## How to align two ellipsoids along center and long axes?

I’m struggling to understand rotation/translation transforms as applied to things like spheres and ellipsoids. Take the following silly example.

Given this pair of random ellipsoids, how do I find a rotation/translation that aligns their centers and their long axes? (no matter that they differ in sizes, only care about their centers and long axes being aligned). Are the two possible geometric transformations equivalent? (Say, moving ellipsoid 1 to be aligned to the fixed ellipsoid 2, and vice versa?)

SeedRandom[1234]; twoRandomEllipsoids =    Table[{Ellipsoid[RandomReal[10, {3}], RandomReal[{1, 2}, 3]]}, {2}]; Graphics3D[{Opacity[0.2], twoRandomEllipsoids}, Axes -> True,   AxesLabel -> {x, y, z}] 

Thanks!

## 3D Arrow path along a surface

I have the following two surfaces, created by the following functions

En1[\[Delta]_,g1_,g2_,k_]:=1/2(-I g1+I g2-Sqrt[-(g1+g2-2k+I \[Delta])(g1+g2+2k+I\[Delta])]) En2[\[Delta]_,g1_,g2_,k_]:=1/2(-I g1+I g2+Sqrt[-(g1+g2-2k+I \[Delta])(g1+g2+2k+I\[Delta])])  g1 = 2;  g2 = -3;  \[Delta]max = 2;  kmax = 1;   a = Plot3D[{Re[En1[\[Delta],g1,g2,k]],Re[En2[\[Delta], g1, g2,k]]},{\[Delta],-\[Delta]max,\[Delta]max}, {k, 0, kmax},AxesLabel->{"\[Delta]", "g"}, PlotStyle->Opacity[0.7],BoxRatios -> {1, 1, 1}];  start = Graphics3D[{Black,Ellipsoid[{-1,0.8,Re[En1[-1,g1,g2,0.8]]},{0.1,0.025,0.075}]}];  end = Graphics3D[{Black,Ellipsoid[{-1,0.8,Re[En2[-1,g1,g2,0.8]]},{0.1,0.025,0.075}]}];   test1=Show[a, start, end, PlotRange -> All] 

Now I would like to make a 3D arrow path by going, along the surface, from one black point to the other one, encircling the origin $$(\delta,k)=(0,0.5)$$.

It seems Mathematica do not have a way to create a path, so I constructed it through a ParametricPlot.

test2 = ParametricPlot3D[{-Cos[t],0.5+kmax Sin[t],Re[En1[-\[Delta]max Cos[t], g1, g2, 0.5 + kmax Sin[t]]]},{t,0,2\[Pi]}];   Show[test1, test2] 

However it seems that this it is not encircling nor being created on the surface. Any thoughts?

## Is the scale of a hex map measured along the edges of the map or across the center?

Various official sources include hex maps with scales on them: the Sword Coast from Dragon of Icespire Peak says "1 hex = 5 miles." Is that 5 miles measured across the center of each hex, or an edge length? I can’t seem to find any clarification in the DMG or otherwise.

## cumulatively calculate the distance along a line

I am using Postgis to:

• calculate the cumulative distance in metres along the line
• store the cumulative distance in "m"

I have a table with points on this line and each row holds lat and lon coordinates. The "m" value is 0 for all rows.

The following code nicely gets me the distance between the first point and the next.

SELECT a.geom FLOOR( ST_Distance(ST_Transform(a.geom, 3857), ST_Transform(b.geom, 3857)) * cosd(42.3521)) AS dist FROM line a LEFT JOIN line b ON a.id = b.id + 1 

however, it does not add up to a total nor does it update the "m" value yet.

I tried a set of permutations of the following, but none of them store a cumulative sum of the previous row’s value for "m" plus the calculation of the distance between this point and the previous.

WITH next AS (     SELECT     ST_Distance(ST_Transform(a.geom, 3857), ST_Transform(b.geom, 3857)) * cosd(42.3521)     AS dist,     a.id AS rowid,     FROM line a     JOIN line b     ON a.id = b.id + 1 ) UPDATE line a SET m = FLOOR(next.m + next.dist) FROM next WHERE a.id = next.rowid RETURNING a.m, next.dist; 

or

update line a     set m = FLOOR(a.m + prev.dist)      from (select             l.*,             ST_Distance(ST_Transform(                 lag(geom) over (order by l.id asc)             , 3857), ST_Transform(geom, 3857)) * cosd(42.3521) as dist             from line l           ) AS prev     where prev.id = a.id - 1     returning prev.m, prev.dist, a.m 

Any ideas?

## Can a spontaneous caster use Arcane Fist along with metamagic?

If a character with the Enlightened Fist prestige class (from Complete Arcane, p.34) comes from a class that casts spontaneously (such as a sorcerer), then is it possible for her to use the Arcane Fist special ability to cast a spell enhanced with metamagic, given that it would itself take a full-round action?

## What deities are common knowledge along the northern parts of the Sword Coast? (post-1480 DR)

Specifically between Neverwinter and Icewind Dale, which religions are common knowledge in the years following 1480 DR?

Common knowledge being that if a player character grew up in this area, they would realistically already know that that deity is a deity and maybe something about their religion.

Example: When asked, the average person in the region may say "Mystra is the deity of Magic" if Mystra and her religion are common knowledge.

