Fabrication of free form structures from planar materials

A method for producing an output surface composed of at least one developable surface element corresponding to a three-dimensional source model surface, comprising the steps of:

• identifying boundaries on the source model surface which divide the source model surface into at least one surface section such that each surface section is bounded by a single closed curve which defines a surface section perimeter;
• representing each surface section perimeter by a plurality of vertices positioned along the surface section perimeter to define a perimeter polygon;
• examining a plurality of alternative divisions of at least one surface section into triangles which connect the vertices of the perimeter polygon, and assigning costs to each of the plurality of alternative divisions such that a better approximation to the source model surface section has a lower cost, and selecting a lowest cost division of the surface section into a plurality of triangles in which at least one suspended triangle extends between vertices on the perimeter polygon such that said suspended triangle has only chords extending between vertices but no edges lying on the perimeter polygon;
• flattening each output perimeter polygon optimal triangulation to define a planar output surface element perimeter;
• and
• outputting instructions for a cutting tool for fabricating from a planar material each output surface element by cutting along its output surface element perimeter.

A process for forming an output object assembled from sheet elements, the output object having a surface which approximates the surface of a source model, comprising the steps of:

• dividing the source model surface into a plurality of surface sections;
• identifying the perimeter of each surface section;
• approximating each surface section perimeter with a perimeter polygon comprised of a plurality of vertices which lie on said surface section perimeter and which are connected by line segments;
• determining a low cost triangulation of each perimeter polygon by comparing a first triangulation to at least one second different triangulation, wherein a cost is determined for each triangulation by applying a function which compares an attribute of each triangle of a possible triangulation with a like attribute of an adjacent triangle of said possible triangulation, or with an attribute of one or more vertices of said triangle on the source model surface section;
• flattening the determined low cost triangulations of all the perimeter polygons;
• outputting the flattened triangulations as sheet elements;
• and
• assembling the output sheet elements to form the output object.

A method for producing an output surface composed of developable surface elements corresponding to an arbitrary three-dimensional source model surface, comprising the steps of:

• identifying boundaries on the source model surface which divide the source model surface into surface sections such that each surface section is bounded by a single closed curve which defines a surface section perimeter;
• for each surface section perimeter, selecting a plurality of vertices spaced along the surface section perimeter, the plurality of vertices defining a perimeter polygon, and each vertex lying on the source model surface section;
• searching for an optimal triangulation comprised of a set of triangles which do not overlap, and which are connected along triangular sides, wherein each triangle has three vertices which are vertices of the perimeter polygon, the triangulation defining a surface which fills the perimeter polygon, wherein the step of searching for an optimal triangulation comprises:
• performing an analysis of at least two different triangulations by using a first function applied to each triangle of the analyzed triangulation to determine a value of the first function for said analyzed triangulation, the optimal triangulation being that one of the at least two different triangulations which has the lowest value, wherein the first function is a function which relates an attribute of each triangle of an analyzed triangulation to an attribute of the source model surface, or to an attribute of an adjacent triangle;
• flattening each optimal triangulation by bringing each triangle of the set of triangles into the same plane by rotation about edges shared by triangles to define a planar output surface element perimeter;
• and
• outputting instructions for a cutting tool for fabricating from a planar material each output surface element by cutting along its output surface element perimeter.

A method for creating a set of at least one developable surface patch corresponding to a three-dimensional source model surface, comprising the steps of:

• identifying boundaries on the source model surface to identify a surface section bounded by a single closed curve which defines a surface section perimeter;
• representing the surface section perimeter by a plurality of vertices positioned along the surface section perimeter to define a perimeter polygon;
• examining a plurality of possible divisions of the surface section into triangles which connect the vertices of the perimeter polygon, and assigning costs to each of the plurality of possible divisions based on a function which relates individual triangles to each other, or to some characteristic of the source surface or both, and comparing the costs of each of said plurality of possible divisions, this comparison serving to determine an optimal triangulation of the perimeter polygon, the optimal triangulation corresponding to a developable surface which has the perimeter polygon as a perimeter;
• and
• deriving the set of at least one developable surface patch corresponding to the optimal triangulation.