## Which plane do Spectators originate from?

MM. p30

A spectator is a lesser beholder that is summoned from another plane of existence by a magical ritual

Which plane do they originally come from?

## Stewart’s formula in plane geometry

In his book “Plane geometry and its groups”, H. Guggenheimer proves the Stewart’s formula:

If $$A$$, $$B$$, and $$C$$ are collinear, then for any point $$Ρ$$ in the plane
$$PA^2 BC + PB^2 CA + PC^2 AB + AB . BC . CA = 0.$$

and states that Stewart’s formula […], in principle, solves all computational problems in plane geometry.

What does he mean ? How do we solve all computational problems in plane geometry using that formula?

I understand that, for example, formulas given by the Gram determinant of the inner product of vectors are fundamental in the sense that two sets of vectors can be deduced from each other using an isometry if and only if these Gram determinants are the same, and this statement can be interpreted in terms of Invariant Theory. But I fail to find such an interpretation to Stewart’s formula. What is the justification to his bold statement?

## mathematics – Clamp quaternion relative to plane normal

How would one be able to clamp a quaternion based on the angle relative to a plane surface (defined by a normal)?

My diagram below provides more of a visual explanation of exactly what I am talking about.

The quaternions are clamped based on their angle between the planes surface, and the quaternion. The quaternions are global quaternions. The plane normal (vector) N defines the upwards direction of the plane surface. Angles that go opposite of the normal vector is negative, whereas in the direction is positive. With that in mind, the quaternion’s angle cannot be greater than MAX degrees and less than MIN degrees. These bounds are visualized by the cones (MIN is -90˚ in the diagram, so the cone is infinitely thin).

Thoughts:

My thought is that this algorithm would have to work in regards to the quaternions delta (unless the quaternion is in the MIN/MAX zones, which I will talk about later). If the change (delta) of the quaternion results in the quaternion entering, or passing, the bounds, the change is capped (clamped) so the final quaternion never has a resulting angle that dissatisfy the bounds.

Clamping the delta could be achieved by determining the beginning and end angles, and obtaining a ratio to be used as the t value (the interpolator) of a Slerp function, so the final quaternion’s angle is that of the bound. The issue I see with this is how the roll would be manipulated due to the Slerp. Would the intended, final roll of the quaternion be lost due to this Slerping?

This idea of change seems more important in the example of if the quaternion was to go from -89˚ to -91˚. The final quaternion should be snapped back to -90˚ (if the bound was -90), instead of considering the new quaternion as -89˚ but from the other side. Without this proper interpretation of the quaternion’s angle, the quaternion could rotate around constantly and consistently due to the angle never being considered less than -90˚ and greater than 90˚.

Edge Case:

If the quaternion was to start inside the MIN/MAX zones (the cones), the quaternion would not have a change to define as the direction that the quaternion needs to move to get back to legal territory/angles (see angle a3 in diagram). My thoughts is that a fallback plane normal that is orthogonal to plane normal N would have to be used, so the quaternion has a plane to rotate around.

Use Case:

Orbiting camera with user or author definable pitch angle clamping, and a varying change in “up”. Example of varying change of “up”: Mario Kart 8 Deluxe.

## mathematics – Rotate quaternion around plane normal

How can one rotate a quaternion so that the rotation is around a plane normal?

A graphic/diagram I made below gives more detail about what exactly I am implying. Note that the tip of the quaternions in the diagram are that of the quaternions local, upwards direction, symbolizing roll (the quaternion’s w component).

English describing the diagram:

The rotation is around plane normal (vector) N with amount θ. The global axes are X, Y, and Z (Y is up). The plane normal and quaternion are both in global space. Notice how the beginning and end of the operation does not use any reference to the global axes, only the plane.

The angle between the start quaternion (q) and end quaternion (q’) and the plane is equal (a = a’). If you were to draw lines from the tips of the quaternions to the plane, the angles of intersection would be equal. In other words, the roll of the quaternions are preserved in relation to that of the plane.

## Distances between planes are ill-defined in D&D 5e

As I cover in this answer, the rules do not clearly spell out how to compute the distance between two points on different planes. However, there are several places where the rules appear to assume that such a definition exists, such as the Wand of Enemy Detection (emphasis added):

For the next minute, you know the direction of the nearest creature hostile to you within 60 feet, but not its distance from you. The wand can sense the presence of hostile creatures that are ethereal, invisible, disguised, or hidden, as well as those in plain sight.

There would be no reason to mention detecting enemies within 60 feet on another plane if distances between the material and Border Ethereal were undefined, and yet a definition for inter-planar distance is never provided by the rules.

There are several other places in the core rules that mention something like this in the core rules, and it seems that the passage quoted in the question is another such example. Explicitly disallowing planar travel entirely sidesteps the issue of whether inter-planar distances are defined.

## It is up to the GM how far apart planes of existence are

The misty step spell allows you to teleport to somewhere you can see; thus, if you are somehow able to see a plane of existence that you are not on, it would appear that you could teleport there. However, the misty step spell states (emphasis mine):

Briefly surrounded by silvery mist, you teleport up to 30 feet to an unoccupied space that you can see.

Whether an entire other plane of existence is 30 feet away or less is something that the GM is going to have to determine.

