What are the least number of lines required to generate a 3D form such that it creates shadows similar to the figure shown below?
Show Hint
For shadow projection problems, think about how lines in 3D project onto 2D planes. A line parallel to the direction of light projects to a point, while a line perpendicular to it projects with its true length. Use a simple coordinate system (like the corners of a cube) to map out the vertices and lines.
Step 1: Understanding the Concept:
The problem asks for the minimum number of straight line segments required to construct a 3D wireframe object that, when projected onto three mutually perpendicular planes (e.g., two walls and a floor), creates a 'Z' shaped shadow on each plane.
Step 2: Detailed Explanation:
Let's define a coordinate system with the origin in the corner where the walls and floor meet. The floor is the XY plane, the left wall is the YZ plane, and the back wall is the XZ plane. A shadow on a plane is the projection of the 3D object onto that plane.
We need to construct a shape with the minimum number of lines that satisfies these three projections. A single 'Z' requires at least 3 lines. It turns out this can be achieved with 5 lines by cleverly arranging two L-shaped segments and a connecting diagonal line.
Consider the following six vertices in 3D space:
V1 = (0, 1, 1)
V2 = (0, 0, 1)
V3 = (1, 0, 1)
V4 = (1, 0, 0)
V5 = (1, 1, 0)
V6 = (0, 1, 0)
Construct a 3D form using these 5 lines:
Line 1: V1 to V2
Line 2: V2 to V3
Line 3: V4 to V5
Line 4: V5 to V6
Line 5: V2 to V5 (This is the diagonal connecting line)
Now, let's check the projections (shadows):
Shadow on XY plane (view from above, projecting Z=0): The vertices project to (0,1), (0,0), (1,0), (1,0), (1,1), (0,1). The lines project to form a 'Z' shape with the top bar from (0,1) to (1,1), the diagonal from (0,0) to (1,1), and the bottom bar from (0,0) to (1,0).
Shadow on XZ plane (view from front, projecting Y=0): The vertices project to (0,1), (0,1), (1,1), (1,0), (1,0), (0,0). The lines project to form a 'Z' shape with the top bar from (0,1) to (1,1), the diagonal from (0,1) to (1,0), and the bottom bar from (1,0) to (0,0).
Shadow on YZ plane (view from side, projecting X=0): The vertices project to (1,1), (0,1), (0,1), (0,0), (1,0), (1,0). The lines project to form a 'Z' shape with the top bar from (1,1) to (0,1), the diagonal from (0,1) to (1,0), and the bottom bar from (1,0) to (0,0).
Since a 3D form with 5 lines can produce the three required 'Z' shadows, and it's not possible with fewer lines, this is the minimum number.
Step 3: Final Answer:
The least number of lines required is 5.
