Plane rotated about line of intersection to another plane?

The plane ax+by=0 is rotated about its line of intersection with the plane z=0 through an angle n. What is the equation of the plane in its new position? I saw a question in stack exchange relating to 3d plane rotation but its answers were of a more advanced level. Would someone please help me with an easy visualisation and solving of this problem?

Finding a plane numerically

Suppose I have three large finite sets $ \{x_i\}$ , $ \{y_i\}$ and $ \{z_i\}$ ; they are obtained by measuring coordinates of a collection of vectors in $ \mathbb{R}^3$ , but I do not know which triples correspond to one vector. Assume I know that all of the vectors lie in a plane, but I do not know which one. I want to reconstruct this plane and the original collection of vectors.

Evidently the problem can be solved and the solution is unique if the original data is generic. But I do not see a reasonable algorithm to solve this problem. (For sure checking all subdivisions into triples is not a good idea.)

Is there a numerical method to find this plane?

and

Did anyone considered this or similar problem; does it have a name?

Growing a chain of unit-area triangles: Fills the plane?

Define a process to start with a unit-area equilateral triangle, and at each step glue on another unit-area triangle.


          n50
          $ 50$ triangles.


Above, the red dot marks the starting triangle $ T_0$ . Then on one edge $ e$ of $ T_0$ , a point $ p$ is chosen to form a unit-area triangle $ T_1$ . Now there are two edges of $ T_1$ for possible growth. One edge is chosen randomly, and another unit-area triangle $ T_2$ is erected on that edge. Then $ T_3$ is built on an edge of $ T_2$ . And so on.

The point $ p$ at each step is chosen according to a normal distribution $ \cal N(0,1)$ centered on the midpoint normal vector to $ e$ , as illustrated below:


          Tri_Normal
          The location of $ p$ follows a normal distribution.


My question is:

Q. Does this process fill the plane in the limit?

I think Yes but am not certain.

Here are two more examples (using different random seeds):


          n500
          $ 500$ triangles.


          n1000
          $ 1000$ triangles.


How to get the villain in prison – on another plane?

Let’s just say I have acquired an Emissary of Barachiel contact, and built a prison on our base plane that can’t be broken out of while he gets to talk to the prisoner unhindered (and use the class’ features to turn them lawful good over time).

That’s all well and good, but sometimes you can’t just haul the villain back the moment you beat them. Maybe there’s more work to be done.

I am looking for ways to land the villain accurately in the prison without going there, or keeping them on my person safely, easily and indefinitely until I can take them there. I can only use items to do it, as the party shifts and my character isn’t a caster nor has room for PrCs.

Can a genie use its Plane Shift ability on unwilling creatures?

A Genie has the SP ability of a limited plane shift.

A genie can enter any of the elemental planes, the Astral Plane, or the Material Plane. This ability transports the genie and up to eight other creatures, provided they all link hands with the genie. It is otherwise similar to the spell of the same name (caster level 13th).

The Plane Shift spell has two versions.

One targets a creature touched, the other several hand-held, willing creatures.

Can the Genie use its Plane Shift on an unwilling creature, or is it limited to the willing version (seeing as it mentions ‘itself and others’, per the willing-group version)?

Find plane within margin of error of >50% of points


There are $ N < 3\times10^4$ 3D points. At least 50% of them lie approximately in the same plane, i.e. the distance between the plane and each point is at most $ p$ . Find such a plane.

Attempt: since the number of points in the plane is at least 50%, we can randomly sample 3 points from the set. They will all be in the plane with probability 12.5%. We build a plane through these 3 points and check that at least 50% of points lie approximately in it. Within 10-20 samples we’ll find the plane.

Problem: because of the margin of error there’s not just one plane going through 3 points, but many possible planes. How do we examine all of them?

How would you tackle this problem?

Schengen visa application in UK requires confirmed plane tickets

I am a non-EU national living in the UK and applying for a Schengen visa. I would like to visit Switzerland and applied through TLS Contact – which is the standard way to apply in the UK. Among their list of documents required is

Copy of transport documentation with applicant’s name as passenger – by plane confirmed booking with airline booking reference number

The TLS center representative confirmed that without a confirmed booking, the file would be considered incomplete. This is also mentioned on the Swiss embassy website

I have applied for Schengen visas before (at Asian embassies) and they only required an itinerary, not confirmed tickets. Any advice on what to do in this case? Book the flights and take the risk or is there a better option?