比较遵照你的方案（基于g:Dscene3d）的方法：

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=intersected_surface_Dscene}
    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 g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-3,3,-2,2},
        viewdir = {"yz",0.75,40}
    }
    local vecN = ld.rotate3d(vecK,-70,{Origin,vecI})
    ld.Hiddenlines = true; ld.Hiddenlinestyle="dashed"
    g:Dscene3d(
        g:addPlane({Origin,vecK},{contrast=0,rectangle={vecI,2,4},edge=true,edgecolor="DarkBlue",hidden=true}),
        g:addPlane({Origin,vecN},{contrast=0,rectangle={vecI,2,2.5},edge=true,edgecolor="Orange",hidden=true}),
        g:addDots({Origin}),
        g:addLabel(
            "$\\ell$",2.5*vecI,{pos="SW"},
            "$P$",Origin,{pos="E",dist=0.1},
            "$\\alpha$",vecI-2*vecJ,{pos="NE",dir={vecJ,-vecI},color="DarkBlue",dist=0},
            "$\\beta$",ld.rotate3d(vecI-vecJ,-70,{Origin,vecI}),{pos="E",dir={-vecI,pt3d.prod(vecN,-vecI)},color="Orange"}
        )
    )
    g:Dpolyline3d({-2.5*vecI,2.5*vecI}, "thick")
    g:Show()
    \end{luadraw}
\end{document}

这里的「灰色」的关键在于 contrast 选项（color的默认值就是white，我是能省就省的，所以我不写）：

Some Notes and Comments

• 我在构造曲面和曲线的时候是比较遵从直角坐标系的，交线的方向就是vecI
• 我采用ld.rotate3d来处理这里的旋转，这样用到的点比较少
• 从原图来看，我不知道你为何要用中心投影，更合理的方法应该是斜投影viewdir = {"yz",0.75,40}（类似斜二测画法）
• 要注意的是，如果一个面是透明的，那么在它后面的线都是可见的，不会变成虚线，只是颜色稍微变暗；相反只有当一个面是不透明的，那么它后面的线才会被ld.Hiddenlines = true; ld.Hiddenlinestyle="dashed"控制。如果一个面是半透明的，那么不会（也不应该）出现虚线

可以对比一下思路 :-)

比较取巧的方法：

由于这里遮挡关系并不算太复杂，可以通过画两次（先遮挡再画虚线）的方法来手动维护遮挡关系：

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=intersected_surface}
    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 g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-3,3,-2,2},
        -- adjust2d = true, 
        -- viewdir = {30,75}
        viewdir = {"yz",0.75,40}
    }
    -- g:Dpolyline3d({{Origin,3*vecI},{Origin,3*vecJ},{Origin,3*vecK}}, "thick,-Stealth")
    g:Dplane({Origin, vecK}, vecI, 2, 4 , ld.all, "DarkBlue,fill=white")
    local vecN = ld.rotate3d(vecK, -70, {Origin,vecI}) -- rotate vecK 30 degree around vecI
    g:Dplane({Origin, vecN}, vecI, 2, 2.5, ld.all, "Orange,fill=white")
    g:Dplane({Origin, vecK}, vecI, 2, 4 , ld.all, "DarkBlue,dashed")
    g:Dplane({vecJ, vecK}, vecI, 2, 2 , ld.all, "DarkBlue,fill=white") -- trick, fill half plane... not elegant actually...
    g:Dplane({Origin, vecN}, vecI, 2, 2.5, ld.all, "Orange,dashed")
    g:Dpolyline3d({-2.5*vecI,2.5*vecI}, "thick")
    g:Ddots3d({Origin})
    g:Dlabel3d(
        "$\\ell$",2.5*vecI,{pos="SW"},
        "$P$",Origin,{pos="E",dist=0.1},
        "$\\alpha$",vecI-2*vecJ,{pos="NE",dir={vecJ,-vecI},node_options="text=DarkBlue",dist=0},
        "$\\beta$",ld.rotate3d(vecI-vecJ, -70, {Origin,vecI}),{pos="E",dir={-vecI,pt3d.prod(vecN,-vecI)},node_options="text=Orange"}
    )
    g:Show()
\end{luadraw}
\end{document}

简单地说：

• window3d控制的是三维空间的边界盒
• window控制的是三维空间边界盒里的内容按照viewdir投影到纸张平面后，2D平面的边界

luadraw在绘制的过程中，_大致_经历了下面的过程：

• 根据3D盒子的边界确定3D空间盒子的边界，一些在盒子外的内容可能会被某些方法切去
• 基于viewdir确定3D物体投影到屏幕投影面的方法和角度
• 基于adjust2d,window和bbox以及size等参数调整屏幕投影面的最终显示效果

(AI总结，仅供参考）

在你的例子中，3D盒子大小是不影响的最终投影结果的，因此：

    local g = ld.graph3d:new{
        window3d = {0,1.5,0,5.5,0,0.6},
        -- window3d = {0,3,0,11,0,1.2},
        window={-1.1,5.5,-1.1,1.5},
        bbox=false,size={9,9},
        viewdir={"yz",0.5,45}
    }

