How can you design a polygonal mesh to represent a rectangular solid using the traditional rectangular coordinate system? Provide a sketch to illustrate your solution.
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To design a polygonal mesh representing a rectangular solid, follow these steps:
Step 1/4:
Start by determining the dimensions of the rectangular solid, denoting the length as L, the width as W, and the height as H.
Step 2/4:
Next, define the vertices of the rectangular solid using the rectangular coordinate system. There will be a total of 8 vertices:
• Vertex 1: (0, 0, 0)
• Vertex 2: (L, 0, 0)
• Vertex 3: (L, W, 0)
• Vertex 4: (0, W, 0)
• Vertex 5: (0, 0, H)
• Vertex 6: (L, 0, H)
• Vertex 7: (L, W, H)
• Vertex 8: (0, W, H)
Step 3/4:
Connect the vertices to form the edges of the rectangular solid. There will be a total of 12 edges:
• Edge 1: Vertex 1 to Vertex 2
• Edge 2: Vertex 2 to Vertex 3
• Edge 3: Vertex 3 to Vertex 4
• Edge 4: Vertex 4 to Vertex 1
• Edge 5: Vertex 5 to Vertex 6
• Edge 6: Vertex 6 to Vertex 7
• Edge 7: Vertex 7 to Vertex 8
• Edge 8: Vertex 8 to Vertex 5
• Edge 9: Vertex 1 to Vertex 5
• Edge 10: Vertex 2 to Vertex 6
• Edge 11: Vertex 3 to Vertex 7
• Edge 12: Vertex 4 to Vertex 8
Step 4/4:
Finally, connect the edges to form the faces of the rectangular solid. There will be a total of 6 faces:
• Face 1: Bottom face (formed by edges 1, 2, 3, and 4)
• Face 2: Top face (formed by edges 5, 6, 7, and 8)
• Face 3: Front face (formed by edges 1, 10, 11, and 9)
• Face 4: Right face (formed by edges 2, 10, 12, and 3)
• Face 5: Back face (formed by edges 3, 11, 12, and 4)
• Face 6: Left face (formed by edges 4, 9, 8, and 7)
The provided sketch visually represents the solution, showcasing the rectangular solid with labeled vertices, edges, and faces.
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