How to refine the code to draw the midsphere of a regular tetrahedron?

[Explorer-cc]

What I want to replicate was as below:

Claim: I want real-3D case, so if there are something wrong with the image above, just correct it with luadraw!

Here below is my code, but...

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=tetrahedron}
    local ld = luadraw
    local pt3d, cpx = ld.pt3d, ld.cpx
    local M, Mc, Ms = pt3d.M, pt3d.Mc, pt3d.Ms
    local Origin, vecI, vecJ, vecK = pt3d.Origin, pt3d.vecI, pt3d.vecJ, pt3d.vecK
    local poly = require 'luadraw_polyhedrons'
    require 'luadraw_spherical'
    local g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-4,4,-4,4},
        -- adjust2d = true, 
        viewdir = {60,80}
    }
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    local P = poly.tetrahedron(M(0,0,0), M(0,0,3))
    local D, B, C, A = table.unpack(P.vertices)
    local T1,T2,T3,T4 = (A+B)/2, (B+C)/2, (C+D)/2, (D+A)/2
    local T5,T6 = (B+D)/2, (A+C)/2
    local R = pt3d.abs(T1)
    g:Define_sphere({radius=R,color="violet!30",edgecolor="violet!75",edgewidth=8,mode = ld.mWireframe,hiddenstyle="dashed", hiddencolor="violet!75"})
    g:DSpolyline({{A,B,C,D,A},{B,D},{A,C}},{width=8})
    local o1,r1,n1 = ld.incircle3d(A,B,C) -- incircle of triangle ABC
    g:DScircle({A,pt3d.prod(C-A,B-A)},{color="violet",style="dashed"})
    local o2,r2,n2 = ld.circumcircle3d(T3,T4,T5) -- circumcircle of triangle T3T4T5
    g:DScircle({T5,pt3d.prod(T3-T5,T4-T5)},{color="violet"})
    g:Dspherical()
    g:Dpolyline3d({{Origin,T3},{T2,T4}},"red,dashed,thick")
    g:Dpolyline3d({D,o1,A},"teal,dashed,semithick")
    g:Dballdots3d({Origin,A,B,C,D,T1,T2,T3,T4,T5,T6,o1,o2},"red",0.75)
    g:Dangle3d(D,T4,T2,0.15,"thick");g:Dangle3d(D,T3,Origin,0.15,"thick");g:Dangle3d(T4,T2,B,0.15,"thick");
    g:Dcircle3d(o1,r1,n1,"draw=none,fill=magenta!75,fill opacity=0.4")
    g:Dcircle3d(o2,r2,n2,"draw=none,fill=magenta!75,fill opacity=0.4")
    g:Dlabel3d(
        "$A$",A,{pos="W"},
        "$B$",B,{pos="S"},
        "$C$",C,{pos="E"},
        "$O$",Origin,{},
        "$O_1$",o1,{},
        "$O_2$",o2,{},
        "$D$",D,{pos="N"},
        "$a$",(A+B)/2,{pos="SW",node_options="text=red"},
        "$R$",(T2+T4)/2,{pos="W",dist=0.1}
    )
    g:Show()
\end{luadraw}
\end{document}

There are not big issue, but some details, I want to refine it:

• Could g:DScircle support fill in the case? I found it cumbersome to manually g:Dcircle3d(o1,r1,n1,"draw=none,fill=magenta!75,fill opacity=0.4") in the case?
• Whether the dashed line correct in my code's result, if I don't want facets with the sphere, I have to consider the drawing order?
• My color choose was awful, in my opinion, I don't know how to make the color scheme more similar to the original one?
• Any further suggestions on my code?

[pfradin]

An obvious initial observation: the original figure is not correct at all...
Another point: if the sphere is opaque (opacity=1), there is no point in rendering the disks inside, since they won't be visible! Consequently, the sphere must be transparent; however, this means there are no longer any hidden edges, as they must be visible through the transparency. Furthermore, points and labels located inside the sphere should not look the same as those outside (again, due to the sphere's transparency).

