System and Method for Computer Modeling of 3D Objects or Surfaces by Mesh Constructions Having Optimal Quality Characteristics and Dynamic Resolution Capabilities

1. A computer-based system for generation of a mesh model of a three-dimensional object or surface, the mesh model having increasing and decreasing resolution of detail, the computer, including a processor coupled to a memory and program elements adapted to accept as input a plurality of data points, with each data point being a coordinate describing the object or surface to be modeled, and construct a mesh to model the object or surface, the system comprising: (a) a mesh construction element to determine the distance each data point has in relation to a preselected reference object and select according to the distance values a set of points for the creation of the mesh; (b) the mesh model construction element further adapted to construct the mesh using the selected set of points and organizing the points into a set of connected faces, each face being a geometric shape with a predetermined number of vertices, and the ones of the points for each face comprising the vertices of that face, the boundary connection between any two adjacent vertices of a face comprising an edge of the face, with a face being connected to another adjacent face in the mesh through a shared edge and shared vertices creating that shared edge; (c) a data point detail ordering element to determine which of the remaining unmeshed data points lies farthest from the mesh; (d) a point insertion element to insert the data point into the mesh face that is closest to that data point by reorganizing the arrangement of vertices of the face for insertion and including the data point for insertion in that rearrangement to create a new set of faces, each new face having the predetermined number of vertices and the new faces sharing edges with neighboring faces to create a surface with the other faces of the mesh; (e) the point insertion element further adapted to create a reference on an insertion list to the data point for insertion in a sequence as the last point inserted; (f) an optimality checking element to determine whether the construction of newly created faces meets a predetermined optimality criteria; (g) a rearranging element to redefine in a sequenced order the boundaries of the newly created faces when one of the newly created faces does not meet the predetermined optimality criteria, the rearranging element further comprised to place an indication on a history list in sequence for each redefinition made for a particular face; and (h) the data point detail ordering element further comprised to redetermine the distance between any unmeshed data point whose closest face was altered either by data point insertion or rearrangement and comparing those redetermined distances with the previously determined distances to identify the next point for insertion.

2. The system of claim 1 configured for downward resolution removal of data points and further comprising a data point removal element to locate the reference to the last inserted data point on the insertion list and delete that data point from the mesh by removing the data point as a vertex from any face and rearranging the vertices of the remaining faces following, in reverse order, the sequence of information stored on the history list to return the mesh to a state of optimal configuration without the removed data point.

3. The system of claim 1 where the initial reference object is a sphere.

4. The system of claim 1 where the initial reference object is a plane.

5. The system of claim 1 configured to continue inserting unmeshed data points to build a mesh model to a level of detail with a specified tolerance, the system comprising: (a) the data point detail ordering element further adapted to determine whether the remaining unmeshed data points are greater in distance to the mesh than a predetermined, tolerance value; and (b) a tolerance insertion element to continue the insertion of data points until the distance that all remaining unmeshed data points have to the closest face of the mesh is less than the tolerance value.

6. The system of claim 1 where the optimality criteria is the Delaunay principle of optimality for an irregular triangulated mesh.

7. A computer-based system for inserting a data point into a mesh model of a three-dimensional object or surface to increase resolution of the mesh detail and removing the data point to downwardly decrease resolution, the computer comprising a processor coupled to a memory, the processor configured to accept as input a plurality of data points, each one of the plurality of data points being a coordinate describing the object or surface to be modeled, the data point to be inserted being one of a subset of the plurality of data points that have not yet been inserted into the mesh, the system comprising: (a) a mesh constructed from ones of the data points and comprising a set of connected faces creating a surface approximating the object or surface, with each face being a triangle determined by ones of the inserted data points, the ones of the data points for each face comprising the vertices of that face with a boundary connection between any two adjacent vertices of a face comprising an edge of the face and each face being connected to another adjacent, neighboring face through a shared edge and the pair of shared vertices creating that shared edge; (b) the computer adapted to determine which one of the unmeshed data points lies farthest from the mesh by locating for each unmeshed data point the mesh face which is closest in distance to that data point and comparing that distance with the distances obtained for the other unmeshed points and identify that point as a point for insertion; (c) the computer adapted to insert the data point into the mesh by arranging the data point for insertion and the vertices of the face that is closest in distance to the point for insertion (the insertion face) to create a set of new faces with the data point for insertion being shared in each new face, the new faces being connected to other faces in the mesh through vertices and edges as were previously established in the insertion face, the processor establishing neighbor relationships for each new face; (d) the mesh insertion element further comprised to place a reference to the data point for insertion on an insertion list sequenced as the last point inserted; (e) an optimality checking element to enable the processor to check one of the newly created faces to determine whether its construction meets optimality criteria, the checking procedure checking a new face and one of its neighbor faces to evaluate its vertice and edge configuration for optimality; (f) a rearrangement element to reconfigure the triangle being checked and the neighbor to redefine their boundaries when one of the newly created faces does not meet the predetermined optimality criteria, with the rearrangement element further comprised to place an indication of the rearrangement on a history list as the next action; (g) the optimality checking element further comprised to check the all the newly created faces; (h) the data point detail ordering element further comprised to redetermine the distance between any unmeshed data point whose closest face was altered either by data point insertion or rearrangement and comparing those redetermined distances with the previously determined distances to identify the next point for insertion; and (i) a data point removal element to locate the reference to the last inserted data point on the insertion list and delete that data point from the mesh by removing the data point as a vertex from any face and rearranging the vertices of the remaining faces following, in reverse order, the sequence of information stored on the history list to return the mesh to a state of optimal configuration without the removed data point.

