Look into using a ‘Projected’ Texture for continuity across a curved surface.
SketchUp draws everything with straight line segments. When you set out to draw a Circle or an Arc you’re confronted with the prompt to choosing how many Sides you’d like… This is the same as choosing how many straight line segments you want to use for drawing the circle/arc.
For Circles and Arc, SU applies some intelligence to the default state of the shape it generates, included in that is a smoothing treatment to help soften the hard edges that would occur otherwise.
The experiment’s you’ve made above are basically showing how smoothing works in different ways. as well as highlighting how an object looses it’s initial (intelligent) status once it’s been exploded.
Cylinder 1: shows SU’s default settings… with automatic smoothing, good wrapping, (and some basic shape intelligence—not completely on display).
Cylinder 2: By changing the softening value you’ve broken SU’s default settings in favor of your own… and SU now treats these edges accordingly.
Cylinder 3: Exploding the cylinder also breaks apart the smoothing automation… But in addition to that—you’re also reducing it’s special “intelligence” as a default SU generated object… for example, after exploding the Circle you can no longer go into the ‘Entity Info’ window and adjust the radius value, or change the Segment (sides) count.
Cylinder 4: is the same issues as cylinder 2 and 3,… but in this case hiding the edges won’t convert the underlying structure of the model back into the pure state of being a Cylinder 1 (SU default Circle)—even though it looks the same from an appearance standpoint.
Where the textures are NOT wrapping well, this is simply based on SU treating the extruded circle as multiple faces, and not one continuous curve. Each face is getting it’s own texture positioning which fill up that face according to whatever size the texture is set to, or any of the other mapping options that are available for any other face/surface/or object.
Mapping a Label to a Cylinder:
Sorry for adding this in last, I always think of the other video first before this one.