和

    local g = ld.graph3d:new{
        -- window3d = {0,1.5,0,5.5,0,0.6},
        window3d = {0,3,0,11,0,1.2},
        window={-1.1,5.5,-1.1,1.5},
        bbox=false,size={9,9},
        viewdir={"yz",0.5,45}
    }

的表现都是相同的：

我们可以通过 Dboxaxes3d 方法来更好地可视化这种差距：

\documentclass[border=8pt]{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
\begin{tabular}{cc}
    \begin{luadraw}{name=cube_small} 
        local ld = luadraw
        local pt3d = ld.pt3d
        local M = pt3d.M
        local g = ld.graph3d:new{
            window3d = {0, 1, 0, 1, 0, 1},
            bbox = false, size = {10, 10},
            viewdir = {"yz", 0.5, 45}
        }
        local Origin, vecI, vecJ, vecK = pt3d.Origin, pt3d.vecI, pt3d.vecJ, pt3d.vecK
        local C = ld.parallelep(Origin, 1.5*vecI, 1.5*vecJ, 1.5*vecK)
        g:Dboxaxes3d({grid = true, gridcolor = "cyan!50", fillcolor = "cyan!20", drawbox = true})
        g:Dpoly(C, {mode = ld.mWireframe})
        g:Show()
    \end{luadraw}
    &
    \begin{luadraw}{name=cube_large} 
        local ld = luadraw
        local pt3d = ld.pt3d
        local M = pt3d.M
        local g = ld.graph3d:new{
            window3d = {0, 2, 0, 2, 0, 2},
            bbox = false, size = {10, 10},
            viewdir = {"yz", 0.5, 45}
        }
        local Origin, vecI, vecJ, vecK = pt3d.Origin, pt3d.vecI, pt3d.vecJ, pt3d.vecK
        local C = ld.parallelep(Origin, 1.5*vecI, 1.5*vecJ, 1.5*vecK)
        g:Dboxaxes3d({grid = true, gridcolor = "cyan!50", fillcolor = "cyan!20", drawbox = true})
        g:Dpoly(C, {mode = ld.mWireframe})
        g:Show()
    \end{luadraw}
    \\
    \texttt{window3d = \{0, 1, 0, 1, 0, 1\}} &
    \texttt{window3d = \{0, 2, 0, 2, 0, 2\}}
\end{tabular}
\end{document}

如果不左右对比，同时不画盒子，只能感受到缩放的差距，因为立方体和2D画布的相对关系是相同的。因为我不怎么会用到那种「盒子外的内容可能会被切去」的方法，所以其实保持默认的 window3d={-5,5,-5,5,-5,5,1,1,1}，只调整window和size其实也是够的...

无论我怎么改变window3d里面的6个参数值，都没有影响到输出的图形,不知道window3d的设置还有什么价值?

是因为只改变「window3d里面的6个参数值」，只会影响3D盒子边界和真实物体的相对大小关系，但是绘制结果是经过上述流程2D投影后得到的，投影后的纸张和物体的投影相对大小关系还是相同

而window的设置可以截取部分图形。我将设置window的代码那行删除,结果右侧显示不完全。

这是因为不写window，就相当使用的是默认值window={-5,5,-5,5}，如果3D投影到2D后还超出了window，那么就会「显示不完全」

我希望是取消留白，只有图形区域,类似border=0pt

那么也许应该使用类似bbox=false的方法....

上面的答案是正确的，我们可以考虑调整角度来说明这一点，下面的代码使用与z轴成90度的「平行方向」来看这个几何体，我们可以看到他们是「对称等高」的：

\documentclass[border=5pt]{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\begin{document}
\begin{luadraw}{name=prism}
    local ld = luadraw
    local pt3d = ld.pt3d
    local Origin, vecK = pt3d.Origin, pt3d.vecK
    local M,Mc=pt3d.M,pt3d.Mc
    local g = ld.graph3d:new{
        window3d={-5,5,-5,5,-5,5},
        window={-5,5,-5,5},
        bbox=false,
        -- adjust2d = true,
        viewdir={0,90} -- <- here
    }
    local A1,B1,C1=Mc(1,-120*ld.deg,0),Mc(1,0,0), Mc(1,120*ld.deg,0)
    local prism1 = ld.prism({A1,B1,C1}, M(-2,0,3))
    g:Dpoly(prism1,{mode=ld.mWireframe,hiddenstyle="dashed"})
    local prism = ld.sym3d(prism1, {Origin,vecK})
    g:Dpoly(prism,{mode=ld.mWireframe,hiddenstyle="dashed"})
    local V = prism.vertices
    g:Dlabel3d(
        "$A$",V[1],{pos="W"},
        "$B$",V[2],{pos="S"},
        "$C$",V[3],{pos="E"},
        "$A_1$",V[4],{pos="W"},
        "$B_1$", V[5],{pos="SE"},
        "$C_1$", V[6],{pos="E"}
    )
    g:Show()
\end{luadraw} 
\end{document}

