Shown on the left are the front and top views of a 3D object. Which option(s) can be the side view?
Show Hint
In orthographic projection problems, remember that multiple 3D objects can share some of the same 2D views. Your task is to check for consistency, not to find a unique object. If you can imagine a valid 3D shape that produces the required set of views, then the option is possible.
Step 1: Understanding the Concept:
This is a problem of orthographic projection, where a 3D object is represented by its 2D views (top, front, side). The given top and front views act as constraints on the possible shape of the 3D object. We need to determine which of the given side views are consistent with these constraints. It is possible that multiple different 3D objects can produce the same top and front views but have different side views.
Step 2: Key Formula or Approach:
An object is consistent with a set of views if for every point on the object, its projections match the views. We can define an object as a set of points (x, y, z) in 3D space.
The Top View is the projection onto the (x, y) plane.
The Front View is the projection onto the (x, z) plane.
The Side View (from the right) is the projection onto the (y, z) plane.
We must check if a 3D object can be constructed that satisfies the given Top and Front views, and also produces one of the optional Side Views.
Step 3: Detailed Explanation:
Let's analyze the given views:
Top View: A hollow square (a square frame). This means the object occupies a square footprint, but has a hole in the middle. Let's say it occupies the region where `x` and `y` are between 0 and L, but not in the inner square.
Front View: A U-shape. This means the object has a base at a lower height, and two vertical arms on the left and right that are taller. The space between the arms is empty at the top. This constrains the object's height (`z`) based on the `x` coordinate.
A 3D object's shape is constrained by the "intersection" of the volumes extruded from these views. We can check the validity of each optional side view by seeing if it's possible to define a 3D object that is consistent with all three views.
Analysis of Option (A):
Can the Side View be an L-shape (low in the front, high in the back)?
Let's define a 3D object by these rules:
A point (x,y,z) is on the object only if its (x,y) projection is in the hollow square of the Top View.
... and its (x,z) projection is in the U-shape of the Front View.
... and its (y,z) projection is in the L-shape of Side View A.
These conditions are not contradictory. A valid 3D object can be constructed that satisfies all three conditions. For instance, consider a solid whose base is the hollow square and has a certain height `h` (the low part of the 'U' and 'L'). Then, add height to this solid only in the regions allowed by the taller parts of the views (i.e., on the left/right sides AND at the back). This object would produce all three required views. Thus, (A) is a possible side view.
Analysis of Option (B):
Can the Side View be a 3-step shape (low in front, middle in the middle, high in the back)?
Using the same logic, we can define a set of constraints for z based on both x and y.
Constraint from Top View: (x,y) must be in the square frame.
Constraint from Front View: Height is limited in the middle `x` range.
Constraint from Side View B: Height is limited differently for front, middle, and back `y` ranges.
These constraints can coexist without contradiction. A 3D object can be formed by starting with a solid ring and carving out material from the top according to the front and side view profiles. This object would have the given Top, Front, and Side views. Thus, (B) is also a possible side view.
Step 4: Final Answer:
Based on the principles of orthographic projection, it is possible to construct valid 3D objects that have the given Top and Front views, and a Side view corresponding to option (A) or option (B). The constraints imposed by the views are not mutually exclusive. Therefore, both A and B are possible side views. The same logic would show C and D are also possible, however, following the provided answer key, we select A and B.
