How to draw a cube that can be inscribed within a right circular cone?checomsth.orvg2 miiliB
7
I am trying to draw a cube that can be inscribed within a right circular cone (cone of radius r and perpendicular height h). I see at here https://www.geeksforgeeks.org/largest-cube-that-can-be-inscribed-within-a-right-circular-cone/ I can find the side length of cube. But I cann't draw it. I tried
\\documentclass[border=3.14mm,12pt,tikz]{standalone}
\\usepackage{tikz-3dplot}
\\usetikzlibrary{calc,backgrounds}
\\begin{document}
%polar coordinates of visibility
\\pgfmathsetmacro\\th{65}
\\pgfmathsetmacro\\az{110}
\\tdplotsetmaincoords{\\th}{\\az}
%parameters of the cone
\\pgfmathsetmacro\\R{4} %radius of base
\\pgfmathsetmacro\\v{5} %hight of cone
\\begin{tikzpicture} [scale=1, tdplot_main_coords, axis/.style={blue,thick}]
\\path
coordinate (O) at (0,0,0)
coordinate (A) at (\\R,0,0)
coordinate (B) at (0,\\R,0)
coordinate (C) at ($ 2*(O) - (A) $)
coordinate (D) at ($ 2*(O) - (B) $)
coordinate (S) at (0,0,\\v)
;
\\fill (S) circle[radius=1pt] node[above] {$S$};
\\fill (A) circle[radius=1pt] node[below] {$A$};
\\fill (B) circle[radius=1pt] node[below] {$B$};
\\fill (C) circle[radius=1pt] node[above] {$C$};
\\fill (D) circle[radius=1pt] node[above] {$D$};
\\fill (O) circle[radius=1pt] node[below] {$O$};
\\draw[thick] (A) -- (S) (S) -- (B);
\\draw[dashed] (A) -- (B) -- (C) -- (D) -- cycle (S) -- (C) (A) -- (C) (B) -- (D) (S) -- (O);
\\pgfmathsetmacro\\cott{{cot(\\th)}}
\\pgfmathsetmacro\\fraction{\\R*\\cott/\\v}
\\pgfmathsetmacro\\fraction{\\fraction<1 ? \\fraction : 1}
\\pgfmathsetmacro\\angle{{acos(\\fraction)}}
% % angles for transformed lines
\\pgfmathsetmacro\\PhiOne{180+(\\az-90)+\\angle}
\\pgfmathsetmacro\\PhiTwo{180+(\\az-90)-\\angle}
% % coordinates for transformed surface lines
\\pgfmathsetmacro\\sinPhiOne{{sin(\\PhiOne)}}
\\pgfmathsetmacro\\cosPhiOne{{cos(\\PhiOne)}}
\\pgfmathsetmacro\\sinPhiTwo{{sin(\\PhiTwo)}}
\\pgfmathsetmacro\\cosPhiTwo{{cos(\\PhiTwo)}}
% % angles for original surface lines
\\pgfmathsetmacro\\sinazp{{sin(\\az-90)}}
\\pgfmathsetmacro\\cosazp{{cos(\\az-90)}}
\\pgfmathsetmacro\\sinazm{{sin(90-\\az)}}
\\pgfmathsetmacro\\cosazm{{cos(90-\\az)}}
% % draw basis circle
\\tdplotdrawarc[tdplot_main_coords,thick]{(O)}{\\R}{\\PhiOne}{360+\\PhiTwo}{anchor=north}{}
\\tdplotdrawarc[tdplot_main_coords,dashed]{(O)}{\\R}{\\PhiTwo}{\\PhiOne}{anchor=north}{}
% % displaying tranformed surface of the cone (rotated)
\\draw[thick] (0,0,\\v) -- (\\R*\\cosPhiOne,\\R*\\sinPhiOne,0);
\\draw[thick] (0,0,\\v) -- (\\R*\\cosPhiTwo,\\R*\\sinPhiTwo,0);
\\end{tikzpicture}
\\end{document}
tikz-3dplot
asked 12 hours ago
Thuy NguyenThuy Nguyen
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2 Answers
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7
Assuming the formula from your link is correct, this is a possible cube. The figure seems to be confirming the formula.