另外，如果我是你，我可能不会考虑这样的构造，我会想办法让「两个棱柱相交的那个面」在「xoy平面」上：

\documentclass[border=5pt]{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
\begin{luadraw}{name=prism}
    local ld = luadraw
    local pt3d = ld.pt3d
    local Origin, vecK = pt3d.Origin, pt3d.vecK
    local M,Mc=pt3d.M,pt3d.Mc
    local g = ld.graph3d:new{
        window3d={-5,5,-5,5,-5,5},
        window={-5,5,-5,5},
        bbox=false,
        -- adjust2d = true,
        viewdir={"yz",0.5,45}
    }
    local A1,B1,C1=Mc(1,-120*ld.deg,0),Mc(1,0,0), Mc(1,120*ld.deg,0)
    local prism1 = ld.prism({A1,B1,C1}, M(-2,0,3))
    g:Dpoly(prism1,{mode=ld.mWireframe,hiddenstyle="dashed"})
    local prism = ld.sym3d(prism1, {Origin,vecK})
    g:Dpoly(prism,{mode=ld.mWireframe,hiddenstyle="dashed"})
    local V = prism.vertices
    g:Dlabel3d(
        "$A_1$",V[1],{pos="W"},
        "$B_1$", V[2],{pos="SE"},
        "$C_1$", V[3],{pos="E"},
        "$A$",V[4],{pos="W"},
        "$B$",V[5],{pos="S"},
        "$C$",V[6],{pos="E"}
    )
    g:Show()
\end{luadraw} 
\end{document}

我对你的代码做了一下简化：

• 0*ld.deg 这里的0就没必要乘了
• node_options="xshift=-0cm" 默认就不移动，不需要这么改，一般改距离可以直接用dist=0.1
• g:Dlabel3d一个命令可以同时画六个点的坐标
• 在这个例子里，完全可以构造一个对称的棱柱，使用ld.sym3d，这可以简化你的不少坐标...

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=prism_sphere}
    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 g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {0,75}
    }
    require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=10, edgecolor="black" }
    local h = 2.25
    local O1,O2 = M(0,0,h),M(0,0,-h)
    local r = math.sqrt(3^2-h^2)
    local t1, t2, t3 = -80*ld.deg, 45*ld.deg, 150*ld.deg
    local A,B,C = Mc(r,t1,-h), Mc(r,t2,-h), Mc(r,t3,-h)
    local A1,B1,C1 = Mc(r,t1,h), Mc(r,t2,h), Mc(r,t3,h)
    g:DScircle({Origin,vecK},{width=6})
    local P = ld.facet2poly( ld.cvx_hull3d({A,B,C,A1,B1,C1}) )
    g:Dcircle3d(O2,r,vecK,"draw=none,fill=blue!25, fill opacity=0.4") -- lower circle
    g:DScircle({O2,vecK},{width=8})
    g:Dpoly(P,{color="violet!30!white",mode=ld.mFlatHidden,edgewidth=8,hiddenstyle="dashed"})
    g:Dcircle3d(O2,r,vecK,"draw=none,fill=blue!25, fill opacity=0.4") -- upper circle
    g:DScircle({O1,vecK},{width=8})
    g:Dspherical()
    g:Dballdots3d({Origin,O1,O2},"red")
    g:Dpolyline3d({C1,O1,O2,A},"blue!75,dashed,semithick")
    g:Dpolyline3d({C1,Origin,A},"red,dashed,semithick")
    g:Dlabel3d(
        "$O$",Origin,{pos="E"},
        "$O_1$",O1,{pos="W"},
        "$O_2$",O2,{pos="E"},
        "$A$",A,{pos="SW"},
        "$B$",B,{pos="SE"},
        "$C$",C,{pos="NE"},
        "$A_1$",A1,{pos="NW"},
        "$B_1$",B1,{pos="SE"},
        "$C_1$",C1,{pos="NE"},
        "$r$",(O1+C1)/2,{pos="",node_options="text=blue"},
        "$r$",A/6+5*O2/6,{},
        "$h$",O1/4,{pos="W"},
        "$R$",C1/2,{pos="E",node_options="text=red"},
        "$R$",2*A/5,{pos="N",dist=0.15}
    )
    g:Show()
\end{luadraw}
\end{document}