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
\begin{luadraw}{name=tetrahedron}
local ld = luadraw
local pt3d, cpx = ld.pt3d, ld.cpx
local M, Mc, Ms = pt3d.M, pt3d.Mc, pt3d.Ms
local Origin, vecI, vecJ, vecK = pt3d.Origin, pt3d.vecI, pt3d.vecJ, pt3d.vecK
local poly = require 'luadraw_polyhedrons'
require 'luadraw_spherical'
local g = ld.graph3d:new{ window = {-4,4,-4,4}, viewdir = {60,80}, size={12,12} }
ld.Hiddenlines = false;ld.Hiddenlinestyle = "dashed"
local P = poly.tetrahedron(M(0,0,0), M(0,0,3))
local D, B, C, A = table.unpack(P.vertices)
local T1,T2,T3,T4 = (A+B)/2, (B+C)/2, (C+D)/2, (D+A)/2
local T5,T6 = (B+D)/2, (A+C)/2
local R = pt3d.abs(T1)
local o1,r1,n1 = ld.incircle3d(A,B,C) -- incircle of triangle ABC
local o2,r2,n2 = ld.circumcircle3d(T3,T4,T5) -- circumcircle of triangle T3T4T5

g:Define_sphere({radius=R, color="violet!30", edgecolor="violet!75", edgewidth=8, 
    mode=ld.mBorder, hiddencolor="violet!75", opacity=0.65})

g:DScircle({Origin,vecK}, {color="violet!75", width=8})
g:DSpolyline( ld.facetedges(P),{width=8})
g:DScircle({A,pt3d.prod(C-A,B-A)},{color="violet"})
g:DSaddinside( ld.circle3db(o1,r1,n1), "draw=violet,fill=magenta!75, fill opacity=0.6")
g:DScircle({T5,pt3d.prod(T3-T5,T4-T5)},{color="violet"})
g:DSaddinside( ld.circle3db(o2,r2,n2), "draw=violet,fill=magenta!75, fill opacity=0.6")
g:DSlabel(
    "$A$",A,{pos="W"},
    "$B$",B,{pos="S"},
    "$C$",C,{pos="E"},
    "$O$",Origin,{},
    "$O_1$",o1,{},
    "$O_2$",o2,{},
    "$D$",D,{pos="N"},
    "$a$",(A+B)/2,{pos="SW",node_options="text=red"},
    "$R$",(T2+T4)/2,{pos="W",dist=0.1}
)
g:DSdots({Origin,T3,T4,T5,T6,o1,o2},{mark_options="red, scale=0.75"})
g:Dspherical()
g:Dpolyline3d({{Origin,T3},{T2,T4}},"red!50!gray,dashed,thick")
g:Dpolyline3d({D,o1,A},"teal!50!gray,dashed,semithick")
g:Dballdots3d({A,B,C,D,T1, T2},"red",0.75)
g:Linecolor("black!80!violet!75")
g:Dangle3d(D,T4,T2,0.15,"thick"); g:Dangle3d(D,T3,Origin,0.15,"thick"); g:Dangle3d(T4,T2,B,0.15,"thick");
g:Show()
\end{luadraw}
\end{document}

[Explorer-cc]

An obvious initial observation: the original figure is not correct at all...
Another point: if the sphere is opaque (opacity=1), there is no point in rendering the disks inside, since they won't be visible! Consequently, the sphere must be transparent; however, this means there are no longer any hidden edges, as they must be visible through the transparency. Furthermore, points and labels located inside the sphere should not look the same as those outside (again, due to the sphere's transparency).

Yes, I know the point with luadraw:

• If the object was opaque, then hidden dashed line would be shown, but also the inner disk would never be seen
• If the object was transparent, then the hidden line could be shown covered by a shading, and the inner disk could be seen

I jhave to say that some illustrative sketches for reaching purpose are fake-3D.... That is annoying for me :-(

Thanks you for implement and refine my code, it looks like my proginal question:

• Could g:DScircle support fill in the case? I found it cumbersome to manually g:Dcircle3d(o1,r1,n1,"draw=none,fill=magenta!75,fill opacity=0.4") in the case?