Here are some other questions about distances between planes:

To quote the highest scoring answer to that last question:

It has been proposed that you might be able to get around this by using a portable hole or some other extraplanar storage see (this Q&A). This might work, except a portable hole is 10 feet deep so removing an item from it would still trigger it.

Also, a DM could argue that the extradimensional space of the hole is an infinite distance away from the current plane. Thus, taking it out or putting it in the hole would break the glyph regardless. So this comes down to a DM decision to make that call.

So it will be up to the GM to determine how far apart two planes really are. For me, allowing this to work gives a fun and interesting use for the spell so I at least would allow it to work (assuming they can actually see this other plane of course).

Something to consider when making a ruling is monsters which have truesight and a teleport, since they could effectively leave any combat or be particularly excellent at ambushing. Another thing to consider is spells like dimension door which have a distance limit but do not require sight; allowing them to go between planes would be a rather significant change to those spells.

For what it’s worth, lead Rules Designer Jeremy Crawford has said the following in a tweet:

If two people are on different planes of existence, they are infinitely far away from each other. For example, if I’m on the Material Plane and you’re on the Ethereal Plane, we’re not within 30 feet of each other.

Of course, this is just what he happened to think at the time he posted that tweet, who knows if it has changed or even why he thought that ruling made sense in the first place.

## Why the image plane is at Z = f in pinhole camera model?

Can someone show me the mathematical interpretation of why the image plane is at Z = f in the pinhole camera model?

I found a possibly useful formula in wiki, as shown in the figure below:

My thought is, when the diameter of the aperture approaching to zero, the thin lens model becomes the pinhole camera model. And according to the DOF formula, the smaller the diameter of the aperture the larger the f-number, and hence the larger the DOF. In this case, everything appears in the focus.

## air travel – Is it allowed to carry steel pipe in checked luggage in plane?

It may sound bizarre, but it’s true…

So I’ve got an order to transport 0,5m (about 3kg) steel pipe with flanges (and parts like washers and bolts) from Poland->Germany using Wizzair. The time needed to do so probably decided for such solution…

People from corps with higher budgets and better logistics, please don’t make fun of me…

Disclaimer: for management guys, it’s always “possible”… “There isn’t any problem”. “Here is my anecdotal story when I’ve transported something illegal (not drugs but metal parts)…”

I’ve checked WizzAir rules:
1. Hand luggage has hardcore restrictions.
2. I can’t find any limitations for checked luggage

## dnd 5e – How many creatures can go to the Ethereal Plane via Etherealness cast at 8th level?

The spell etherealness allows you (the caster) to travel to the Ethereal Plane for the duration of the spell (8 hours). It has a range of Self, so you cannot target someone else if cast at its base level (7th level).

However, if you upcast it with an 8th level spell slot, it says (PHB, p. 238):

At Higher Levels. When you cast this spell using a spell slot of 8th level or higher, you can target up to three willing creatures (including you) for each slot level above 7th. The creatures must be within 10 feet of you when you cast the spell.

Where it says “including you”, does this mean that:

1. you can only target two other creatures, because one of these three creatures must be you, or
2. you can target three other creatures, but you’re going with them (i.e. you can’t just dump these three creatures on the Ethereal Plane whilst you remain on the plane you were all on moment ago)?

Since I’m a cleric in a party of 4, I want to know if I can bring the entire party (me plus my 3 allies) into the Ethereal Plane or whether one of my allies has to be left behind…

## dnd 5e – If I place a Bag of Holding inside another Bag of Holding, can I use Misty Step to get away before the gate to the Astral Plane opens?

I’d very much put this conundrum into the ‘Should bring disaster of one sort or another onto the character’ category.

You mention whether you can cast the spell ‘before’ the effect, depends on whether there’s a somantic component to the spell I suppose, you can’t start to cast the spell, then get your bags ready, then finish the spell, then… but I suppose you could use a foot to drop one into the other…

In terms of game mechanics, obviously the timing of the spell would affect where the character was when one bag went into another (whether the whole bag or just the first thread of it needs to go in the produce the effect), ultimately the destruction/gate effect either happens before or after the spell activates, at a nano-scale of timing so either the character is still where the bags are if they do it by hand, or if they use some kind of gravity/delay mechanism to have one fall into another then sure, they’re wherever they are and the bags are… well, no longer wherever they were when it happens. Same as if they cast the spell and a nearby character puts the bag into the other bag before or after the spell activates – when you get to a sufficient level of time slicing, nothing is simultaneous.

Either way, Misty Step is short range, quite possibly less than the reach/size/spell/psionic range/maw size of something horrific that I’d be pretty sure to decide emerges from the gate to see what all the commotion is.

Opening a random gate to the Astral plane by such shenanigans would invariably be a bad thing in my campaign… it doesn’t mean you can step through it or benefit from it in any way, it’s an Astral gate, it’s not under your control and doesn’t want to be your friend.

And where are all these bags of holding coming from ?? – seems such a waste.

Call me a purist but this kind of thing smacks of trying to exploit wording in texts that the characters themselves don’t have access to – there aren’t game rule books in the D&D universe, just in the real world and in the end, the DM decides how they want to interpret the rules. I’d suggest just fighting the monster and using the 2 bags of holding to carry its loot away.

There’s more pleasure to be had in playing the game than playing the rules so as a DM I discourage this sort of play.