8. A computer-based system for inserting data points into a Delaunayian triangulated mesh model of a three-dimensional object or surface to increase resolution of the mesh detail and permit downward resolution removal of points, the computer comprising a processor coupled to a memory, the processor configured to accept as input a plurality of data points, each one of the plurality of data points being a three dimensional coordinate describing the object or surface to be modeled, the data point to be inserted being one of a subset of the plurality of data points that have not yet been inserted into the mesh, the system comprising: (a) a mesh model constructed from ones of the data points and comprising a set of adjacent faces creating a surface approximating the object or surface, with each face being a triangle determined by ones of the inserted data points, the ones of the data points for each face comprising the vertices of that face, the vertices for each face being ordered sequentially to have first, second and third vertices with a boundary connection between any two adjacent vertices of a face comprising an edge of the face, with a face being connected to another adjacent, neighboring face through a shared edge and the pair of shared vertices creating that shared edge, the neighbors of the face being ordered following a sequential order corresponding to the ordering of the face's vertices; (b) a data point detail ordering element, comprised to enable the processor to determine which one of the unmeshed data points lies farthest from the mesh by locating for each unmeshed data point the mesh face which is closest in distance to that data point and comparing that distance with the distances obtained for the other unmeshed points; (c) a mesh insertion element, comprised to enable the processor to insert the data point determined to lie farthest from the mesh into the mesh by incorporating the data point for insertion and reorganizing the arrangement of vertices of the face that is closest in distance to the point for insertion (the insertion face) to create a set of three new triangles, each having its vertices ordered in a sequential manner with the data point for insertion being the first point in each new triangular face, the new triangles being connected to other triangles in the mesh through vertices and edges as were previously established in the insertion face, the new faces creating a continuous surface with the other faces of the mesh, the neighbors for each new face being ordered following the sequential order and corresponding to ordering of the face's vertices; (d) the mesh insertion element further comprised to place a reference to the data point for insertion on an insertion list as the next point inserted; (e) an optimality checking element to enable the processor to check one of the newly created faces to determine whether their construction meets Delaunayian optimality criteria, by checking the newly created face with a neighbor triangle to determine whether the non-shared vertex of the neighbor lies within a circumcircle determined by the vertices of the newly created triangle; (f) a rearrangement element to reconfigure the shared edge between the triangle being checked and the neighbor to create two new flipped triangles, if the combination does not pass Delaunayian optimality; (g) the rearrangement element further comprised to place an indication of the flip on a history list as the next action; (h) the optimality checking element further comprised to recheck the triangle previously being checked if it was rearranged, using the rearrangement element to flip and indicate the flip history, if the new configuration does not meet Delaunayian optimality; (i) the optimality checking element further comprised to proceed to a neighbor of the triangle being checked according to a preselected sequence and perform a check for optimality; (j) the optimality checking element further comprised to proceed checking the first neighbor of all checked triangles until it arrives again to the first triangle checked; and (k) the data point detail ordering element further comprised to redetermine the distance between any unmeshed data point whose closest face was altered either by data point insertion or optimality checking and comparing those redetermined distances with the previously determined distances to identify the next point for insertion.

9. The system of claim 8 further comprising a point deletion element comprised to reference the insertion list to locate the last point inserted and remove the point by reconfiguring the faces to maintain optimality by following the indications in the history list in reverse order.

10. The system of claim 8 further comprising: (a) the point insertion element, further comprised to enable the processor to insert the data point determined to lie farthest from the mesh into the mesh by rearranging the face for insertion by replacing the first vertex of that triangle with the data point for insertion, creating a new right neighbor triangle having the data point for insertion as a first vertex, the first face point as the second vertex and the second face point as the third vertex, and creating a new left face with the data point for insertion as the first vertex, the third face point as the second vertex and the first face point as the third vertex, the newly organized triangles being referred to as insertion face, left and right, and the processor configured to adjust the identification of neighboring triangles to follow the order of vertices; and (b) an optimality checking element further comprised with a flipping element to rearrange the shared edge between the triangle being checked and the first neighbor, if the combination does not pass Delaunayian optimality, the flipping element enabling the processor to rearrange the triangles by redefining the vertices for the first neighbor triangle by setting the first vertex to be the first vertex of the triangle being checked, setting the second vertex to be the first neighbor's former first vertex and setting the third vertex to be the first neighbor's former second vertex and redefining the vertices of the triangle being checked to change its third vertex to be the former first vertex of the first neighbor.

