My unrolling theory

ZOBACZ TEN POST PO POLSKU

A developable surface is a ruled surface which in every point has a Gaussian curvature equal to 0 .

That is a definition. In fact that means that developable surface can be developed to a plane without stretching or cutting .
 

Cone side development .

An example can be side of a cone or a cylinder .
You can also say that cone is an pyramid with an infinite number of base sides .Cylinder is a prism with an infinite count of base sides . That means that every prism or pyramid can be developed .
That leads to the conclusion that surface is developable if it consist from multiple flat surfaces , and every surface is connected to others with not more than 2 sides .

Surfaces spanned between spline an arc,
and polylines that interpolate basic curves.
Surfaces modelled with quads and triangles.

So if You have two curves, You can interpolate it with a polylines with the same sides count . On these polylines You can create a surface that consist from multiple quadrangles .
But not every quadrangle must be flat . To solve this problem, You can divide every quadrangle into two triangles , which solves the problem because every triangle is a flat figure .

How to develop 3-dimensional triangulated surface ?



Definition curves, spanned surface and it's development .


We have 2 polylines that will define our surface, lets mark their points with A1-A5 and B1-B5 .
Create a triangulated surface spanned on those 2 polylines ( see picture to the left ) . Such a surface you can develop using simple geometrical construction.

Construction  of a triangle using 3 known sides .

If You know length of all 3 sides of a triangle , You can draw one side , and then draw from both ends arcs with a radios equal to length of other sides . Intersection between those arcs gives You third point of a triangle. If You repeat the same construction for every triangle , in the end You get a developed surface .




Next task is to make the whole process automatic ?

To be continued ...