% https://ask.latexstudio.net/ask/question/18066.html
\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=cylinder_sphere}
    local ld = luadraw
    local pt3d = ld.pt3d
    local M, Mc, Origin, vecK = pt3d.M, pt3d.Mc, pt3d.Origin, pt3d.vecK
    local g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {0,80} -- should be fixed
    }
    require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=10, edgecolor="black" }
    local h = 2
    local O1,O2 = M(0,0,h),M(0,0,-h)
    local r = math.sqrt(3^2-h^2)
    local A,B = Mc(r,140*ld.deg,h), Mc(r,45*ld.deg,-h)
    g:DScircle({Origin,vecK});g:DScircle({O1,vecK},{width=8});g:DScircle({O2,vecK},{width=8})
    g:Dcylinder(O1,r,O2,{color="violet!75", edgestyle="noline",hiddenstyle="noline"}) -- hide the frustum border and hidden lines
    local M1, N1, M2, N2 = table.unpack(g:Cylinder_tangency(O1, r, O2)) -- get the tangency points of the cylinder
    g:DSpolyline({{M1,M2},{N1,N2}},{width=8}) -- draw the 
    --
    g:Dspherical()
    g:Dangle3d(A,O1,O2,0.3,"very thick")
    g:Dpolyline3d({B,Origin,A,O1,O2},"blue,dashed,thick")
    g:Dballdots3d({Origin,O1,O2,A,B},"red",0.75)
    g:Dlabel3d(
        "$O$",Origin,{pos="W"},
        "$h$",O2/4,{},
        "$O_1$",O1,{},
        "$O_2$",O2,{},
        "$A$",A,{pos="NE"},
        "$B$",B,{pos="SE"},
        "$r$",(O1+A)/2,{pos="S",node_options="text=red"},
        "$h/2$",O1/2,{pos="W"},
        "$R$",3*A/5,{pos="E"}
    )
    g:Show()
\end{luadraw}
\end{document}

注意右手定则，看注释...

\documentclass[border=5pt]{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}

\begin{document}
\begin{luadraw}{name=prism}
    local ld = luadraw
    local pt3d = ld.pt3d
    local M = pt3d.M
    local g = ld.graph3d:new{
        window3d={-5,5,-5,5,-5,5},
        windows={-5,5,-5,5},
        size={8,8},
        bbox=false,
        viewdir={"yz",0.5,45}
        -- viewdir={30,70}
    }

    -- local base = { M(0,0,0), M(2,0,0), M(0,0,2) }
    local base = { M(0,0,0), M(0,0,2), M(2,0,0) } -- 右手定则
    -- M(0,0,2), M(2,0,0)在右手法则下应该与 (0,3,0) 通向
    local prism = ld.prism(base, M(0,3,0))
    
    g:Dpoly(prism,{
        mode=ld.mWireframe,
        hiddenstyle="dashed",
        -- edgestyle="solid"
    })
    g:Ddots3d(M(0,0,2),"blue")

    g:Show()
\end{luadraw}
\end{document}

更新你的 latex3-backend:

在终端运行：

tlmgr update --all --self

应该就行了...

https://tex.stackexchange.com/q/765234/322482

原图又在骗

如果「D」点确实是截面圆与视界圆的切点的话，那么根本不可能看到如上角度的蓝色三角形：

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=cuboid_sphere}
    local ld = luadraw
    local pt3d = ld.pt3d
    local M, Origin, vecK = pt3d.M, pt3d.Origin, pt3d.vecK
    local g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {20,75} 
    }
   require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=8, edgecolor="black" }
    local sqrt = math.sqrt
    local h = 1.75
    local O1, P = M(0,0,-h), sM(0,0)
    local Out = {}
    g:DScircle({O1,vecK},{out=Out});
    local D, _ = table.unpack(Out)
    local A = ld.rotate3d(D, 90, {O1, vecK})
    local B = ld.rotate3d(A, 90, {O1, vecK})
    local C = ld.rotate3d(B, 90, {O1, vecK})
    local vecn = pt3d.prod(A-P,D-P)
    local O2 = ld.proj3d(Origin,{P,vecn})
    local h1,h2 = pt3d.abs(Origin-O1),pt3d.abs(Origin-O2)
    local r1,r2 = sqrt(3^2-h1^2),sqrt(3^2-h2^2)
    g:DScircle({Origin,vecK})
    local T = ld.facet2poly( ld.cvx_hull3d({A,B,C,D,P}) )
    g:Dcircle3d(O1,r1,vecK,"draw=none,fill=blue!25, fill opacity=0.4") 
    g:Dcircle3d(O2,r2,vecn,"draw=none,fill=blue!25, fill opacity=0.4") 
    g:DScircle({O1,vecK}); g:DScircle({O2,vecn})
    g:Dpoly(T,{color="violet!30!white",mode=ld.mFlatHidden,edgewidth=8,hiddenstyle="dashed"})
    g:Dspherical()
    g:Dpolyline3d({P,O1,A,Origin},"blue,dashed,semithick")
    g:Dballdots3d({Origin,O1,A,B,C,D,P},"red",0.75)
    g:Dlabel3d(
        "$O$",Origin,{pos="W"},
        "$O_1$",O1,{},
        "$P$",P,{pos="N"},
        "$A$",A,{pos="NE"},
        "$B$",B,{pos="W"},
        "$C$",C,{pos="S"},
        "$D$",D,{pos="E"},
        "$h$",O1/2,{pos="W"},
        "$R$",P/2,{pos="",node_options="text=blue"},
        "$R$",A/2,{},
        "$r$",(O1+A)/2,{pos="E"}
    )
    g:Show()
\end{luadraw}
\end{document}