## How far can a 9th level monk move along a vertical surfaces and across liquids without falling?

The monk class feature Unarmored Movement says:

At 9th level, you gain the ability to move along vertical surfaces and across liquids on your turn without falling during the move.

I am looking for an answer based in the rules as written, although interpretations and opinions are welcome.

I assert the rules support a notion of turns as meta-constructs useful for examining the game world at intervals, as opposed to a game world law of nature. As an example, if a character jumps and “runs out” of jump at the end of their turn while still jumping, I view the character as ending their round in mid-air, and that the jump gets continued on the character’s next turn. I don’t actually have rules citations to support my assertion, unfortunately.

This is important because I think it affects how far the monk can move along vertical or liquid surfaces. I don’t know how, though.

As an example, let’s assume a monk with 30′ movement is 10′ away from a 30′ wide body of water. The monk moves 10′ on land and then 20′ on water and then the monk’s turn is over. Does the monk fall in the water?

A slightly different example. Let’s assume the monk has 30′ of movement and is at the edge of a 1000′ wide body of water. How far can the monk move before falling in?

One answer might be the monk can move its movement, then it is no longer able to stay above the water and gets dunked. If this is the case, can the monk get back up on the water and keep moving?

One perhaps unintuitive answer is that there is actually no limit on how far the monk can move without falling, which means the monk could move across liquid until the monk needs to stop for some reason, say exhaustion, then they fall in.

Not to be absurd, but in this case, I think the monk could “ice” skate on the water, and do swirls and tricks and stuff. Okay, yes, that’s actually absurd.

To summarize: how far can a 9th level monk move along a vertical surfaces and across liquids without falling?

1. to the end of the monk’s turn, then the monk needs to be on solid ground or fall

2. the monk’s movement distance, even if that leaves the monk suspended on the vertical surface or the liquid between turns

3. as far as the monk wants, as long as the monk keeps moving

4. something else?

## Are there any tabletop RPGs in which players advance along a technology tree in the course of the game? [closed]

There are many video games that involve a player constructing an industrial base and improving their technology (Factorio comes to mind). Are there any tabletop RPGs that incorporate a similar mechanic?

## Correct CRL and OSCP URIs along certificate chain

Merely because of private interest and usage in my own network, I’m creating a certificate chain (Root CA → Intermediate CA → Server cert) using openssl. I’d like the certificate chain to be traceable and also being able to revoke certificates.

At the moment, I am not sure, which CRL distribution Points (crlDistributionPoints in openssl config speech) and OSCP URIs (authorityInfoAccess = OCSP;URI: ... and authorityInfoAccess = OCSP;caIssuers: ...) are the correct ones to set when creating a certificate. Examining the certificates of some public websites, it seems to me, that the following is the way to go. So, would you please check whether I’m right?

Root CA certificate:

• CRL: Root CA CRL or none at all
• OCSP URI: OCSP URI of Root CA or none at all
• CA Issuer: URI of Root CA certificate or none at all

Intermediate CA certificate:

• CRL: Root CA CRL
• OCSP URI: OCSP URI of Root CA
• CA Issuer: URI of Root CA

Server certificate:

• CRL: Intermediate CA CRL
• OCSP URI: OCSP URI of Intermediate CA*)
• CA Issuer: URI of Intermediate CA

*) It seems, one OCSP responder could handle OSCP requests for the Root and the intermediate CA. If so, I could also use the Root CA’s OSCP URI, right?

So, every certificates data have to point “one level up”, to the location, where its own validity can be checked. Is that correct?

And btw, is there any OCSP responder you can recommend? I tried openssl’s own one and the one from openca, but both had disadvantages for me.

## How did (in particular) Americans just go along with the concept of having to own and show “photo id” everywhere? [closed]

I remember hearing or seeing a documentary or something a “long” time ago, likely in the early 2000s, about how a lot of Americans (USAians) refused to show an “id” even when voting, where it might possibly be justified to demand some form of identification in order to prevent duplicate votes (if you believe in the concept of democracy in the first place, which is not on-topic for this question).

I also know for a fact that they never had to show IDs to board cruise ships and stuff, for the longest time.

Nowadays, you have to scan and send a photo id to many sites or services online, you can’t as much as collect a package at the post office (pre-paid, with the notification number with you) where I live (Europe), and you’d never be allowed to board any kind of ship or airplane or anything like that without them “verifying” exactly who you are. This seems to very much go for Americans as well.

They even have “facial recognition” and hi-tech x-rays showing you fully naked to the personnel and whoever gets access to their computer systems (massive leaks happen every day).

I wonder: how did this happen without anyone putting up any kind of fight? I could really, really use a vacation for the first time in my life, and it would be interesting to go on one of those huge cruise ships, but given that they not only charge through the nose for it, but demand that I show them photo id to board it (not just my ticket), it’s an impossibility to me.

I literally don’t own a photo id, and getting one would involve having to go to some government building and all kinds of scary (to me) stuff. It’s way more than not wanting to be tracked, which in itself should be enough as a reason for not playing along with this. I feel fundamentally violated having to “prove who I am”. I just have zero interest in doing so. I’m violently disinterested in being tracked by anyone, for any reason, anywhere.

So how did Americans, who traditionally have been extremely “freedom-loving”, just go along with this? I’m furious and steaming from just thinking about this, and I live in the “merry old oppressive Europe”… I can only imagine how many Americans must feel completely “trapped” wherever they live because everything requires you to have these photo ids.