11. A system for generation of a mesh model of a three-dimensional object or surface with increasing and decreasing the resolution of detail, the system configured to accept as input a plurality of data points, each data point being a coordinate describing the object or surface to be modeled, the system comprising: (a) means for determining the distance each of the data points has in relation to a preselected initial reference object and select based on those determined distances a set of points to create an initial mesh, the set of points being descriptive of the basic shape of the object modeled; (b) means for creating an initial mesh model of the object using the selected set of points for the initial mesh by organizing the points into a set of connected faces, with each face being a geometric shape with a predetermined number of vertices and the ones of the data points for each face comprising the vertices of that face, with a boundary connection between any two vertices of a face comprising an edge of the face and each face being connected to another adjacent face in the mesh through a shared edge and shared vertices creating that shared edge; (c) means for determining which of the remaining unmeshed data points lies farthest from the mesh by locating for each remaining unmeshed data point the mesh face which is closest in distance to that data point and comparing that distance with the distances obtained for the other unmeshed points to identify from the comparison the next point for insertion and identify the closest face to that point as the face for insertion; (d) means for inserting the data point into the mesh by reorganizing the arrangement of vertices of the face that defines the mesh and including the data point for insertion in that rearrangement to create a new set of faces from those points, each new face having the predetermined number of vertices, and the new faces creating a continuous surface with the other faces of the mesh; (e) means for referencing the data point for insertion on an insertion list as the next point inserted, and placing an indication on a history list of the sequence of steps taken during reconfiguration for the data point insertion; and (f) means for determining which of the remaining unmeshed data points lies farthest from the mesh by determining a new distance for each data point whose face used in previous distance calculations was altered during insertion and sorting those recaptured values with the previously computed values to find the unmeshed data point that lies farthest from the mesh and identifying the closest face to that point as the face for insertion.

12. A computer-based system for increasing and decreasing the resolution of a mesh model of an object or surface, the computer, including a processor coupled to a memory, adapted to accept as input a plurality of data points comprising coordinates that describe the object or surface and further construct and deconstruct the mesh using the data points, the system comprising: (a) the computer adapted to identify from the unmeshed data points the next data point to insert into the mesh by determining which unmeshed data point lies farthest in distance from the mesh, that data point being the next point of detail; (b) the computer further adapted to enable the processor to take predetermined reorganizing steps to connect the data point for insertion into the mesh; (c) the computer further adapted to store in sequence a history of the reorganization steps taken in relation to the point for insertion on a history list; and (d) the computer further adapted to remove the last point inserted by reversing the reorganization steps as indicated on the history list.

13. The system of claim 12 further comprising: (a) the computer further adapted after inserting the data point to continue increasing mesh resolution through the insertion of data points by identifying the next data point for insertion by finding the data point that lies farthest from the mesh and inserting that point into the mesh, by following the reorganization steps and storing in sequence the history of the reorganization steps; and (b) the computer further adapted after removing the last data point to continue removing data points in reverse order of their insertion by following the indications on the history list to reverse the steps of their organization until there is one unmeshed data point whose distance from the mesh is greater than a predetermined tolerance value.

14. A mesh data structure used on a computer for generating a mesh model of an object or surface from a plurality of data points, the computer comprising a processor to manipulate the mesh data structure and a memory to store the mesh data structure, each data point being a coordinate describing the object or surface to be modeled, the mesh data structure comprising: (a) a face data record with vertex slots to hold references to data points that make up the vertices of a face in the mesh, the face in the mesh being a geometric shape with a predetermined number of vertices, the ones of the data points referenced in the slots comprising the vertices of the face and data points being placed in the slots in a predetermined order to traverse the vertices in an ordered sequence, with the boundary edges of the face being determined by any two adjacent vertex slots and each face being connected to another adjacent, neighboring face through a shared edge and a pair of shared vertices creating that shared edge; and (b) the face data record being further comprised with neighbor slots to reference face data records representing neighboring faces in the mesh, the neighbor slots being configured to link a neighbor face reference to a particular vertex in the vertex slot and follow the predetermined order of the vertices for the face.

15. A computer-based system for inserting a data point into a mesh model of a three-dimensional object or surface, the computer comprising a processor and memory, the processor configured to accept as input a plurality of data points, each one of the plurality of data points being a three dimensional coordinate describing the object or surface to be modeled, the mesh model being constructed from ones of the data points and comprising a set of adjacent faces creating a surface with each face being a geometric shape having the same predetermined number of vertices and having a boundary determined by ones of the inserted data points, the ones of the data points for each face comprising the vertices of that face and a boundary connection between any two vertices of a face comprising an edge of the face and each face being connected to another adjacent face in the mesh through a shared edge and the shared vertices creating that shared edge, the data point to be inserted being one of a subset of the plurality of data points that have not yet been inserted into the mesh, the system comprising: (a) an ordering element, comprised to enable the processor to identify which one of the unmeshed data points lies farthest from the mesh by locating for each unmeshed data point the mesh face which is closest in distance to that data point and comparing that distance with the distances obtained for the other unmeshed points; (b) a mesh building element, comprised to enable the processor to insert the data point into the mesh by reorganizing the arrangement of vertices of the face and including the data point for insertion in that rearrangement to create a new set of faces from those points, each new face having the predetermined number of vertices, and the vertices and edges of the new faces arranged to create a continuous surface with the other faces of the mesh; (c) an optimality checking element to enable the processor to check the set of newly created faces to determine whether their construction meets a predetermined optimality criteria; and (d) a rearranging element to redefine the boundaries of the newly created faces when one of the newly created faces does not meet the predetermined optimality criteria.

