP: 27
  • Destiny p 398 – Scrose – Level: 7 HP: 30

There are many, many more examples in Cypher Systems, OG-Numenera, Discovery, Destiny, The Strange and Predation. And they are not unique or are used generously, HP inflation is extremely common. As you can only see from this small list, the creatures range from boss encounters to low random animals with no rhyme or reason that I can perceive. Across all level areas.

My question is why? Is there a systematic process for this? Is the default HP suggested in the "Creature" section just too low? I'm looking for designer comments or even personal GM experience to assess how much HP should be given to the fighters.