Equation of a Sphere


\begin{tikzpicture} % MERC

%% some definitions
\def\R{3} % sphere radius
\def\angEl{25} % elevation angle
\def\angAz{-100} % azimuth angle
\def\angPhiOne{-50} % longitude of point P
\def\angPhiTwo{-35} % longitude of point Q
\def\angBeta{33} % latitude of point P and Q

%% working planes
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhiOne}
\LongitudePlane[qzplane]{\angEl}{\angPhiTwo}
\LatitudePlane[equator]{\angEl}{0}

%% draw background sphere
\fill[ball color=grey!10] (0,0) circle (\R); % 3D lighting effect

%% characteristic points
\path[pzplane] (\angBeta:\R) coordinate [label={[black]above left:$P$}] (P);
\drawpoint{P}{.5mm}{black};
\coordinate [label={[black]above left:$O$}] (O) at (0,0);
\drawpoint{O}{.5mm}{black};

\path[pzplane] (\R,0) coordinate (PE);
\path[xzplane] (\R,0) coordinate (XE);
\path[pzplane] (\R,0) coordinate [label={[black]above right:$R$}] (R);
\drawpoint{R}{.5mm}{black};
\draw[dotted,black] (O) -- (R);
\drawbrace{O}{R}{2mm}{black}{$r$}{0}{-4mm}{mirror};

%% meridians and latitude circles
\DrawLatitudeCircle[\R]{0} % equator

%% draw lines and put labels
%\draw (-\R,-\H) -- (-\R,2*\R) (\R,-\H) -- (\R,2*\R);
\draw[->] (O) -- (P);

% z axis
\draw[->,dashed,blue] (O) -- +(0,1.5*\R) node[above] {$z$};
% x axis
\path[xzplane] (0:\R) node[below,red] {$x$} coordinate (x);
\draw[->,dashed,red] (O) -- (x);
% y axis
\path[pyplane] (0:\R) node[left,green!50] {$y$} coordinate (y);
\draw[->,dashed,green!50] (O) -- (y);

% projection P
\path[pzplane] (0.8*\R,0) coordinate [label={[black]above right:$Q$}] (Q);
\drawpoint{Q}{.5mm}{black};
\draw[->,dotted] (P) -- (Q);

%angles
\draw[pzplane,->,thin] (0:0.5*\R) to[bend right=15]
    node[midway,right] {$\alpha$} (\angBeta:0.5*\R);
\draw[equator,->,thin] (\angAz:0.5*\R) to[bend right=30]
    node[pos=0.4,below] {$\beta$} (\angPhiOne:0.5*\R);

\end{tikzpicture}

\begin{eqnarray*}
x &=& r * \cos{\alpha} * \cos{\beta} & \\
y &=& r * \cos{\alpha} * \sin{\beta} & \\
z &=& r * \sin{\alpha}
\end{eqnarray*}

Volume of a Sphere


\begin{tikzpicture}
%% some definitions
\def\R{3} % sphere radius
\def\angEl{25} % elevation angle
\def\angAz{-100} % azimuth angle
\def\angPhiOne{-50} % longitude of point P
\def\angPhiTwo{-35} % longitude of point Q
\def\angBeta{33} % latitude of point P and Q

%% working planes
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhiOne}
\LongitudePlane[qzplane]{\angEl}{\angPhiTwo}
\LatitudePlane[equator]{\angEl}{0}

%% draw background sphere
\fill[ball color=grey!10] (0,0) circle (\R); % 3D lighting effect

% circles
\foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} }
%\foreach \t in {-5,-35,...,-175} { \DrawLongitudeCircle[\R]{\t} }

% z axis
\draw[->,dashed,blue] (O) -- +(0,\R) node[above] {$z$} coordinate (z);
% x axis
\path[xzplane] (0:\R) node[below,red] {$x$} coordinate (x);
\draw[->,dashed,red] (O) -- (x);
% y axis
\path[pyplane] (0:\R) node[left,green!50] {$y$} coordinate (y);
\draw[->,dashed,green!50] (O) -- (y);