16. A computer-based system for increasing resolution of a mesh model of an object or surface, the computer, including a processor coupled to a memory, adapted to accept as input a plurality of data points comprising coordinates that describe the object or surface and further construct the mesh using the data points, the system comprising: (a) the computer adapted to identify from the unmeshed data points the next point of detail to insert into the mesh by determining which unmeshed point lies farthest in distance from the mesh; and (b) the computer further adapted to connect the data point for insertion into the mesh and follow a predetermined path using the mesh structure to organize the mesh according to a predetermined optimality criteria.

17. The system of claim 16 further comprising the computer further adapted to continue identifying the next data point for insertion by finding the data point which lies farthest from the mesh, inserting that point into the mesh, and reordering the unmeshed data points to find the next point to insert until the distance of all the remaining unmeshed data points falls below a predetermined tolerance value.

18. A computer-based system for decreasing the resolution of a mesh model of an object or surface, the computer including a processor coupled to a memory, adapted to accept as input a plurality of data points comprising coordinates that describe the object or surface and further construct and manipulate the mesh based on the data points, the system comprising: (a) an insertion list to sequentially record the order of insertion into the mesh of the data points as the processor constructs a mesh; (b) a history list to record any information concerning the sequence of alterations made by the processor to the mesh as the data points were inserted into the mesh during construction; (c) the processor adapted to remove data points from the mesh following in reverse sequential order as indicated on the insertion list; and (d) the computer further adapted to reverse any changes to the mesh having resulted from the initial insertion of the data point being removed, as indicated on the history list.

19. A computer based system for checking optimality, based on Delaunayian principles, of data points being inserted into a triangulated 3D mesh model, the computer including a processor coupled to a memory, configured to accept as input a plurality of data points, with each data point being a coordinate describing the object or surface to be modeled, and the computer further configured to construct a mesh to model the object or surface, the system comprising: (a) a data structure to describe the mesh faces to include each data point in the mesh as a vertex of at least one triangular face of the mesh, sequentially order the vertices of each face of the mesh, and describe the relationship of each face being connected to another adjacent, neighboring face through a shared edge and the pair of shared vertices creating that shared edge; (b) the computer adapted to insert a new data point into the mesh by reorganizing the arrangement of vertices of a face for insertion to create a set of three new faces, each new face having a data structure similar to the existing faces; (c) the computer further adapted to check the mesh faces for Delaunayian optimality by determining a circumcircle from the vertices of the newly created triangles and determine if any other mesh data point lies within the circumcircle; (d) the computer further adapted to rearrange the mesh if any other data points fall within the created circumcircle; and (e) the computer further adapted to continue checking all the faces affected by the input of the new data point, the check to proceed in an order determined by the ordering of the vertices in the face data structure.

20. A computer based system for checking optimality, based on Delaunayian principles, of data points being inserted into a triangulated 3D mesh model, the computer including a processor coupled to a memory, configured to accept as input a plurality of data points, with each data point being a coordinate describing the object or surface to be modeled, and the computer further configured to construct a mesh to model the object or surface, the system comprising: (a) a data structure to describe the mesh faces such that each data point in the mesh is a vertex of at least one triangular face of the mesh, each face of the mesh has its vertices ordered sequentially to have first, second and third vertices with a boundary connection between any two adjacent vertices of a face comprising an edge of the face and any face being connected to another adjacent, neighboring face through a shared edge and the pair of shared vertices creating that shared edge, the neighbors of the face being ordered following the sequential order corresponding to the ordering of the face's vertices with a first neighbor face, second neighbor face and third neighbor face; (b) the computer adapted to insert a new data point into an existing face and alter the data structure such that the existing face is reorganized to create a set of three new faces, each having the new data point as one vertex and the remaining two vertices being vertices of the existing face, the vertices of the new faces being ordered in a sequence consistent with the existing face's vertices, where the new data point is the first ordered vertex in each of the three new faces, and the neighbors of the new faces being ordered following a predetermined sequential order corresponding to the ordering of the new face's vertices; (c) the computer further adapted to begin a checking sequence at one of the new faces created by the insertion of the new data point, by creating a circumcircle from the vertices of the face, and checking the face's first neighbor for Delaunayian optimality; (d) the computer further adapted to reorient the vertices of the first neighbor face in the data structure, if the combination does not pass Delaunayian optimality, so that the vertices of the first neighbor are ordered such that the first vertex is opposite the face being checked; (e) the computer further adapted to rearrange the shared edge between the face being checked and the first neighbor, if the combination does not pass Delaunayian optimality, the rearrangement occurring by redefinition in the data structure of the vertices for the first neighbor face by setting the first vertex to be the first vertex of the face being checked, setting the second vertex to be the first neighbor's former first vertex and setting the third vertex to be the first neighbor's former second vertex and redefining in the data structure the vertices of the face being checked to change its third vertex to be the former first vertex of the first neighbor; (f) the computer further adapted to redefine the neighbors of the rearranged faces in the data structure based on the new ordering of the vertices; and (g) the computer further adapted to recheck the face previously being checked if it was rearranged, the checking on the face proceeding until it passes the Delaunayian optimality criteria.