\\documentclass[border=3.14mm,12pt,tikz]{standalone}
\\usepackage{tikz-3dplot}
\\begin{document}
%polar coordinates of visibility
\\pgfmathsetmacro\\th{65}
\\pgfmathsetmacro\\az{110}
\\tdplotsetmaincoords{\\th}{\\az}
%parameters of the cone
\\begin{tikzpicture}[scale=1, tdplot_main_coords, axis/.style={blue,thick}]
\\pgfmathsetmacro\\R{4} %radius of base
\\pgfmathsetmacro\\v{5} %hight of cone
\\path (0,0,0) coordinate (O)
(\\R,0,0) coordinate (A)
(0,\\R,0) coordinate (B)
(-\\R,0,0) coordinate (C)
(0,-\\R,0) coordinate (D)
(0,0,\\v) coordinate (S);
\\foreach \\point/\\pos in {S/above,A/below,B/below,C/above,D/above,O/below}
{\\fill (\\point) circle[radius=1pt] node[\\pos] {$\\point$};}
\\draw[thick] (A) -- (S) (S) -- (B);
\\draw[dashed] (A) -- (B) -- (C) -- (D) -- cycle (S) -- (C) (A) -- (C) (B) -- (D) (S) -- (O);
\\pgfmathsetmacro\\cott{{cot(\\th)}}
\\pgfmathsetmacro\\fraction{\\R*\\cott/\\v}
\\pgfmathsetmacro\\fraction{\\fraction<1 ? \\fraction : 1}
\\pgfmathsetmacro\\angle{{acos(\\fraction)}}
% % angles for transformed lines
\\pgfmathsetmacro\\PhiOne{180+(\\az-90)+\\angle}
\\pgfmathsetmacro\\PhiTwo{180+(\\az-90)-\\angle}
% % coordinates for transformed surface lines
\\pgfmathsetmacro\\sinPhiOne{{sin(\\PhiOne)}}
\\pgfmathsetmacro\\cosPhiOne{{cos(\\PhiOne)}}
\\pgfmathsetmacro\\sinPhiTwo{{sin(\\PhiTwo)}}
\\pgfmathsetmacro\\cosPhiTwo{{cos(\\PhiTwo)}}
\\tdplotdrawarc[tdplot_main_coords,dashed]{(O)}{\\R}{\\PhiTwo}{\\PhiOne}{anchor=north}{}
\\pgfmathsetmacro{\\a}{\\v*\\R*sqrt(2)/(\\v+sqrt(2)*\\R)} % edge
\\pgfmathsetmacro{\\mya}{sqrt(1/2)*\\a} % half diagonal
\\pgfmathsetmacro{\\myalpha}{90}
\\begin{scope}[canvas is xy plane at z=0]
\\path foreach \\X in {1,...,4}{ (\\myalpha+90*\\X:\\mya) coordinate (P\\X)};
\\end{scope}
\\begin{scope}[canvas is xy plane at z=\\a]
\\path foreach \\X in {1,...,4}{ (\\myalpha+90*\\X:\\mya) coordinate (Q\\X)};
\\end{scope}
\\draw[dashed] (P4) -- (P1) -- (P2) (P1) -- (Q1);
\\draw (P4) -- (P3) -- (P2) -- (Q2) -- (Q3) -- (Q4) -- (Q1) -- (Q2)
(Q4) -- (P4) (P4) (P3) -- (Q3);
% % draw basis circle
\\tdplotdrawarc[tdplot_main_coords,thick]{(O)}{\\R}{\\PhiOne}{360+\\PhiTwo}{anchor=north}{}
% % displaying transformed surface of the cone (rotated)
\\draw[thick] (0,0,\\v) -- (\\R*\\cosPhiOne,\\R*\\sinPhiOne,0);
\\draw[thick] (0,0,\\v) -- (\\R*\\cosPhiTwo,\\R*\\sinPhiTwo,0);
\\end{tikzpicture}
\\end{document}
If you rotate the cube, it still fits.