把底面微调成长方形，修改这里的旋转角度：

    local D, _ = table.unpack(Out)
    local A = ld.rotate3d(D, 75, {O1, vecK}) -- <- 75 degree?
    local B = ld.rotate3d(A, 115, {O1, vecK})
    local C = ld.rotate3d(B, 75, {O1, vecK})

完整代码如下：

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=cuboid_sphere}
    local ld = luadraw
    local pt3d = ld.pt3d
    local M, Origin, vecK = pt3d.M, pt3d.Origin, pt3d.vecK
    local g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {20,75} 
    }
   require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=8, edgecolor="black" }
    local sqrt = math.sqrt
    local h = 1.75
    local O1, P = M(0,0,-h), sM(0,0)
    local Out = {}
    g:DScircle({O1,vecK},{out=Out});
    local D, _ = table.unpack(Out)
    local A = ld.rotate3d(D, 75, {O1, vecK}) -- <- 75 degree?
    local B = ld.rotate3d(A, 115, {O1, vecK})
    local C = ld.rotate3d(B, 75, {O1, vecK})
    local vecn = pt3d.prod(A-P,D-P)
    local O2 = ld.proj3d(Origin,{P,vecn})
    local h1,h2 = pt3d.abs(Origin-O1),pt3d.abs(Origin-O2)
    local r1,r2 = sqrt(3^2-h1^2),sqrt(3^2-h2^2)
    g:DScircle({Origin,vecK})
    local T = ld.facet2poly( ld.cvx_hull3d({A,B,C,D,P}) )
    g:Dcircle3d(O1,r1,vecK,"draw=none,fill=blue!25, fill opacity=0.4") 
    g:Dcircle3d(O2,r2,vecn,"draw=none,fill=blue!25, fill opacity=0.4") 
    g:DScircle({O1,vecK}); g:DScircle({O2,vecn})
    g:Dpoly(T,{color="violet!30!white",mode=ld.mFlatHidden,edgewidth=8,hiddenstyle="dashed"})
    g:Dspherical()
    g:Dpolyline3d({P,O1,A,Origin},"blue,dashed,semithick")
    g:Dballdots3d({Origin,O1,A,B,C,D,P},"red",0.75)
    g:Dlabel3d(
        "$O$",Origin,{pos="W"},
        "$O_1$",O1,{},
        "$P$",P,{pos="N"},
        "$A$",A,{pos="NE"},
        "$B$",B,{pos="W"},
        "$C$",C,{pos="S"},
        "$D$",D,{pos="E"},
        "$h$",O1/2,{pos="W",dist=0.1},
        "$R$",P/2,{pos="",dist=0,node_options="text=blue"},
        "$R$",A/2,{},
        "$r$",O1/3+2*A/3,{pos="S"}
    )
    g:Show()
\end{luadraw}
\end{document}

我们可以获得比较贴近原图的效果...

真的别用伪3D来骗了😡😡😡😡😡修改和分析成本很大，打咩！

Analysis

需要指出的是，原图的坐标也是糊弄的...

这里的四个点并不是真正的交点，原图就是在糊弄....!!!

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=cuboid_sphere}
    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 g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {0,75}
    }
   require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=8, edgecolor="black" }
    local h = 1.25
    local O1, O2 = -h*vecK, h*vecK
    local Out1, Out2 = {}, {}
    g:DScircle({O1,vecK},{out=Out1});g:DScircle({O2,vecK},{out=Out2});
    local D,B = table.unpack(Out1);
    local B1,D1 = table.unpack(Out2)
    g:Dlabel3d(
        "$O$",Origin,{pos="SW"},
        "$A$",A,{pos=""},
        "$A_1$",A1,{},
        "$B$",B,{},
        "$B_1$",B1,{},
        "$C$",C,{},
        "$C_1$",C1,{},
        "$D$",D,{},
        "$D_1$",D1,{}
    )    
    g:Dspherical()
    local R = math.sqrt(3^2-h^2) 
    -- g:Dballdots3d({Origin,B,B1,D,D1,O1,O2},"red",0.75)
    -- g:Dballdots3d({M(0,-R,-h),M(0,R,-h),M(0,R,h),M(0,-R,h)},"blue",0.75)
    g:Dpolyline3d({B,D,D1,B1},true,"red")
    g:Dpolyline3d({M(0,-R,-h),M(0,R,-h),M(0,R,h),M(0,-R,h)},true,"blue")
    g:Show()
\end{luadraw}
\end{document}

蓝色点并非水平截面圆与视界圆的交点！！！

Code

下面的代码也只能糊弄了，否则无法取到和原图类似的显示效果...