21. The system of claim 20 further comprising the computer adapted to proceed, after completing a Delaunayian optimality check on a face, to the first neighbor of the face and perform a Delaunayian optimality check on the first neighbor, the computer proceeding thereafter to each succeeding first neighbor face and performing a Delaunayian optimality check at each said face until it arrives at the first face checked.

22. The system of claim 20 further comprising the computer adapted to project the 3D mesh model onto a 2D reference object creating a 2D triangulated mesh on the reference object that is representative of the original 3D mesh model, and performing the checking on the 2D projected mesh.

23. The system of claim 20 further comprising the computer adapted to retain the equation of the circumcircle for a particular face on a list with reference to a particular face and using that stored equation when that face is checked a second time.

24. The system of claim 20 where the reorientation of the first neighbor's vertices is accomplished by assigning the values 0,1,2 to the first, second and third vertices and neighbors respectively, storing a variable(k) equal to the neighbor value of the face being checked in the data structure of the first neighbor face, the first neighbor's vertices are redefined such that the vertex k value is assigned to the original vertex (3-k), the original vertex (3-k) value is assigned to the original vertex with value 0, and the value 0 is assigned to the original vertex k, resulting in a reorientation of the first neighbor vertices such that the vertex assigned value 0 is the first vertex, the vertex assigned value 1 is the second vertex, and vertex assigned value 2 is the third vertex.

25. A computer based system for increasing and decreasing the resolution of a mesh model of an object or surface, the computer including a processor coupled to a memory, adapted to accept as input a plurality of data points comprising coordinates that describe the object or surface, and construct a mesh to model the surface or object, the system comprising: (a) a history list containing information identifying a sequential list of entries comprising the sequence of operations executed by the processor during a mesh construction, to generate a mesh model containing the complete set of data points describing the object or surface; (b) a data pointer to reference entries in the history list of the operation taken in the construction of the mesh; (c) the processor further comprised to reset the data pointer to an initial location in the sequence of entries in the history list; (d) the mesh construction element further adapted to enable the processor to advance through the entries of the history list and build a mesh model following the sequence of operations on the history list; and (e) the mesh construction element further adapted to enable the processor to move the data pointer in reverse order through the sequence of entries in the history list to remove data points from the mesh model.

26. A computer based system for ordering a plurality of data points to be inserted into a mesh model of an object or surface comprised of geometric faces, the computer comprising a processor to manipulate the order of the data points and a memory to store the order of the data points, each data point being a coordinate describing the object or surface to be modeled, the system comprising: (a) a location element to enable the processor to determine the mesh face which lies closest to each unmeshed data point; (b) a distance computing element to enable the processor to determine the distance each data point lies from its associated face; and (c) an ordering element to enable the processor to determine based on the distances the data point that is farthest from the mesh.

27. A computer based system for ordering a plurality of data points to be inserted into a mesh model of an object or surface, the computer comprising a processor and a memory, the mesh model comprising a set of connected faces, each face being a geometric shape with a predetermined number of vertices, with each face being connected to another adjacent, neighboring face through a shared edge of adjacent vertices, the system comprising: (a) the computer comprised to locate the closest mesh face to each data point by drawing a line from the center point of a current face to the data point, and determining if a projection of that line intersects with the edge of the current face; (b) the computer comprised to determine if no intersection exists that the data point is associated with the current face; (c) the computer further comprised to move from the current face, if there is an intersection with an edge, to the face which shares the edge of intersection with the original face and perform the same check recursively until the closest mesh face is determined when no intersection exists; (d) the computer further adapted to determine the distance from each data point to its closest face; (e) the computer further adapted to order the data points based on the distances to locate the data point that is farthest from the mesh; and (f) the computer further adapted to re-execute the locate and distance determination procedures after each data point insertion.

28. The system of claim 27, further comprising: (a) a face alteration storage element to record during point insertion which mesh faces become altered; and (b) the computer further comprised to reexecute the locate and distance determination procedure only for those unmeshed points whose previous distance calculation would be effected in point insertion.

