MAT223 Lecture 02
\begin{document}
\begin{tikzpicture}[x={(0.8cm,-0.2cm)}, y={(0.5cm,0.4cm)}, z={(0cm,1cm)}, scale=0.9]
% Case 1: Two planes intersecting in a line
\begin{scope}[shift={(0,0,0)}]
\node at (0,0,2.5) {\textbf{Two Planes: Line Intersection}};
% Horizontal Plane
\filldraw[color=blue, fill opacity=0.3] (-1.5,-2,0) -- (1.5,-2,0) -- (1.5,2,0) -- (-1.5,2,0) -- cycle;
% Vertical Plane
\filldraw[color=red, fill opacity=0.3] (0,-2,-1.2) -- (0,2,-1.2) -- (0,2,1.2) -- (0,-2,1.2) -- cycle;
% Intersection Line
\draw[ultra thick, color=black] (0,-2,0) -- (0,2,0);
\end{scope}
% Case 2: Two parallel planes (no intersection)
\begin{scope}[shift={(6,0,0)}]
\node at (0,0,2.5) {\textbf{Two Parallel Planes}};
% Bottom Plane
\filldraw[color=blue, fill opacity=0.3] (-1.5,-1.5,0) -- (1.5,-1.5,0) -- (1.5,1.5,0) -- (-1.5,1.5,0) -- cycle;
% Top Plane
\filldraw[color=red, fill opacity=0.3] (-1.5,-1.5,1.2) -- (1.5,-1.5,1.2) -- (1.5,1.5,1.2) -- (-1.5,1.5,1.2) -- cycle;
\end{scope}
% Case 3: Three planes intersecting in a common line
\begin{scope}[shift={(0,-6,0)}]
\node at (0,0,2.5) {\textbf{Three Planes: Common Line}};
% Plane 1 (0 degrees)
\filldraw[color=blue, fill opacity=0.3] (0,-1.5,-1.2) -- (1.5,-1.5,-1.2) -- (1.5,1.5,-1.2) -- (0,1.5,-1.2) -- (0,1.5,1.2) -- (1.5,1.5,1.2) -- (1.5,-1.5,1.2) -- (0,-1.5,1.2) -- cycle;
% Plane 2 (120 degrees)
\filldraw[color=red, fill opacity=0.3] (0,-1.5,-1.2) -- (-0.75, 1.3, -1.5) -- (-0.75, 1.3, 1.5) -- (0,-1.5,1.2) -- cycle;
% Plane 3 (240 degrees)
\filldraw[color=green, fill opacity=0.3] (0,-1.5,-1.2) -- (-0.75, -1.3, -1.5) -- (-0.75, -1.3, 1.5) -- (0,-1.5,1.2) -- cycle;
% Common Line
\draw[ultra thick, color=black] (0,-1.5,-1.2) -- (0,1.5,1.2);
% Note: The coordinates are adjusted to show the "spine" clearly
\draw[ultra thick, color=black] (0,-1.5,-1.2) -- (0,1.5,1.2);
\end{scope}
% Case 4: Three planes intersecting in a point
\begin{scope}[shift={(6,-6,0)}]
\node at (0,0,2.5) {\textbf{Three Planes: Common Point}};
% XY Plane
\filldraw[color=blue, fill opacity=0.3] (0,0,0) -- (1.8,0,0) -- (1.8,1.8,0) -- (0,1.8,0) -- cycle;
% YZ Plane
\filldraw[color=red, fill opacity=0.3] (0,0,0) -- (0,1.8,0) -- (0,1.8,1.8) -- (0,0,1.8) -- cycle;
% XZ Plane
\filldraw[color=green, fill opacity=0.3] (0,0,0) -- (1.8,0,0) -- (1.8,0,1.8) -- (0,0,1.8) -- cycle;
% Intersection Point
\fill[color=black] (0,0,0) circle (2.5pt);
\end{scope}
\end{tikzpicture}
\end{document}
\begin{document}
\begin{tikzpicture}[x={(-0.5cm,-0.5cm)}, y={(1cm,0cm)}, z={(0cm,1cm)}, scale=1.2]
% Define coordinates for a rectangular prism
\coordinate (A) at (0,0,0);
\coordinate (B) at (0,4,0);
\coordinate (C) at (2,4,0);
\coordinate (D) at (2,0,0);
\coordinate (E) at (0,0,3);
\coordinate (F) at (0,4,3);
\coordinate (G) at (2,4,3);
\coordinate (H) at (2,0,3);
% Draw the frame of the prism with dashed lines for 3D context
\draw[dashed, gray] (A) -- (B) -- (C) -- (D) -- cycle;
\draw[dashed, gray] (E) -- (F) -- (G) -- (H) -- cycle;
\draw[dashed, gray] (A) -- (E);
\draw[dashed, gray] (B) -- (F);
\draw[dashed, gray] (C) -- (G);
\draw[dashed, gray] (D) -- (H);
% Draw Line 1 (L1) along the front bottom edge
\draw[very thick, color=blue] (-1,4,0) -- (3,4,0) node[right] {$L_1$};
% Draw Line 2 (L2) along the back-left vertical edge
\draw[very thick, color=red] (0,0,-1) -- (0,0,4) node[above] {$L_2$};
% Add a small note about the skew relationship
\node[anchor=north, align=center, font=\small] at (1,2,-1.5) {Lines $L_1$ and $L_2$ are skew;\\no single plane contains both.};
\end{tikzpicture}
\end{document}
mat223_1_1.pdf