Computer-assisted design system for flattening a three-dimensional surface and for wrapping a flat shape to a three-dimensional surface

A method for developing a two-dimensional shape that approximates a three-dimensional surface region mapped to a flat plane using a computer-assisted design system, said three-dimensional surface region capable of spanning a plurality of mathematically defined surfaces, said surface region having at least one curve defining a discontinuity in said surface region that forms a boundary of said surface region, including the steps of:

• growing a regional mesh on said surface region capable of spanning a plurality of mathematically defined surfaces wherein said mesh is grown independent of the boundaries of said mathematically defined surfaces, said mesh conforming to the topology of the surface region by growing a ribbon of polygonal elements conforming to the topology of the surface region along each said curve and growing interior polygonal element conforming to the topology of the surface region to said ribbons until said polygonal elements substantially cover said surface region, said mesh including nodal points interconnected by links defining said polygonal elements between said links;
• mapping at least one of said nodal points to a location in said flat plane;
• and
• mapping said polygonal elements to a location in said flat plane where said polygonal elements will share at least one nodal point with other ones of said polygonal elements previously mapped to said flat plane in a manner that said polygonal elements are located in ring portions generally concentric with said at least one of said nodal points in order to develop a two-dimensional shape.

A method of growing a mesh conforming to the topology of two-dimensional and three-dimensional surface regions using a computer-assisted design system, said surface regions capable of spanning a plurality of mathematically defined surfaces and having at least one edge therein and at least one vertex defining a point on said edge, said mesh including nodal points interconnected by links defined triangular elements between said links, including the steps of:

• distributing nodal points along each edge of a surface region capable of spanning a plurality of mathematically defined surfaces;
• growing a ribbon of triangular elements along each said edge by at least partially enclosing said edge nodal points with triangular elements and thereby establishing additional nodal points, said ribbon conforming to the topology of the surface region, said ribbon grown independent of the boundaries of said mathematically defined surfaces;
• and
• growing interior triangular elements conforming to the topology of the surface region by at least partially enclosing said additional points with triangular elements, said interior triangular elements grown independent of the boundaries of said at least one mathematically defined surface.

A method for wrapping a shaped surface from a two-dimensional plane to a three-dimensional destination surface region capable of spanning a plurality of mathematically defined surfaces and unwrapping a shaped surface from a three-dimensional surface region capable of spanning a plurality of mathematically defined surfaces to a flat destination surface using a computer assisted design system, including the steps of:

• growing a mesh conforming to the topology of the shaped surface, said mesh including nodal points interconnected by links defining polygonal elements between said links;
• defining a growing surface distinct from the three-dimensional surface region capable of spanning a plurality of mathematically defined surfaces;
• mapping said nodal point and a plurality of said polygonal elements enclosing said nodal point to said growing surface;
• rotating and translating said growing surface;
• and
• mapping said nodal point and said plurality of said polygonal elements enclosing said nodal point from said growing surface to the destination surface.

A method for growing a mesh conforming to the topology of a three-dimensional surface region using a computer-assisted design system, said three-dimensional surface region capable of spanning a plurality of mathematically defined surfaces, said mesh including nodal points interconnected by links defining polygonal elements between said links, including the steps of:

• distributing nodal points along each edge of a three-dimensional surface region capable of spanning a plurality of mathematically defined surfaces;
• growing a ribbon of triangular elements along each said edge by at least partially enclosing said edge nodal points with polygonal elements and thereby establishing additional nodal points, said ribbon conforming to the topology of the three-dimensional surface region, said ribbon grown independent of the boundaries of said mathematically defined surfaces;
• growing interior polygonal elements conforming to the topology of the three-dimensional surface region by at least partially enclosing said additional points with polygonal elements, said interior polygonal elements grown independent of the boundaries of said mathematically defined surfaces;
• and
• wherein said steps of growing includes projecting an existing nodal point to a growing surface distinct from said three-dimensional surface, establishing new nodal points on said growing surface by enclosing the particular existing nodal point with polygonal elements, and mapping the existing and new nodal points from the growing surface normally to the three-dimensional surface.

A computer-assisted design apparatus for generating a two-dimensional pattern image that is a close approximation of a flattened three-dimensional surface region, said three dimensional surface region capable of spanning a plurality of mathematically defined surfaces and including at least one curve establishing a discontinuity in said three-dimensional surface region that forms a boundary of said region comprising:

• a mesh generator adapted to growing a regional mesh on said three-dimensional surface region capable of spanning a plurality of mathematically defined surfaces wherein said mesh is grown independent of the boundaries of said mathematically defined surfaces, said mesh conforming to the topology of the three-dimensional surface region by growing a ribbon of polygonal elements conforming to the topology of the three-dimensional surface region along each said curve and growing interior polygonal elements conforming to the topology of the surface region to said ribbons until said polygonal elements cover said surface region, said mesh including nodal points interconnected by links defining said polygonal elements between said links;
• and
• means for mapping said nodal points to a flat plane and for locating nodal points in said flat plane, said means for mapping being adapted to locating at least one of said nodal points in said flat plane and said polygonal elements in said flat plane in a manner at said polygonal elements will share at least one nodal point with other ones of said polygonal elements previously mapped to said flat plane and are in ring portions generally concentric with said at least one of said nodal points in order to develop a two-dimensional shape.

In a computer assisted design system, an apparatus that is adapted to growing a mesh conforming to the topology of a three-dimensional surface region, said three-dimensional surface region capable of spanning a plurality of mathematically defined surfaces, said mesh including nodal points interconnected by links defining polygonal elements between said links, said apparatus comprising:

• a perimeter nodal point generator that is adapted to generating and distributing nodal points along each said edge of a three-dimensional surface region capable of spanning a plurality of mathematically defined surfaces and growing a ribbon of polygonal elements along each said edge by at least partially enclosing said edge nodal points with polygonal elements and thereby establishing additional nodal points, said ribbon conforming to the topology of the three-dimensional surface region, said ribbon grown independent of the boundaries of said mathematically defined surfaces;
• an interior nodal point generator that is adapted to developing polygonal elements conforming to the topology of the surface by at least partially enclosing said additional points with polygonal elements, said polygonal elements developed independent of the boundaries of said mathematically defined surfaces;
• and
• wherein said nodal point generators include means for projecting an existing nodal point to a growing surface distinct from the three-dimensional surface region for establishing new nodal points on said growing surface by enclosing the particular existing nodal point with polygonal elements, and for mapping the existing and new nodal points from the growing surface to the three-dimensional surface.