29. A method to operate a computer for generation of a mesh model of a three-dimensional object or surface, the mesh model being constructed from a plurality of data points and having increasing and decreasing resolution of detail and constructed from a plurality of data points, the computer, including a processor coupled to a memory and program elements adapted to accept as input a plurality of data points, with each data point being a coordinate describing the object or surface to be modeled, and construct a mesh to model the object or surface, the method comprising the steps of: (a) determining the distance each data point has in relation to a preselected reference object and selecting according to the distance values a set of points for the creation of the mesh; (b) constructing the mesh using the selected set of points and organizing the points into a set of connected faces, each face being a geometric shape with a predetermined number of vertices, and the ones of the points for each face comprising the vertices of that face, the boundary connection between any two adjacent vertices of a face comprising an edge of the face, with a face being connected to another adjacent face in the mesh through a shared edge and shared vertices creating that shared edge; (c) determining which of the remaining unmeshed data points lies farthest from the mesh; (d) inserting the data point into the mesh face that is closest to that data point by reorganizing the arrangement of vertices of the face for insertion and including the data point for insertion in that rearrangement to create a new set of faces, each new face having the predetermined number of vertices and the new faces sharing edges with neighboring faces to create a surface with the other faces of the mesh; (e) creating a reference on an insertion list to the data point for insertion in a sequence as the last point inserted; (f) checking to determine whether the construction of newly created faces meets a predetermined optimality criteria; (g) redefining in a sequenced order the boundaries of the newly created faces when one of the newly created faces does not meet the predetermined optimality criteria and placing an indication on a history list in sequence for each redefinition made for a particular face; and (h) redetermining the distance between any unmeshed data point whose closest face was altered either by data point insertion or rearrangement and comparing those redetermined distances with the previously determined distances to identify the next point for insertion.

30. The method of claim 29 wherein decreasing the resolution of the mesh is accomplished by locating the reference to the last inserted data point on the insertion list and deleting that data point from the mesh by removing the data point as a vertex from any face and rearranging the vertices of the remaining faces following, in reverse order, the sequence of information stored on the history list to return the mesh to a state of optimal configuration without the removed data point.

31. The method of claim 29 where the initial reference object is a sphere.

32. The method of claim 29 where the initial reference object is a plane.

33. The method of claim 29 further comprising the steps of: (a) checking whether the remaining unmeshed points are greater in distance to the mesh than a predetermined, tolerance value; and (b) inserting data points until the distance that all remaining unmeshed data points have to the closest face of the mesh is less than the tolerance value.

34. The method of claim 29 where the optimality criteria is the Delaunay principle of optimality for an irregular triangulated mesh.

35. A method for computer-based insertion of data points into a Delaunayian triangulated mesh model of a three-dimensional object or surface to increase resolution of the mesh detail and permit downward resolution removal of points, the computer comprising a processor coupled to a memory, the processor configured to accept as input a plurality of data points, each one of the plurality of data points being a three dimensional coordinate describing the object or surface to be modeled, the data points to be inserted being ones of a subset of the plurality of data points that have not yet been inserted into the mesh, the mesh model constructed from ones of the data points and comprising a set of adjacent faces creating a surface approximating the object or surface, with each face being a triangle determined by ones of the inserted data points, the ones of the data points for each face comprising the vertices of that face, the vertices for each face being ordered sequentially to have first, second and third vertices with a boundary connection between any two adjacent vertices of a face comprising an edge of the face, with a face being connected to another adjacent, neighboring face through a shared edge and the pair of shared vertices creating that shared edge, the neighbors of the face being ordered following a sequential order corresponding to the ordering of the face's vertices, the method comprising the steps of: (a) determining which one of the unmeshed data points lies farthest from the mesh by locating for each unmeshed data point the mesh face which is closest in distance to that data point and comparing that distance with the distances obtained for the other unmeshed points; (b) inserting the data point determined to lie farthest from the mesh into the mesh by incorporating the data point for insertion and reorganizing the arrangement of vertices of the face that is closest in distance to the point for insertion (the insertion face) to create a set of three new triangles, each having its vertices ordered in a sequential manner with the data point for insertion being the first point in each new triangular face, the new triangles being connected to other triangles in the mesh through vertices and edges as were previously established in the insertion face, the new faces creating a continuous surface with the other faces of the mesh, the neighbors for each new face being ordered following the sequential order and corresponding to ordering of the face's vertices; (c) placing a reference to the data point for insertion on an insertion list as the next point inserted; (d) checking one of the newly created faces to determine whether their construction meets Delaunayian optimality criteria, by checking the newly created face with a neighbor triangle to determine whether the non-shared vertex of the neighbor lies within a circumcircle determined by the vertices of the newly created triangle; (e) reconfiguring the shared edge between the triangle being checked and the neighbor to create two new flipped triangles, if the combination does not pass Delaunayian optimality; (f) placing an indication of the flip on a history list as the next action; (g) rechecking the triangle previously being checked if it was rearranged and indicating a flip in the history list, if the new configuration does not meet Delaunayian optimality; (h) proceeding to a neighbor of the triangle being checked according to a preselected sequence and performing a check of the neighbor triangle for optimality, reconfiguring and indicating the flip on the history list, if the reconfiguration does not meet Delaunayian optimality; (i) the optimality checking element further comprised to proceed checking the first neighbor of all checked triangles until it arrives again to the first triangle checked; and (j) the data point detail ordering element further comprised to redetermine the distance between any unmeshed data point whose closest face was altered either by data point insertion or optimality checking and comparing those redetermined distances with the previously determined distances to identify the next point for insertion.

