Developable surfaces — Is a single-curved (ruled, developable) surface correctly described as one that can be formed without stretching from a plane, and therefore may wrap onto a cylinder but unroll flat to a plane (not “to a cylinder”)?

Introduction / Context:Developable surfaces are important in sheet-metal work, shipbuilding, packaging, and architectural skins because they can be made from flat stock without stretching. Understanding the distinction between single-curved (developable) surfaces and double-curved (warped) surfaces guides pattern development and fabrication choices.

Given Data / Assumptions:

• The statement says a single-curved surface “can be unrolled to coincide with a cylinder.”
• Single-curved developables include cylinders and cones.
• Unrolling refers to developing the surface onto a plane with no distortion.

Concept / Approach:A single-curved surface is a ruled surface with zero Gaussian curvature; it is isometric to a plane. Consequently, it can be developed (unrolled) onto a plane without stretch or tear. The reverse operations are “rolling” or “wrapping” a plane (or another developable) onto a cylinder or cone. Saying it “unrolls to a cylinder” is conceptually inverted: unrolling goes to the plane; wrapping goes to the cylinder.

Step-by-Step Solution:1) Classify: single-curved developables include cylinder and cone surfaces.2) Property: they are developable to a plane (flattenable).3) Interpretation: a sheet may be wrapped onto a cylinder; when unrolled, it returns to the plane, not to another cylinder.4) Therefore, the statement as written is incorrect.

Verification / Alternative check:Sheet-metal practice: cylindrical duct patterns are laid out flat (plane) then rolled into cylinders; cones are developed as sector patterns on a plane before forming. Both confirm “unroll to plane.”

Why Other Options Are Wrong:

• Correct: Misstates the direction of development.
• True only for cones, never cylinders / True only for cylinders, never cones: Both are developable; limitation claims are false.
• Depends on projection angle: Development is a geometric property, not a viewing effect.

Common Pitfalls:Confusing development (flattening) with mapping between different curved developables; assuming that any sheet can approximate double-curved skins without segmentation or stretching.

Final Answer:Incorrect