Developments and intersections: identify the most common real-world application where pattern developments and intersection geometry are routinely required.

Introduction / Context:
Developments convert 3D surfaces into flat patterns that can be cut and formed. Intersections determine the curves where two shapes meet (for example, a duct branch). These topics are fundamental in fabrication industries that form material from flat stock.

Given Data / Assumptions:

• We are choosing the most common field relying on developments and intersections.
• Materials are often thin and formable (sheet stock).
• Accuracy in unrolled patterns reduces waste and rework.

Concept / Approach:
Sheet metal fabrication of HVAC ducts, hoppers, transitions, and enclosures depends on precise developments (patterns) and accurate intersection curves (for example, where a round branch meets a rectangular trunk). While piping and general mechanical design use these concepts, sheet metal work uses them daily for layout.

Step-by-Step Solution:

Verification / Alternative check:
Use CAD sheet-metal unfolding to validate manual development dimensions; check that formed pieces meet within tolerance.

Why Other Options Are Wrong:

• Piping: Uses intersections for branches, but not as universally for flat developments as sheet metal does.
• Architecture / Mechanical engineering: Broad fields; developments are situational rather than the dominant daily task.

Common Pitfalls:
Ignoring bend allowances and K-factors; even developable surfaces need allowance to avoid fit-up errors.

Final Answer:
Sheet metal construction