36. The method of claim 35 comprising the further step of deleting points by referencing the insertion list to locate the last point inserted and removing the point by reconfiguring the faces to maintain optimality by following the indications in the history list in reverse order.

37. The method of claim 35 comprising the further steps of: (a) inserting the data point determined to lie farthest from the mesh into the mesh and rearranging the face for insertion by replacing the first vertex of the face for insertion with the data point, creating a new right neighbor triangle having the data point for insertion as a first vertex, the first data point of the face for insertion as the second vertex and the second data point of the face for insertion as the third vertex, creating a new left face with the data point for insertion as the first vertex, the third data point of the face for insertion as the second vertex and the first data point of the face for insertion as the third vertex, and the processor configured to adjust the identification of neighboring triangles to follow the ordering of vertices; and (b) checking the newly created triangles for Delaunayian optimality by selecting one of the newly created triangles and rearranging the shared edge between the triangle being checked and the first neighbor, if the combination does not pass Delaunayian optimality, the rearranging of the triangles by redefining the vertices for the first neighbor triangle by setting the first vertex to be the first vertex of the triangle being checked, setting the second vertex to be the first neighbor's former first vertex and setting the third vertex to be the first neighbor's former second vertex and redefining the vertices of the triangle being checked to change its third vertex to be the former first vertex of the first neighbor.

38. A method for increasing and decreasing the resolution of a mesh model of an object or surface, the mesh constructed from a plurality of data points comprising coordinates that describe the object or surface, the method comprising the steps of: (a) identifying from the unmeshed data points the next data point to insert into the mesh by determining which unmeshed data point lies farthest in distance from the mesh, that data point being the next point of detail; (b) executing predetermined reorganizing steps to connect the data point for insertion into the mesh; (c) storing in sequence a history of the reorganization steps taken in relation to the point for insertion on a history list; and (d) removing the last point inserted by reversing the reorganization steps as indicated on the history list.

39. The method of claim 38 comprising the further steps of: (a) continuing, after inserting the data point, to increase mesh resolution through the insertion of data points by identifying the next data point for insertion, finding the data point that lies farthest from the mesh inserting that point into the mesh, by following the history of the reorganized storing in sequence the history of the reorganization steps; and (b) removing data points in reverse order of their insertion by following the indications on the history list to reverse the steps of their organization until there is one unmeshed data point whose distance from the mesh is greater than a predetermined tolerance value.

40. A method for inserting a data point into a mesh model of a three-dimensional object or surface, the mesh model created from plurality of data points, each one of the plurality of data points being a three dimensional coordinate describing the object or surface to be modeled, the mesh model being constructed from ones of the data points and comprising a set of adjacent faces creating a surface with each face being a geometric shape having the same predetermined number of vertices and having a boundary determined by ones of the inserted data points, the ones of the data points for each face comprising the vertices of that face and a boundary connection between any two vertices of a face comprising an edge of the face and each face being connected to another adjacent face in the mesh through a shared edge and the shared vertices creating that shared edge, the data point to be inserted being one of a subset of the plurality of data points that have not yet been inserted into the mesh, the method comprising the steps of: (a) identifying which one of the unmeshed data points lies farthest from the mesh by locating for each unmeshed data point the mesh face which is closest in distance to that data point and comparing that distance with the distances obtained for the other unmeshed points; (b) inserting the data point into the mesh by reorganizing the arrangement of vertices of the face that defines the mesh and including the data point for insertion in that rearrangement to create a new set of faces from those points, each new face having the predetermined number of vertices, and the vertices and edges of the new faces arranged to create a continuous surface with the other faces of the mesh; (c) checking the set of newly created faces to determine whether their construction meets a predetermined optimality criteria; and (d) rearranging the boundaries of the newly created faces when one of the newly created faces does not meet the predetermined optimality criteria.

41. A method for increasing resolution of a mesh model of an object or surface, the mesh created from a plurality of data points comprising coordinates that describe the object or surface and further constructing the mesh using the data points, the method comprising the steps of: (a) identifying from the unmeshed data points the next point of detail to insert into the mesh by determining which unmeshed point lies farthest in distance from the mesh; and (b) connecting the data point for insertion into the mesh and follow a predetermined path using the mesh structure to organize the mesh according to a predetermined optimality criteria.

42. The method of claim 41 comprising the further step of identifying the next data point for insertion by finding the data point which lies farthest from the mesh, inserting that point into the mesh, and reordering the unmeshed data points to find the next point to insert until the distance of all the remaining unmeshed data points falls below a predetermined tolerance value.