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=cuboid_sphere}
    local ld = luadraw
    local pt3d = ld.pt3d
    local M, Origin, vecK = pt3d.M, pt3d.Origin, pt3d.vecK
    local g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {0,75} -- should be fixed
    }
   require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=8, edgecolor="black" }
    local C,C1 = sM(40,120), sM(40,60)
    local O1,O2 = ld.proj3d(Origin,{C,vecK}),ld.proj3d(Origin,{C1,vecK})
    local h = pt3d.abs(O1-O2)/2
    local r = math.sqrt(3^2-h^2)
    local B,B1 = M(0,-r,-h),M(0,-r,h)
    local D1,D = M(0,r,h),M(0,r,-h)
    local A,A1 = ld.psym3d(C,O1),ld.psym3d(C1,O2)
    g:DScircle({Origin,vecK})
    local P = ld.facet2poly( ld.cvx_hull3d({A,B,C,D,A1,B1,C1,D1}) )
    g:Dcircle3d(O1,r,vecK,"draw=none,fill=blue!25, fill opacity=0.4") -- lower circle
    g:DScircle({C,vecK})
    g:Dpoly(P,{color="violet!30!white",mode=ld.mFlatHidden,edgewidth=8,hiddenstyle="dashed"})
    g:Dcircle3d(O2,r,vecK,"draw=none,fill=blue!25, fill opacity=0.4") -- upper circle
    g:DScircle({C1,vecK})
    g:Dspherical()
    g:Dpolyline3d({A1,Origin,D},"blue,dashed")
    g:Dballdots3d({Origin,A,A1,B,B1,C,C1,D,D1},"red",0.75)
    g:Dlabel3d(
        "$O$",Origin,{pos="W"},
        "$A$",A,{pos="NW"},
        "$A_1$",A1,{pos="N"},
        "$B$",B,{pos="W"},
        "$B_1$",B1,{},
        "$C$",C,{pos="S"},
        "$C_1$",C1,{pos="SE"},
        "$D$",D,{pos="E"},
        "$D_1$",D1,{},
        "$a$",(B+C)/2,{pos="S",node_options="text=red"},
        "$b$",(C+D)/2,{pos="N"},
        "$c$",(C+C1)/2,{pos="E"},
        "$R$",D/2,{pos="N"},
        "$R$",A1/3,{}
    )
    g:Show()
\end{luadraw}
\end{document}

中规中矩吧.....

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
\begin{luadraw}{name=insphere_cone}
    local ld = luadraw
    local pt3d = ld.pt3d
    local M, Origin, vecK = pt3d.M,  pt3d.Origin, pt3d.vecK
    require 'luadraw_spherical'
    local g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        windows = {-4,4,-4,4},
        viewdir = {0,80}, size={10,10}
    }
    local sM = ld.sM
    local A,B,P = sM(-30,140), sM(150,140), sM(0,0)
    local O1 = (A+B)/2
    ld.Hiddenlines = true; ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "",edgecolor = "black",edgewidth = 8 }
    -- g:DSpolyline({A,B},{style="dashed",width=8})
    -- g:DSpolyline({{O1,P},{Origin,A}},{color="red",style="dashed",width=8})
    local vecn = pt3d.prod(A,B)
    g:Dcone(O1,pt3d.abs(B-A)/2,P,{edgewidth=8, hiddenstyle="dashed", color="magenta", opacity=0.5})
    g:DScircle({Origin,vecn});g:DScircle({Origin,vecK},{width=8});
    g:Dspherical()
    g:Dcircle3d(Origin,3,vecn,"draw=none,fill=blue!25, fill opacity=0.4")
    g:Dpolyline3d({A,B},"dashed,thick")
    g:Dpolyline3d({{O1,P},{Origin,A}},"dashed,red,thick")
    g:Dballdots3d({Origin,O1,A,B,P},"red")
    g:Dlabel3d(
        "$O$",Origin,{pos="E"},
        "$O_1$",O1,{pos="SE"},
        "$B$",B,{},
        "$A$",A,{pos="SW"},
        "$r$",(O1+A)/2,{pos="N"},
        "$P$",P,{},
        "$R$",P/2,{pos="E",node_options="text=red"},
        "$R$",A/2,{pos="W"},
        "$h$",O1/2,{pos="E"}
    )
    g:Show()
\end{luadraw}
\end{document}

这里又学会一些小技巧：

• ld.poly2facet可以把「poly」转为「facets」
• ld.facetedges可以把「facets」转为「edges」直接作为参数列表传入「绘制」函数

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=spherical_cube}
    local ld = luadraw
    local pt3d, cpx = ld.pt3d, ld.cpx
    local Z = cpx.Z
    local M, Mc, Ms = pt3d.M, pt3d.Mc, pt3d.Ms
    local Origin, vecI, vecJ, vecK = pt3d.Origin, pt3d.vecI, pt3d.vecJ, pt3d.vecK
    local g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {15,75}
    }
    local sqrt = math.sqrt
    require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "magenta!25",edgewidth=8, edgecolor="black" }
    local a = 3/sqrt(2)
    local S = M(-a,-a,-a)
    local O1, O2= a*vecK, -a*vecK
    local P = ld.poly2facet(ld.parallelep(S,2*a*vecI,2*a*vecJ,2*a*vecK))
    local edges = ld.facetedges(P)
    g:DScircle({Origin, vecK},{color="violet",width=8})
    g:DSpolyline(edges,{width=8})
    g:DScircle({a*vecK, vecK},{color="violet",width=8})
    g:DScircle({-a*vecK, vecK},{color="violet",width=8})
    g:Dspherical()
    g:Dcircle3d(O1,a,vecK,"draw=none, fill=magenta!75, fill opacity=0.4") -- not quite good
    g:Dcircle3d(O2,a,vecK,"draw=none, fill=magenta!75, fill opacity=0.4")
    