Could be handled by g:DSaddinside:

g:DScircle({A,pt3d.prod(C-A,B-A)},{color="violet"})
g:DSaddinside( ld.circle3db(o1,r1,n1), "draw=violet,fill=magenta!75, fill opacity=0.6")
g:DScircle({T5,pt3d.prod(T3-T5,T4-T5)},{color="violet"})
g:DSaddinside( ld.circle3db(o2,r2,n2), "draw=violet,fill=magenta!75, fill opacity=0.6")

However, I have my last question:

My color choose was awful, in my opinion, I don't know how to make the color scheme more similar to the original one?

I have to admit that this might be a very individualized aesthetic issue. But let's consider the following two series of images, just focus on the style, but whether they are correct in the following:

Group 1 Bright and Brilliant:

Group 2 A little bit depressing and gloomy(might be caused by opacity)

I have to say that was quite personally, but I prefer the group 1. I wonder whether I could do something(maybe choose some bright colors) to make the transparent object looks more bright? Many thanks!

[pfradin]

The real difference between the two groups is transparency. With a completely opaque sphere, here is what I would do:

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
\begin{luadraw}{name=tetrahedron}
local ld = luadraw
local pt3d, cpx = ld.pt3d, ld.cpx
local M, Mc, Ms = pt3d.M, pt3d.Mc, pt3d.Ms
local Origin, vecI, vecJ, vecK = pt3d.Origin, pt3d.vecI, pt3d.vecJ, pt3d.vecK
local poly = require 'luadraw_polyhedrons'
require 'luadraw_spherical'
local g = ld.graph3d:new{ window = {-4,4,-4,4}, viewdir = {60,80}, size={12,12} }
ld.Hiddenlines = true; ld.Hiddenlinestyle = "dashed"
local P = poly.tetrahedron(M(0,0,0), M(0,0,3))
local D, B, C, A = table.unpack(P.vertices)
local T1,T2,T3,T4 = (A+B)/2, (B+C)/2, (C+D)/2, (D+A)/2
local T5,T6 = (B+D)/2, (A+C)/2
local R = pt3d.abs(T1)
local o1,r1,n1 = ld.incircle3d(A,B,C) -- incircle of triangle ABC
local o2,r2,n2 = ld.circumcircle3d(T3,T4,T5) -- circumcircle of triangle T3T4T5

g:Define_sphere({radius=R, color="violet!30", edgecolor="violet!75", edgewidth=8, 
    mode=ld.mBorder, hiddencolor="violet!75"})

g:DScircle({Origin,vecK}, {color="violet!75", width=8})
g:DSpolyline( ld.facetedges(P),{width=8})
g:DScircle({A,pt3d.prod(C-A,B-A)},{color="violet"})
g:DScircle({T5,pt3d.prod(T3-T5,T4-T5)},{color="violet"})
g:DSlabel(
    "$O$",Origin,{pos="E"},
    "$O_1$",o1,{},
    "$O_2$",o2,{},
    "$a$",(A+B)/2,{pos="SW",node_options="text=red"},
    "$R$",(T2+T4)/2,{pos="W",dist=0.1}
)
g:DSdots({Origin,T3,T4,T5,T6,o1,o2},{mark_options="red, scale=0.75"})
g:Dspherical()
g:Dpolyline3d({{Origin,T3},{T2,T4}},"red!50!gray,dashed,thick")
g:Dpolyline3d({D,o1,A},"teal!50!gray,dashed,semithick")
g:Dballdots3d({A,B,C,D,T1, T2},"red",0.75)
g:Dlabel3d(
    "$A$",A,{pos="W"},
    "$B$",B,{pos="S"},
    "$C$",C,{pos="E"},
    "$D$",D,{pos="N"}
)
g:Linecolor("black!80!violet!75"); g:Linestyle("dashed")
g:Dangle3d(D,T4,T2,0.15,"thick"); g:Dangle3d(D,T3,Origin,0.15,"thick"); g:Dangle3d(T4,T2,B,0.15,"thick");
g:Show()
\end{luadraw}
\end{document}

[Explorer-cc]

The real difference between the two groups is transparency.

I have to say that was quite personally, but I prefer the group 1. I wonder whether I could do something(maybe choose some bright colors) to make the transparent object looks more bright?

So the "glmooy"-style was caused by transparency, I still wonder there exist color named "SuperBrightViolet" to make it more bright?

