How to map out and run an encounter set in a predominantly vertical area online?

Upcoming in one of my D&D 5E games, the player characters are going to have to venture down a dwarven mineshaft while being attacked by dwarf zombies. The shaft itself is an entirely vertical pit, with layers of stairs and scaffolding around the edge creating a path to the bottom.

I’ve researched into ways to do this, including this question here, and I would be happy to attempt to build a 3D model of the terrain, however the issue is that the game is entirely online. Usually for maps, we’ve used Roll20 which offers a nice top-down 2D perspective, and I can get some limited 3D out of it by placing things in layers. However, this method is already clunky, and with how tall the shaft is going to be… it’s just really not gonna work.

So that leads me to my question, what are some ways I can represent this vertical area and it’s associated encounter in an online-friendly way?

Analysis of kd-tree, how is the vertical line L’s intersect areas equivalent to sqrt(N)?

I’m trying to understand how the number of intersected areas by a vertical line in a KD-tree is equivalent to sqrt(n)

If you draw a balanced KD-tree with 7 nodes.
enter image description here

And then draw a vertical line l.

enter image description here

The number of areas this line intersects should be equivalent to sqrt(N) where N is amount of nodes. (7)

When I count the areas the vertical line L intersects I get 5. But sqrt(7) = 2,6 not even close.

Both sources get to the recurrence:

enter image description here

Which solves to O(sqrt(N)).

What am I doing wrong?

Sources:

Source 1

Source 2

1D Wave Equation: Vertical Rod and Displacement vs. Textbook Solution

I am trying to setup Mathematica to analyze a vertical round rod under its own weight, fixed on one end free on the other. I have the 1D wave equation and a distributed load to represent the self weight of the round rod.

Vertical Rod Layout

The problem is when I compare the Mathematica solution to the textbook solution the two do not agree.

Sample problem is given below.

Y = 199*^9; (*Young's modulus in Pa *) \[Rho] = 7860; (* Steel density in kg/m^3*) dia = 1/39.37; (* 1" dia converted to meters*) c = Sqrt[Y/\[Rho]]; len = 1000; (*length in meters*) tmax = 5; (* Max time for analysis*) area = \[Pi]*dia^2/4; (*Round rod cross sectional area*) wtfactor = \[Rho]*9.81*area/len;  frwt[x_] := \[Rho]*    area*9.81*(1 -       x/len); (*Rod Self weight imposed as a distributed load*) nsol6 = NDSolve[{\!\( \*SubscriptBox[\(\[PartialD]\), \({t, 2}\)]\(z[x, t]\)\) == c^2*\!\( \*SubscriptBox[\(\[PartialD]\), \({x, 2}\)]\(z[x, t]\)\) + frwt[x] +       NeumannValue[0, x == len],    z[0, t] == 0}, z[x, t], {x, 0, len}, {t, 0, tmax},   Method -> {"FiniteElement",      "MeshOptions" -> {"MaxCellMeasure" -> 10}}   ] fnnsol6[x_, t_] = nsol6[[1, 1, 2]] Plot3D[fnnsol6[x, t], {x, 0, len}, {t, 0, tmax},   PlotLabels -> Automatic, AxesLabel -> Automatic]  deltaL = ((\[Rho]*9.81*len^2)/(  Y*2)) (*Textbook elongation for a vertical rod under self weight*) calcdeltaL =   fnnsol6[len,    5] (*Calculated delta Length from PDE solution.  Should match \ textbook*)  deltaLfunc[x_, l_] := \[Rho]*9.81*   x*(2*len - x)/(2*Y) (*Verified Correct*) xydata = Thread[{Range[0, 1000, 100],      deltaLfunc[x, 1] /. {x -> Range[0, 1000, 100]}}]; xydata2 =   Thread[{Range[0, 1000, 100],     Reverse[a]}]; (*Same answer different calc format for debugging*) Show[Plot[fnnsol6[x, 0], {x, 0, len}, PlotLabels -> {"PDE Val"},    PlotRange -> All   ],  ListLinePlot[xydata2, PlotStyle -> Green, PlotLabels -> {"Correct"}]] 

If you’ve read this far, thank you.