% Ox
%\drawbrace{O}{x}{2mm}{black}{$r$}{4mm}{0}{};
% Oy
%\drawbrace{O}{y}{2mm}{black}{$r$}{0}{4mm}{};
% Oz
%\drawbrace{O}{z}{2mm}{black}{$r$}{-4mm}{0}{};

% right-angle
%\node[square,minimum size=1mm, dotted] at (0.1,0.1) [draw,fill] (O) [magenta] {};
%\coordinate[label={[magenta]above right:$90^{\circ}$}] (90) at (0.2, 0.2);

\coordinate [label={[black]left:$O$}] (O) at (0,0);
\drawpoint{O}{.5mm}{black};

\path[pzplane] (\angBeta:\R) coordinate [label={[black]right:$P$}] (P);
\drawpoint{P}{.5mm}{black};

\draw[->,black] (O) -- (P);
\drawbrace{O}{P}{2mm}{black}{$r$}{0}{-4mm}{mirror};

\coordinate [label={[black]left:$Q$}] (Q) at (0,0.5*\R);
\drawpoint{Q}{.5mm}{black};

\draw[dotted] (P) -- (Q);

\end{tikzpicture}

\begin{eqnarray*}
V &=& \frac{4}{3}\pi*r^{3}
\end{eqnarray*}

Area of a Sphere


\begin{tikzpicture}
%% some definitions
\def\R{3} % sphere radius
\def\angEl{25} % elevation angle
\def\angAz{-100} % azimuth angle
\def\angPhiOne{-50} % longitude of point P
\def\angPhiTwo{-35} % longitude of point Q
\def\angBeta{33} % latitude of point P and Q

%% working planes
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhiOne}
\LongitudePlane[qzplane]{\angEl}{\angPhiTwo}
\LatitudePlane[equator]{\angEl}{0}

%% draw background sphere
\fill[ball color=gray!10] (0,0) circle (\R); % 3D lighting effect
\fill[ball color=gray!75] (0,0) circle (0.75*\R); % 3D lighting effect

% circles
%\foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} }

% z axis
\draw[->,dashed,blue] (O) -- +(0,\R) node[above] {$z$} coordinate (z);
% x axis
\path[xzplane] (0:\R) node[below,red] {$x$} coordinate (x);
\draw[->,dashed,red] (O) -- (x);
% y axis
\path[pyplane] (0:\R) node[left,green!50] {$y$} coordinate (y);
\draw[->,dashed,green!50] (O) -- (y);

% Ox
%\drawbrace{O}{x}{2mm}{black}{$r$}{4mm}{0}{};
% Oy
%\drawbrace{O}{y}{2mm}{black}{$r$}{0}{4mm}{};
% Oz
%\drawbrace{O}{z}{2mm}{black}{$r$}{-4mm}{0}{};

% right-angle
%\node[square,minimum size=1mm, dotted] at (0.1,0.1) [draw,fill] (O) [magenta] {};
%\coordinate[label={[magenta]above right:$90^{\circ}$}] (90) at (0.2, 0.2);

\coordinate [label={[black]left:$O$}] (O) at (0,0);
\drawpoint{O}{.5mm}{black};

\path[pzplane] (\angBeta:\R) coordinate [label={[black]right:$P$}] (P);
\drawpoint{P}{.5mm}{black};

\draw[->,black] (O) -- (P);
\drawbrace{O}{P}{2mm}{black}{$r$}{0}{-4mm}{mirror};

\coordinate [label={[black]left:$Q$}] (Q) at (0,0.5*\R);
\drawpoint{Q}{.5mm}{black};

\draw[dotted] (P) -- (Q);

\end{tikzpicture}

\begin{eqnarray*}
A &=& 4\pi r^{2}
\end{eqnarray*}


fuss/mathematics/geometry/bodies/spheres.txt ยท Last modified: 2017/02/22 18:30 (external edit)

Access website using Tor Access website using i2p


For the copyright, license, warranty and privacy terms for the usage of this website please see the license, privacy and plagiarism pages.