Methods for determining if boundary distance constraints between shapes are satisfied

Abstract

This disclosure describes methods for determining if arrangements of shapes satisfy exact or approximate analytical function representations of boundary distance constraints among those shapes. The first embodiment includes the following steps: (a) constructing a function which is an exact or approximate analytical representation of the constraint between two shapes; and (b) evaluating the function of step (a) to determine if the constraint between the two shapes is exactly or approximately satisfied. The second embodiment includes the following steps: (a) constructing a function which is an exact or approximate analytical representation of the constraint between two shapes; (b) assigning values to variables and/or parameters of the function constructed in step (a); and (c) evaluating the function of step (a) using the assignments of step (b) to determine if the constraint between the two shapes is exactly or approximately satisfied.

I claim the following invention: 
     
         1 . A method to manufacture one or more articles of manufacture that determines if a boundary distance constraint between two shapes is exactly or approximately satisfied comprising:
 (a) constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more external boundary surface of closest approach functions where at least one of the external boundary surfaces of closest approach is a non-n-sphere and each external boundary surface of closest approach is formed from two shapes and, optionally, additional boundary distance constraints between the shapes and is determined using the addition of two or three functions where one of the three functions represents the first shape and another function represents the second shape and, optionally, a third function represents the additional boundary distance constraints between the shapes;   (b) evaluating the function of step (a) to determine if the constraint between the two shapes is exactly or approximately satisfied.   
     
     
         2 . The claim according to  claim 1  wherein said step (a) further comprises constructing functions using (1) one or more analytical compositions and, (2) one or more superellipsoid shape function approximations for the interior or exterior boundary surface of closest approach between a pair of axis aligned orthotopes with axis-aligned orthotopic boundary distance constraints. 
     
     
         3 . The claim according to  claim 1  wherein said step (a) further comprises constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more analytical compositions. 
     
     
         4 . The claim according to  claim 1  wherein said step (a) further comprises constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more superellipsoid shape function approximations for the interior or exterior boundary surface of closest approach between a pair of axis aligned orthotopes with axis-aligned orthotopic boundary distance constraints. 
     
     
         5 . A method to manufacture one or more articles of manufacture that determines if a boundary distance constraint between two shapes is exactly or approximately satisfied comprising:
 (a) constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more external boundary surface of closest approach functions where at least one of the external boundary surfaces of closest approach is a non-n-sphere and each external boundary surface of closest approach is formed from two shapes and, optionally, additional boundary distance constraints between the shapes and is determined using the addition of two or three functions where one of the three functions represents the first shape and another function represents the second shape and, optionally, a third function represents the additional boundary distance constraints between the shapes;   (b) assigning values to variables and/or parameters of the function constructed in step (a);   (c) evaluating the function of step (a) using the assignments of step (b) to determine if the constraint between the two shapes is exactly or approximately satisfied.   
     
     
         6 . The claim according to  claim 5  wherein said step (a) further comprises constructing functions using (1) one or more analytical compositions and, (2) one or more superellipsoid shape function approximations for the interior or exterior boundary surface of closest approach between a pair of axis aligned orthotopes with axis-aligned orthotopic boundary distance constraints. 
     
     
         7 . The claim according to  claim 5  wherein said step (a) further comprises constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more analytical compositions. 
     
     
         8 . The claim according to  claim 5  wherein said step (a) further comprises constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more superellipsoid shape function approximations for the interior or exterior boundary surface of closest approach between a pair of axis aligned orthotopes with axis-aligned orthotopic boundary distance constraints. 
     
     
         9 . A non-transitory program storage device readable by a computing device that tangibly embodies a program of instructions executable by the computing device to perform a method to manufacture one or more articles of manufacture that determines if a boundary distance constraint between two shapes is exactly or approximately satisfied comprising:
 (a) constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more external boundary surface of closest approach functions where at least one of the external boundary surfaces of closest approach is a non-n-sphere and each external boundary surface of closest approach is formed from two shapes and, optionally, additional boundary distance constraints between the shapes and is determined using the addition of two or three functions where one of the three functions represents the first shape and another function represents the second shape and, optionally, a third function represents the additional boundary distance constraints between the shapes;   (b) evaluating the function of step (a) to determine if the constraint between the two shapes is exactly or approximately satisfied.   
     
     
         10 . The claim according to  claim 9  wherein said step (a) further comprises constructing functions using (1) one or more analytical compositions and, (2) one or more superellipsoid shape function approximations for the interior or exterior boundary surface of closest approach between a pair of axis aligned orthotopes with axis-aligned orthotopic boundary distance constraints. 
     
     
         11 . The claim according to  claim 9  wherein said step (a) further comprises constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more analytical compositions. 
     
     
         12 . The claim according to  claim 9  wherein said step (a) further comprises constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more superellipsoid shape function approximations for the interior or exterior boundary surface of closest approach between a pair of axis aligned orthotopes with axis-aligned orthotopic boundary distance constraints. 
     
     
         13 . A non-transitory program storage device readable by a computing device that tangibly embodies a program of instructions executable by the computing device to perform a method to manufacture one or more articles of manufacture that determines if a boundary distance constraint between two shapes is exactly or approximately satisfied comprising:
 (a) constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more external boundary surface of closest approach functions where at least one of the external boundary surfaces of closest approach is a non-n-sphere and each external boundary surface of closest approach is formed from two shapes and, optionally, additional boundary distance constraints between the shapes and is determined using the addition of two or three functions where one of the three functions represents the first shape and another function represents the second shape and, optionally, a third function represents the additional boundary distance constraints between the shapes;   (b) assigning values to variables and/or parameters of the function constructed in step (a);   (c) evaluating the function of step (a) using the assignments of step (b) to determine if the constraint between the two shapes is exactly or approximately satisfied.   
     
     
         14 . The claim according to  claim 13  wherein said step (a) further comprises constructing functions using (1) one or more analytical compositions and, (2) one or more superellipsoid shape function approximations for the interior or exterior boundary surface of closest approach between a pair of axis aligned orthotopes with axis-aligned orthotopic boundary distance constraints. 
     
     
         15 . The claim according to  claim 13  wherein said step (a) further comprises constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more analytical compositions. 
     
     
         16 . The claim according to  claim 13  wherein said step (a) further comprises constructing a function which is an exact or approximate analytical representation of the constraint between two shapes using one or more superellipsoid shape function approximations for the interior or exterior boundary surface of closest approach between a pair of axis aligned orthotopes with axis-aligned orthotopic boundary distance constraints.