System and method for generating a low discrepancy curve on an abstract surface
View Patent ↗A system and method for generating a curve, such as a Low Discrepancy Curve, on a surface, such as an abstract surface with a Riemannian metric. The system may comprise a computer which includes a CPU and a memory medium which is operable to store one or more programs executable by the CPU to perform the method. The method may: 1) parameterize the surface; 2) select a curve, such as a Low Discrepancy Curve, in a parameter space, for example, a simple space such as a unit square; 3) re-parameterize the surface, for example, re-parameterize the surface such that a ratio of line and area elements of the surface based on a Riemannian metric is constant; and 4) map the curve onto the surface using the re-parameterization. The method may also generate output comprising information regarding the mapped curve, for example, displaying the mapped curve on a display device.
1. A method for generating a curve on a surface, comprising:
selecting a parameterization of the surface, wherein the parameterization corresponds to a parameter space;
selecting a first curve in the parameter space, wherein the first curve is a Low Discrepancy Curve in the parameter space;
determining a re-parameterization of the surface based on a metric of the surface; and
mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve, wherein the generated curve is a Low Discrepancy Curve on the surface;
wherein the curve is useable in analyzing the surface.
2. The method of claim 1 , further comprising:
generating output comprising the generated curve on the surface.
3. The method of claim 1 , wherein the parameter space comprises a 2D rectangle.
4. The method of claim 1 , wherein the parameter space comprises a unit n-dimensional cube, wherein the dimensionality n is greater than 4.
5. The method of claim 1 , wherein the parameter space comprises one of a unit square, a unit cube, or a unit hyper-cube.
6. The method of claim 1 , wherein the metric of the surface is a Riemannian metric.
7. The method of claim 1 , wherein the surface comprises a space with dimensionality greater than two.
8. The method of claim 1 ,
wherein the surface comprises a six-dimensional space comprising three position degrees of freedom and three orientation degrees of freedom for an object; and
wherein the generated curve comprises a scan path useable in scanning the space.
9. The method of claim 8 , further comprising:
scanning the space, wherein said scanning the space is performed to determine an optimum position and orientation for the object.
10. The method of claim 1 , wherein the parameter space comprises a 2D rectangle.
11. The method of claim 1 , wherein the parameter space comprises one of a unit square, a unit cube, or a unit hyper-cube.
12. The method of claim 1 , wherein the parameter space comprises a unit n-dimensional cube, wherein the dimensionality n is greater than 4.
13. The method of claim 1 , wherein the metric of the surface is a Riemannian metric.
14. The method of claim 1 , wherein the surface comprises a space with dimensionality greater than two.
15. A method for generating a curve on a surface of an object, comprising:
selecting a parameterization of the surface, wherein the parameterization corresponds to a parameter space;
selecting a first curve in the parameter space, wherein the first curve is a Low Discrepancy Curve in the parameter space;
determining a re-parameterization of the surface based on a metric of the surface; and
mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve, wherein the generated curve is a Low Discrepancy Curve on the surface;
wherein the curve is useable in analyzing the object.
16. The method of claim 15 , further comprising:
generating output comprising the generated curve on the surface.
17. The method of claim 15 , further comprising:
analyzing the object using the curve.
18. The method of claim 11 , further comprising:
determining a property of the object using the curve.
19. A memory medium which is operable to store program instructions for generating a curve on a surface, wherein said program instructions are executable to perform:
storing a parameterization of the surface, wherein the parameterization corresponds to a parameter space;
storing a first curve in the parameter space, wherein the first curve is a Low Discrepancy Curve in the parameter space;
determining a re-parameterization of the surface based on a metric of the surface; and
mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve, wherein the generated curve is a Low Discrepacy Curve on the surface;
wherein the curve is useable in analyzing the surface.
20. The memory medium of claim 19 , wherein the program instructions are further executable to perform:
generating output comprising the generated curve on the surface.
21. The memory medium of claim 19 , wherein the parameter space comprises one of a 2D rectangle, a unit square, a unit cube, or a unit hyper-cube.
22. The memory medium of claim 19 , wherein the parameter space comprises a unit n-dimensional cube, wherein the dimensionality n is greater than 4.
23. The memory medium of claim 19 , wherein the metric of the surface is a Riemannian metric.
24. The memory medium of claim 19 , wherein the surface comprises a space with dimensionality greater than two.
25. The memory medium of claim 19 ,
wherein the surface comprises a surface of an object;
wherein the curve comprises a scan path useable to scan the object;
wherein the program instructions are further executable to perform:
scanning the object to determine one or more properties of the object.
26. The memory medium of claim 19 ,
wherein the surface comprises a six-dimensional space comprising three position degrees of freedom and three orientation degrees of freedom for an object;
wherein the generated curve comprises a scan path useable in scanning the six-dimensional space;
wherein the program instructions are further executable to perform:
scanning the six-dimensional space to determine an optimum position and orientation of the object.
27. A system for generating a curve on a surface, comprising:
a CPU; and
a memory medium which is operable to store one or more software programs;
wherein said CPU is operable to execute said one or more software programs to perform:
selecting a parameterization of the surface, wherein the parameterization corresponds to a parameter space;
selecting a first curve in the parameter space, wherein the first curve is a Low Discrepancy Curve in the parameter space;
determining a re-parameterization of the surface based on a metric of the surface; and
mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve, wherein the generated curve is a Low Discrepancy Curve on the surface;
wherein the curve is useable in analyzing the surface.
28. The system of claim 27 , wherein the CPU is further operable to execute said one or more software programs to perform:
generating output comprising the generated curve on the surface.
29. The system of claim 27 , wherein the parameter space comprises one of a 2D rectangle, a unit square, a unit cube, a unit hyper-cube, or a unit n-dimensional cube, wherein the dimensionality n is greater than 4.
30. The system of claim 27 , wherein the metric of the surface is a Riemannian metric.
31. The system of claim 27 , wherein the surface comprises a space with dimensionality greater than two.
32. A method for generating a curve on a surface, comprising:
selecting a parameterization of the surface, wherein the surface comprises a space with dimensionality greater than two, and wherein the parameterization corresponds to a parameter space;
selecting a first curve in the parameter space;
determining a re-parameterization of the surface based on a metric of the surface; and
mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve;
wherein the curve is useable in analyzing the surface.
33. A method for generating a curve on a surface of an object, comprising:
selecting a parameterization of the surface, wherein the surface comprises a space with dimensionality greater than two, wherein the parameterization corresponds to a parameter space;
selecting a first curve in the parameter space;
determining a re-parameterization of the surface based on a metric of the surface; and
mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve;
wherein the curve is useable in analyzing the object.
34. A memory medium which is operable to store program instructions for generating a curve on a surface, wherein said program instructions are executable to perform:
storing a parameterization of the surface, wherein the surface comprises a space with dimensionality greater than two, wherein the parameterization corresponds to a parameter space;
storing a first curve in the parameter space;
determining a re-parameterization of the surface based on a metric of the surface; and
mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve;
wherein the curve is useable in analyzing the surface.
35. A system for generating a curve on a surface, comprising:
a CPU; and
a memory medium which is operable to store one or more software programs;
wherein said CPU is operable to execute said one or more software programs to perform:
selecting a parameterization of the surface, wherein the surface comprises a space with dimensionality greater than two, wherein the parameterization corresponds to a parameter space;
selecting a first curve in the parameter space;
determining a re-parameterization of the surface based on a metric of the surface; and
mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve;
wherein the curve is useable in analyzing the surface.