In summary my question is: Is this a Mathematica issue or a PDE setup problem? The PDE is right out of a textbook so I don’t think that’s the problem but Mathematica gives no errors and I am out of troubleshooting ideas so looking for some help.

Thank You

How do I graph a vertical isocline?

The function r/a is a verticle isocline. I need to find a way to graph it. Right now the line is not showing, even when it equals y. I have experimented with Epilogue, but I am too unfamiliar with it to incorporate it with the stream plot.

————————HERE IS MY CODE———————————–

    Manipulate[  Show[Plot[{d/(f*a), r/a}, {x, 0, 100},     PlotStyle -> {RandomChoice, Thick}, ImageSize -> Full,     PlotRange -> {{0, 100}, {0, 100}}],    StreamPlot[{{(r*x - (a*x*y)), (f*a*x*y - d*y)}}, {x, 0, 100}, {y, 0,      100}], AxesLabel -> {"N", "P"},    LabelStyle -> Directive[Blue, Bold], ContentSize -> {100, 100}],   {{r, 25}, 0, 100, Appearance -> "Labeled"},  {{N, 50}, 0, 100, Appearance -> "Labeled"},  {{a, .5}, 0, 1, Appearance -> "Labeled"},  {{f, .5 }, 0, 1, Appearance -> "Labeled"},  {{d, .5}, 0, 1, Appearance -> "Labeled"}] 

Must the Wind Wall spell be vertical, or can it be angled or even curved?

Our group has two different interpretations of Wind Wall.

A wall of strong wind rises from the ground at a point you choose within range. You can make the wall up to 50 feet long, 15 feet high, and 1 foot thick. You can shape the wall in any way you choose so long as it makes one continuous path along the ground.

Interpretation 1: The wall must be vertical, and the wind blows straight upward.

Interpretation 2: The wall can be angled or even curved, as long as the wind is blowing at some upward angle, even if it’s just a few degrees above horizontal.

So, in the following diagram, is the leftmost configuration the only acceptable configuration, or do the two rightmost configurations also work?

enter image description here

How do I add a vertical plane in Plot3D?

I use the following code to generate a plot:

a[r_] := A  r^3/( r^3 + A^3 L1^2); Rg[r_] := (2 ma r^3)/( r^3 + 2 ma^3 L^2); \[Rho][r_] := r^2 + a[r]^2*Cos[\[Theta]]^2; \[CapitalDelta][r_] := r^2 - Rg[r] r + a[r]^2;  g = {{-(1 - (Rg[r] r)/\[Rho][r])(*G[r]*), 0,      0, -((a[r] Sin[\[Theta]]^2 r Rg[r])/\[Rho][r])}, {0, \[Rho][      r]/\[CapitalDelta][r], 0, 0}, {0, 0, \[Rho][r],      0}, {-((a[r] Sin[\[Theta]]^2 r Rg[r])/\[Rho][r]), 0, 0,      Sin[\[Theta]]^2 (r^2 + a[r]^2 + (        a[r]^2 Rg[r] r Sin[\[Theta]]^2)/\[Rho][r])}};  v1 = {Sqrt[-(g[[1, 1]] - g[[1, 4]]^2/g[[4, 4]])], 0, 0, 0} ;  (v1.g.v1) /. {A -> 6, ma -> 10, L1 -> 0.1, L -> 0.1};  Plot3D[%,{r,0,20},{\[Theta],0,Pi}] 

and I get the plot:

enter image description here

Now I want to add a vertical plane at r=18. How can I do that?

Best Practice for Vertical scrolling through long collapsible submenus

We’re redesigning some of our landing pages. In this case each landing page has 6-8 course categories each of which contain 3-4 courses by date, time and price. Current Layout

Each course has a CTA button and is contained under collapsible course menu. The landing page currently opens with all menus expanded, which introduces the issue of needing to scroll down a lot to find the right course. We also believe we can organize the data better.

For this we’ve come up with a floating bar at the right side which, when clicked upon, scrolls to the right course category (e.g, category 4). Propose layout

We’re wondering if there’s a better way to do this. Suggestions to improve the layout would also be welcome.