\\documentclass[border=3.14mm,12pt,tikz]{standalone}
\\usepackage{tikz-3dplot}
\\begin{document}
%polar coordinates of visibility
\\pgfmathsetmacro\\th{65}
\\pgfmathsetmacro\\az{110}
\\tdplotsetmaincoords{\\th}{\\az}
\\foreach \\Angle in {0,2,...,88}
{\\begin{tikzpicture}[scale=1, tdplot_main_coords, axis/.style={blue,thick}]
\\pgfmathsetmacro\\R{4} %radius of base
\\pgfmathsetmacro\\v{5} %hight of cone
\\path (0,0,0) coordinate (O)
(\\R,0,0) coordinate (A)
(0,\\R,0) coordinate (B)
(-\\R,0,0) coordinate (C)
(0,-\\R,0) coordinate (D)
(0,0,\\v) coordinate (S);
\\foreach \\point/\\pos in {S/above,A/below,B/below,C/above,D/above,O/below}
{\\fill (\\point) circle[radius=1pt] node[\\pos] {$\\point$};}
\\draw[thick] (A) -- (S) (S) -- (B);
\\draw[dashed] (A) -- (B) -- (C) -- (D) -- cycle (S) -- (C) (A) -- (C) (B) -- (D) (S) -- (O);
\\pgfmathsetmacro\\cott{{cot(\\th)}}
\\pgfmathsetmacro\\fraction{\\R*\\cott/\\v}
\\pgfmathsetmacro\\fraction{\\fraction<1 ? \\fraction : 1}
\\pgfmathsetmacro\\angle{{acos(\\fraction)}}
% % angles for transformed lines
\\pgfmathsetmacro\\PhiOne{180+(\\az-90)+\\angle}
\\pgfmathsetmacro\\PhiTwo{180+(\\az-90)-\\angle}
% % coordinates for transformed surface lines
\\pgfmathsetmacro\\sinPhiOne{{sin(\\PhiOne)}}
\\pgfmathsetmacro\\cosPhiOne{{cos(\\PhiOne)}}
\\pgfmathsetmacro\\sinPhiTwo{{sin(\\PhiTwo)}}
\\pgfmathsetmacro\\cosPhiTwo{{cos(\\PhiTwo)}}
\\tdplotdrawarc[tdplot_main_coords,dashed]{(O)}{\\R}{\\PhiTwo}{\\PhiOne}{anchor=north}{}
\\pgfmathsetmacro{\\a}{\\v*\\R*sqrt(2)/(\\v+sqrt(2)*\\R)} % edge
\\pgfmathsetmacro{\\mya}{sqrt(1/2)*\\a} % half diagonal
\\pgfmathsetmacro{\\myalpha}{\\Angle}
\\begin{scope}[canvas is xy plane at z=0]
\\path foreach \\X in {1,...,4}{ (\\myalpha+90*\\X:\\mya) coordinate (P\\X)};
\\end{scope}
\\begin{scope}[canvas is xy plane at z=\\a]
\\path foreach \\X in {1,...,4}{ (\\myalpha+90*\\X:\\mya) coordinate (Q\\X)};
\\end{scope}
\\draw (P4) -- (P3) -- (P2) -- (Q2) -- (Q3) -- (Q4) -- (Q1) -- (Q2)
(Q4) -- (P4) (P4) (P3) -- (Q3) (P4) -- (P1) -- (P2) (P1) -- (Q1);
% % draw basis circle
\\tdplotdrawarc[tdplot_main_coords,thick]{(O)}{\\R}{\\PhiOne}{360+\\PhiTwo}{anchor=north}{}
% % displaying transformed surface of the cone (rotated)
\\draw[thick] (0,0,\\v) -- (\\R*\\cosPhiOne,\\R*\\sinPhiOne,0);
\\draw[thick] (0,0,\\v) -- (\\R*\\cosPhiTwo,\\R*\\sinPhiTwo,0);
\\end{tikzpicture}}
\\end{document}
answered 12 hours ago
Schrödinger's catSchrödinger's cat
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Are style of all sides of cube dashed? – minhthien_2016 11 hours ago
-
@minhthien_2016 One could of course dash them. It depends on your conventions. – Schrödinger's cat 10 hours ago
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5
I also use the formula from your link.