The case I used faced with opacity was that, when we are expected to get the following result(For the moment, let's not worry about whether he is truly 3d case):

A common scenario is that, says that I want a "megenta" sphere(that mean I want the final main color was "magenta", that is #FF00FF in HTML form):

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
\begin{luadraw}{name=test}
ld = luadraw
require 'luadraw_spherical'
local g = ld.graph3d:new{size={10,10}}
g:Define_sphere({color="magenta"})
g:Dspherical()
g:Show()
\end{luadraw}
\end{document}

That is what I expeceted:

However, if I need opacity=0.5 to show the items inside the sphere, I have:

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
\begin{luadraw}{name=test}
ld = luadraw
require 'luadraw_spherical'
local g = ld.graph3d:new{size={10,10}}
g:Define_sphere({color="magenta",opacity=0.5})
g:Dspherical()
g:Show()
\end{luadraw}
\end{document}

I know that is correct, but that is "gloomy", and the color was roughly #953f95, far away from what I want(the referenced example skectch's color scheme). Let's consider the case that, example image was magenta(#FF00FF), but for the sake of opacity, I have to see #953f95, the color scheme was quite different, and that might highly affected the reader's reading experience(sorry, quite personally, I would be more likely to be happy with bright color, but no so happy with gloomy color scheme). Could I have some options to inhibit the shading color affected the original color to be darken, or I could choose a "super-bright-magenta" to make sure I could see #FF00FF even with opacity finally? If my explanations and requirements are incorrect or unreasonable, please accept my most sincere apology.

[pfradin]

It is not luadraw that handles the ball color and fill opacity options, but TikZ, since it is responsible for the rendering. I therefore suggest you look into TikZ.

[Explorer-cc]

It is not luadraw that handles the ball color and fill opacity options, but TikZ, since it is responsible for the rendering. I therefore suggest you look into TikZ.

Okay, thank you! ;-)

[Explorer-cc]

A follow-up TikZ discussion: https://tex.stackexchange.com/q/766439

[cfr42]

IMHO, luadraw could use a transparency group here. Right now, you interpret opacity as fill opacity, but that does not adjust the transparency of the shading, because that's added separately. So you render with 50% magenta and then the darkening is added, with no modification, as if the fill were opaque. A transparency group would ensure the object as a whole is transparent.

Or you could use the transparent package, since you don't have to worry about engine compatibility. That might be easier to add, I'm not sure.

[pfradin]

Indeed, adding a transparency group right before drawing the sphere solves the problem. However, I need to implement this internally, so the change will take effect in the next version.

On the other hand, I am not sure the transparent package is useful here.

[cfr42]

I mentioned it because I could not easily figure out from the code which might be easiest to implement. Since your package is Lua-specific, there wouldn't be any disadvantage to using the transparency stack when \DocumentMetadata is used. But transparency groups are perfectly fine too and much more tikz-like. I just mentioned it in case adding a scope around the object was problematic, since you could avoid using a group.

[Explorer-cc]

For comparsion, I would add a POV-Ray version with my previous comment #335 (reply in thread)

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\newcommand{\povball}[2]{g:Pov_sphere(pt3d.M(#1,0,0),1.5,{color=ld.Magenta,opacity=#2,shadow=false})}
\begin{document}
\begin{luadraw}{name=opacity-balls}
local ld = luadraw
local pt3d = ld.pt3d
local g = ld.graph3d:new{
    window3d={-10,10,-2.5,2,-2,2},window={-10,10,-2.5,2},
    viewdir="xOy",size={15,3.375}
}
require "luadraw_povray"
g:Pov_new()
\povball{-8}{0.05}
\povball{-4}{0.25}
\povball{0}{0.5}
\povball{4}{0.75}
\povball{8}{1}
g:Pov_exec()
g:Pov_show()
local Z = ld.cpx.Z
g:Dlabel(
    "opacity=0.05",Z(-8,-1.65),{pos="S"},
    "opacity=0.25",Z(-4,-1.65),{pos="S"},
    "opacity=0.5",Z(0,-1.65),{pos="S"},
    "opacity=0.75",Z(4,-1.65),{pos="S"},
    "opacity=1",Z(8,-1.65),{pos="S"}
)
g:Show()
\end{luadraw}
\end{document}