43. A computer based method for checking optimality, based on Delaunayian principles, of a mesh model after data points have been inserted, the computer including a processor coupled to a memory, configured to accept as input a plurality of data points, with each data point being a coordinate describing the object or surface to be modeled, and the computer further configured to construct a mesh to model the object or surface, the method comprising the steps of: (a) organizing a data structure to describe the mesh faces such that each data point in the mesh is a vertex of at least one triangular face of the mesh, each face of the mesh has its vertices ordered sequentially to have first, second and third vertices with a boundary connection between any two adjacent vertices of a face comprising an edge of the face and any face being connected to another adjacent, neighboring face through a shared edge and the pair of shared vertices creating that shared edge, the neighbors of the face being ordered following the sequential order corresponding to the ordering of the face's vertices with a first neighbor face, second neighbor face and third neighbor face; (b) inserting a new data point into an existing face and alter the data structure such that the existing face is reorganized to create a set of three new faces, each having the new data point as one vertex and the remaining two vertices being vertices of the existing face, the vertices of the new faces being ordered in a sequence consistent with the existing face's vertices, where the new data point is the first ordered vertex in each of the three new faces, and the neighbors of the new faces being ordered following a predetermined sequential order corresponding to the ordering of the new face's vertices; (c) checking one of the new faces created by the insertion of the new data point for Delaunayian optimality, by creating a circumcircle from the vertices of the face, and checking the face's first neighbor for Delaunayian optimality; (d) reorienting the vertices of the first neighbor face in the data structure, if the combination does not pass Delaunayian optimality, so that the vertices of the first neighbor are ordered such that the first vertex is opposite the face being checked; (e) rearranging the shared edge between the face being checked and the first neighbor, if the combination does not pass Delaunayian optimality, the rearrangement occurring by redefining the vertices for the first neighbor face by setting the first vertex to be the first vertex of the face being checked, setting the second vertex to be the first neighbor's former first vertex and setting the third vertex to be the first neighbor's former second vertex and redefining the vertices of the face being checked to change its third vertex to be the former first vertex of the first neighbor; (f) redefining the neighbors of the rearranged faces in the data structure based on the new ordering of the vertices; and (g) rechecking the face previously being checked if it was rearranged, the checking on the face proceeding until it passes the Delaunayian optimality criteria.

44. The method of claim 43 comprising the further step of proceeding after completing a Delaunayian optimality check on a face, to the first neighbor of the face and performing a Delaunayian optimality check on the first neighbor, the computer proceeding thereafter to each succeeding first neighbor face and performing a Delaunayian optimality check at each said face until it arrives at the first face checked.

45. The method of claim 43 comprising the further step of projecting the 3D mesh model onto a 2D reference object creating a 2D triangulated mesh on the reference object that is representative of the original 3D mesh model, and performing the checking on the 2D projected mesh.

46. The method of claim 43 comprising the further step of retaining the equation of the circumcircle for a particular face on a list with reference to a particular face and using that stored equation when that face is checked a second time.

47. The method of claim 43 where the reorientation of the first neighbor's vertices is accomplished by assigning the values 0,1,2 to the first, second and third vertices and neighbors respectively, storing a variable#(k) equal to the neighbor value of the face being checked in the data structure of the first neighbor face, the first neighbor's vertices are redefined such that the vertex k value is assigned to the original vertex (3-k), the original vertex (3-k) value is assigned to the original vertex with value 0, and the value 0 is assigned to the original vertex k, resulting in a reorientation of the first neighbor vertices such that the vertex assigned value 0 is the first vertex, the vertex assigned value 1 is the second vertex, and vertex assigned value 2 is the third vertex.

48. A method to operate a computer to increase and decrease the resolution of a mesh model of an object or surface, the computer including a processor coupled to a memory, adapted to accept as input a plurality of data points comprising coordinates that describe the object or surface, and construct a mesh to model the surface or object, the method comprising the steps of: (a) storing information on a history list, identifying a sequential list of entries comprising the sequence of operations executed by the processor during a mesh construction, to generate a mesh model containing the complete set of data points describing the object or surface; (b) setting a data pointer to an initial location in the sequence of entries in the history list; (c) advancing through the entries of the history list and building a mesh model following the sequence of operations on the history list; and (d) moving the data pointer in reverse order through the sequence of entries in the history list to remove data points from the mesh model.

49. A computer based method for ordering a plurality of data points to be inserted into a mesh model of an object or surface comprised of geometric faces, the computer comprising a processor to manipulate the order of the data points and a memory to store the order of the data points, each data point being a coordinate describing the object or surface to be modeled, the method comprising the steps of: (a) determining the mesh face which lies closest to each unmeshed data point; (b) determining the distance each data point lies from its associated face; and (c) determining based on the distances the data point that is farthest from the mesh.

50. A computer based method for ordering a plurality of data points to be inserted into a mesh model of an object or surface, the computer comprising a processor and a memory, the mesh model comprising a set of connected faces, each face being a geometric shape with a predetermined number of vertices, with each face being connected to another adjacent, neighboring face through a shared edge of adjacent vertices, the method comprising the steps of: (a) locating the closest mesh face to each data point by drawing a line from the center point of a current face to the data point, and determining if a projection of that line intersects with the edge of the current face; (b) determining if no intersection exists that the data point is associated with the current face; (c) moving from the current face, if there is an intersection with an edge, to the face which shares the edge of intersection with the original face and perform the same check recursively until the closest mesh face is determined when no intersection exists; (d) determining the distance from each data point to its closest face; (e) ordering the data points based on the distances to locate the data point that is farthest from the mesh; and (f) re-executing the locate and distance determination procedures after each data point insertion.