    local t1,t2,t3,t4  = M(0,-a,a), M(0,a,a), M(-a,0,a), M(a,0,a)
    local m1,m2,m3,m4  = M(a,a,0), M(a,-a,0), M(-a,a,0), M(-a,-a,0)
    local b1,b2,b3,b4  = M(0,-a,-a), M(0,a,-a), M(-a,0,-a), M(a,0,-a)
    for _, p in ipairs{t3,m3,b3,b4} do
        g:Dpolyline3d({Origin,p},"teal,thick,dashed")
    end
    g:Dpolyline3d({Origin,O1,t1},true,"red,thick,dashed")
    g:Dballdots3d({Origin,O1,t1,t2,t3,t4,m1,m2,m3,m4,b1,b2,b3,b4},"red")
    g:Dlabel3d(
        "$O$",Origin,{pos="NE"},
        "$C_1$",M(-a,a,a),{},
        "$H$",M(0,-a,a),{pos="NW"},
        "$A_1$",M(a,-a,a),{pos="W"},
        "$D$",M(-a,-a,-a),{},
        "$D_1$",M(-a,-a,a),{pos="N"},
        "$O_1$",O1,{},
        "$A$",M(a,-a,-a),{pos="SW"},
        "$B$",M(a,a,-a),{pos="E"},
        "$C$",M(-a,a,-a),{},
        "$B_1$",M(a,a,a),{},
        "$a$",M(a,0,-a),{pos="S"},
        "$R$",M(0,-a,a)/2,{node_options="text=red"}
    )
    g:Show()
\end{luadraw}
\end{document}

又学到一个 ld.cvx_hull3d()...

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=spherical_circle}
    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 g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {0,78}
    }
    require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=8, edgecolor="black" }
    local phi1, phi2 = 75, 45
    local A,B,C = sM(90,phi1), sM(-140,phi1),sM(-30,phi1)
    local A1,B1,C1 = sM(90,phi2), sM(-140,phi2),sM(-30,phi2)
    ---------
    -- https://github.com/pfradin/luadraw/discussions/341#discussioncomment-18530158
    local P = ld.facet2poly( ld.cvx_hull3d({A,B,C,A1,B1,C1}) )
    ---------
    local O1,O2 = ld.proj3d(Origin,{A,vecK}), ld.proj3d(Origin,{A1,vecK})
    local R1,R2 = pt3d.abs(A-O1), pt3d.abs(A1-O2)
    g:DScircle({Origin, vecK});g:DScircle({A, vecK})
    g:DScircle({A1, vecK})
    g:Dcircle3d(O1,R1,vecK,"draw=none, fill=cyan!30, fill opacity=0.4")
    g:Dpoly(P,{color="violet!30!white",mode=ld.mFlatHidden, edgewidth=8})
    g:Dcircle3d(O2,R2,vecK,"draw=none, fill=cyan!30, fill opacity=0.4")
    g:Dspherical()
    g:Dpolyline3d({A,Origin,A1},"dashed,red,thick")
    g:Dpolyline3d({{Origin,O2,A1},{O1,A}},"dashed,blue!75")
    g:Dballdots3d({Origin,O1,O2,A,A1},"red",0.75)
    g:Dlabel3d(
        "$O$",Origin,{pos="S"},
        "$O_1$",O1,{pos="W"},
        "$O_2$",O2,{},
        "$A$",A,{pos="E"},
        "$A_1$",A1,{},
        "$R$",A/2,{pos="S",node_options="text=red"},
        "$R$",2*A1/3,{pos="W"},
        "$h$",(O1+O2)/2,{pos="E",node_options="text=blue"},
        "$r_1$",(O1+A)/2,{pos=""},
        "$r_2$",0.2*A1+0.8*O2,{}
    )
    g:Show()
\end{luadraw}
\end{document}

学到一些新东西...

查看原图，这里定位的方式最佳的方式应该不是直接通过极坐标给定A,B,C,D四个点的坐标(实际上也很难确定)，注意到其实这里的B,C,D应该是截面圆和视界圆的交点，所以我使用了out这个小特性：

   local A = sM(-140,120)
    local Out1, Out2 = {}, {}
    g:DScircle({A,vecK},{out=Out1});
    local B,C = table.unpack(ld.reverse(Out1))
    local M = (A+B)/2
    local vecN = pt3d.prod(B-A,vecK)
    g:DScircle({M,vecN},{out=Out2});
    local _,D = table.unpack(Out2)