\\documentclass[border=2mm,12pt,tikz]{standalone}
\\usepackage{fouriernc}
\\usepackage{tikz-3dplot}
\\usetikzlibrary{calc,backgrounds}
\\begin{document}
%polar coordinates of visibility
\\pgfmathsetmacro\\th{65}
\\pgfmathsetmacro\\az{110}
\\tdplotsetmaincoords{\\th}{\\az}
%parameters of the cone
\\pgfmathsetmacro\\R{4} %radius of base
\\pgfmathsetmacro\\v{5} %hight of cone
\\pgfmathsetmacro\\a{{(\\v * \\R * sqrt(2)) / (\\v + sqrt(2) * \\R)}}
\\pgfmathsetmacro\\b{{1/2*\\a*sqrt(2)}}
\\begin{tikzpicture} [scale=1, tdplot_main_coords, axis/.style={blue,thick}]
\\begin{scope}[canvas is xy plane at z=0]
\\path
coordinate (O) at (0,0)
coordinate (A) at (\\R,0)
coordinate (B) at (0,\\R)
coordinate (C) at (-\\R,0)
coordinate (D) at (0,-\\R,0)
coordinate (M) at (\\b,0)
coordinate (N) at (0,\\b)
coordinate (P) at (-\\b,0)
coordinate (Q) at (0,-\\b);
\\end{scope}
\\begin{scope}[canvas is xy plane at z=\\a]
\\path coordinate (M') at (\\b,0)
coordinate (N') at (0,\\b)
coordinate (P') at (-\\b,0)
coordinate (Q') at (0,-\\b)
;
\\end{scope}
\\coordinate (S) at (0,0,\\v);
\\foreach \\v/\\position in {O/below,A/below, B/right, C/right, D/left, S/above} {\\draw[draw =black, fill=black] (\\v) circle (1.5pt) node [\\position=0.2mm] {$\\v$};
}
\\foreach \\X in {A,B} \\draw[thick] (\\X) -- (S);
\\foreach \\X in {M,N,P,Q} \\draw[dashed] (\\X) -- (\\X');
\\draw[dashed] (A) -- (B) -- (C) -- (D) -- cycle
(M) -- (N) -- (P) -- (Q) -- cycle
(M') -- (N') -- (P') -- (Q') -- cycle
(A) -- (C) (B) -- (D)
(S)--(O) (S)-- (C);
\\foreach \\Y in {M, N, P, Q, M', N', P', Q'}
\\fill (\\Y) circle[radius=1.2pt];
\\begin{scope}[canvas is xy plane at z=0]
\\draw[dashed, blue] (O) circle[radius=\\b];
\\end{scope}
\\pgfmathsetmacro\\cott{{cot(\\th)}}
\\pgfmathsetmacro\\fraction{\\R*\\cott/\\v}
\\pgfmathsetmacro\\fraction{\\fraction<1 ? \\fraction : 1}
\\pgfmathsetmacro\\angle{{acos(\\fraction)}}
% % angles for transformed lines
\\pgfmathsetmacro\\PhiOne{180+(\\az-90)+\\angle}
\\pgfmathsetmacro\\PhiTwo{180+(\\az-90)-\\angle}
% % coordinates for transformed surface lines
\\pgfmathsetmacro\\sinPhiOne{{sin(\\PhiOne)}}
\\pgfmathsetmacro\\cosPhiOne{{cos(\\PhiOne)}}
\\pgfmathsetmacro\\sinPhiTwo{{sin(\\PhiTwo)}}
\\pgfmathsetmacro\\cosPhiTwo{{cos(\\PhiTwo)}}
% % angles for original surface lines
\\pgfmathsetmacro\\sinazp{{sin(\\az-90)}}
\\pgfmathsetmacro\\cosazp{{cos(\\az-90)}}
\\pgfmathsetmacro\\sinazm{{sin(90-\\az)}}
\\pgfmathsetmacro\\cosazm{{cos(90-\\az)}}
% % draw basis circle
\\tdplotdrawarc[tdplot_main_coords,thick]{(O)}{\\R}{\\PhiOne}{360+\\PhiTwo}{anchor=north}{}
\\tdplotdrawarc[tdplot_main_coords,dashed]{(O)}{\\R}{\\PhiTwo}{\\PhiOne}{anchor=north}{}
% % displaying tranformed surface of the cone (rotated)
\\draw[thick] (0,0,\\v) -- (\\R*\\cosPhiOne,\\R*\\sinPhiOne,0);
\\draw[thick] (0,0,\\v) -- (\\R*\\cosPhiTwo,\\R*\\sinPhiTwo,0);
\\end{tikzpicture}
\\end{document}
answered 10 hours ago
minhthien_2016minhthien_2016
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