Anatomy of a Cube
A cube is the familiar three-dimensional solid with eight vertices, six faces and twelve edges. I have been working with cubes recently in posts about both the Menger sponge fractal and the 4-by-4 matrix from computer graphics.
Contents
Vertices
Cartesian coordinates, V, for the eight vertices of a cube can be generated from the binary representation of 0:7.
j = (0:7)'
j = 0 1 2 3 4 5 6 7
k = dec2bin(j)
k = 8×3 char array '000' '001' '010' '011' '100' '101' '110' '111'
V = double(k-'0')
V = 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
Edges
The twelve edges of a cube are described by the adjacency matrix A of connections between its vertices.
A = adjacency(V) spy(A)
A = 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0
Wireframe
A plot of the graph of A provides a wireframe view of our cube.
G = graph(A); p = plot(G, ... NodeLabel = string(k), ... NodeFontSize = 12, ... XData = V(:,3), ... YData = V(:,2), ... ZData = V(:,1)); axis([-1 4 -1 4 -1 4]/3) axis square off vis3d view(3)
Let's replace the node labels with 1-based indices for rows of V.
p.NodeLabel = string(j+1);
Faces
A cube has six square faces. This array F provides the indices in V of the coordinates of the corners of each face. The ordering ensures that the normal to each face points out of the cube.
F = [ 1 5 7 3 3 7 8 4 1 3 4 2 2 4 8 6 1 2 6 5 5 6 8 7 ]
F = 1 5 7 3 3 7 8 4 1 3 4 2 2 4 8 6 1 2 6 5 5 6 8 7
If you Google "rgb gold", you will get links to Web sites offering red-green-blue values for dozens of shades of the color gold. My forthcoming post about the complement of the Menger sponge fractal uses just two shades.
gold = [212 175 55]/256 dark = gold/2
gold = 0.8281 0.6836 0.2148 dark = 0.4141 0.3418 0.1074
The single patch formed from V and F is just the skin enclosing our cube; its inside is hollow.
cla patch(Faces = F, ... Vertices = V, ... FaceColor = gold, ... EdgeColor = dark, ... LineWidth = 1.5); axis([-1 4 -1 4 -1 4]/3) axis square off vis3d view(3)
- カテゴリ:
- Graphics,
- Graphs,
- Programming
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