以下是我的完整代码尝试：

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=perpendicular_circle}
    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 g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {0,70}
    }
    require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=8, edgecolor="black" }
    g:DScircle({Origin, vecK})
    local A = sM(-140,120)
    local Out1, Out2 = {}, {}
    g:DScircle({A,vecK},{out=Out1});
    local B,C = table.unpack(ld.reverse(Out1))
    local M = (A+B)/2
    local vecN = pt3d.prod(B-A,vecK)
    g:DScircle({M,vecN},{out=Out2});
    local _,D = table.unpack(Out2)
    local O1,O2 = ld.proj3d(Origin,{M,vecK}),ld.proj3d(Origin,{M,vecN})
    local r1,r2 = pt3d.abs(C-O1), pt3d.abs(D-O2)
    g:Dcircle3d(O1,r1,vecK,"draw=none, fill=violet!50, fill opacity=0.4")
    g:Dcircle3d(O2,r2,vecN,"draw=none, fill=violet!50, fill opacity=0.4")
    g:Dspherical()
    g:Dpolyline3d({{A,B,C},{A,B,D},{C,D}},true,"dashed,red,thick")
    g:Dpolyline3d({{Origin,O1,M,O2},{A,Origin},{A,O1},{A,O2}},true,"dashed,blue!60,thick")
    g:Dballdots3d({Origin,O1,O2,A,B,C,D,M},"red")
    g:Dangle3d(Origin,O2,M,0.2,"thick");g:Dangle3d(Origin,O1,M,0.2,"thick");g:Dangle3d(O1,M,O2,0.2,"thick")
    g:Dlabel3d(
        "$O$",Origin,{pos="N"},
        "$O_2$",O2,{},
        "$D$",D,{},
        "$O_1$",O1,{pos="S"},
        "$M$",M,{pos="S"},
        "$A$",A,{pos="NE"},
        "$B$",B,{pos="SW"},
        "$C$",C,{pos="E"},
        "$\\ell$",M,{pos="W",node_options="text=red"},
        "$R$",A/2,{pos="",node_options="text=blue"},
        "$r_1$",(A+O1)/2,{},
        "$r_2$",(A+O2)/2,{}
    )
    g:Show()
\end{luadraw}
\end{document}

同样的，一些地方应该用g:DSpolyline而不是g:Dployline3d，但是不想改，我认为这是「教学性3D」，不用强制追求「真实3D」(哼)...

没有完全复刻，具体角度和位置，以及一些线条之间颜色遮挡的细节并未考虑得十分完备...有缘再讨论吧;-)

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier-otf}
\begin{document}
    \begin{luadraw}{name=spherical_circle}
    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 g = ld.graph3d:new{
        window3d = {-5,5,-5,5,-5,5},
        window = {-5,5,-5,5},
        -- adjust2d = true, 
        viewdir = {0,77.5}
    }
    require 'luadraw_spherical'
    local sM = ld.sM
    ld.Hiddenlines = true;ld.Hiddenlinestyle = "dashed"
    g:Define_sphere{ color = "", edgewidth=8, edgecolor="black" }
    local A,B,M,N = sM(-10,120), sM(-115,120),sM(150,120), sM(0,0)
    g:DScircle({Origin, vecK})
    local vecN = pt3d.prod(A-N,B-N)
    local O1,O2 = ld.proj3d(Origin,{M,vecK}), ld.proj3d(Origin,{N,vecN})
    local r1,r2 = pt3d.abs(M-O1), pt3d.abs(N-O2)
    g:DScircle({M,vecK});g:DScircle({N,vecN})
    g:Dcircle3d(O1,r1,vecK,"draw=none, fill=violet!50, fill opacity=0.4")
    g:Dcircle3d(O2,r2,vecN,"draw=none, fill=violet!50, fill opacity=0.4")
    g:Dspherical()
    g:Dpolyline3d({{A,B,M},{A,B,N}},true,"dashed,blue!75,thick")
    local D = (A+B)/2
    g:Dpolyline3d({Origin,O1,D,O2},true,"dashed,red,thick")
    g:Dpolyline3d({B,Origin,D},"dashed,thick")
    g:Dballdots3d({Origin,O1,O2,A,B,M,N},"red",0.75)
    g:Dballdots3d({D},"blue",0.75)
    g:Dangle3d(Origin,O2,D,0.2,"thick");g:Dangle3d(Origin,O1,D,0.2,"thick");g:Darc3d(O1,D,O2,0.3,1,"red,very thick")
    g:Dlabel3d(
        "$O$",Origin,{pos="E"},
        "$O_1$",O1,{},
        "$M$",M,{pos="N"},
        "$N$",N,{},
        "$O_2$",O2,{pos="N"},
        "$A$",A,{pos="S"},
        "$D$",D,{},
        "$B$",B,{pos="W"},
        "$\\ell$",(B+D)/2,{pos="S",node_options="text=blue!75"},
        "$R$",0.8*B,{pos="N",node_options="text=red"},
        "$n$",0.2*D+0.8*O2,{pos="W"},
        "$m$",0.4*D+0.6*O1,{pos="S"},
        "$\\theta$",D,{pos="NE",dist=0.15}
    )
    g:Show()
\end{luadraw}
\end{document}