Exact evaluation of subdivision surfaces generalizing box splines at arbitrary parameter values

A computer-based method of evaluating any location of a computer generated surface, the location being specified by a set of parameters, the surface being described by a set of control vertices having a corresponding set of subdivision rules and admitting a parameterization of regular sets of control vertices, one of the control vertices being a vertex having an extraordinary valence, the method comprising:

• receiving input specifying coordinates of the control vertices;
• determining whether one of the control vertices comprises a vertex having an extraordinary valence;
• performing an exact surface evaluation of any desired location of the surface as a function of the control vertices using predefined eigenbasis functions without explicitly subdividing the control vertices in accordance with the subdivision rules;
• and
• storing the evaluated location in a computer memory.

A computer program product, disposed on a computer readable medium, for evaluating a property of any location on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices having a corresponding set of subdivision rules and admitting a regular parameterization for regular patches corresponding to regular sets of control vertices, the computer program product comprising instructions for causing a computer to:

• receive input specifying coordinates of the control vertices defining a patch of the surface;
• determine whether one of the control vertices comprises a vertex having an extraordinary valence;
• project the specified control vertices into a space of predefined eigenbasis functions corresponding to eigenvectors derived from a matrix representation of the subdivision rules;
• exactly evaluate any desired location on the patch, without the need to modify the patch by subdividing, as a function of a valence of one of the control vertices, the location in the parameter space of the patch, and the set of projected control vertices;
• and
• store the evaluated location in a computer memory.

A computer-based method for evaluating any location on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices having a corresponding set of subdivision rules, the method comprising:

• receiving input specifying coordinates of the control vertices defining a patch of the surface;
• determining whether one of the control vertices comprises a vertex having an extraordinary valence;
• performing an exact surface evaluation of any desired location on the patch without subdividing the patch using predefined eigenbasis functions;
• and
• storing the evaluated location in a computer memory.

A computer-based method for evaluating any point on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices, the method comprising:

• receiving input specifying coordinates of the control vertices defining a patch of the surface;
• determining whether one of the control vertices comprises a vertex having an extraordinary valence;
• performing an exact surface evaluation of any desired point on the patch without the need to modify the patch by subdividing using predefined eigenbasis functions;
• and
• storing the evaluated location in a computer memory.

A computer apparatus for evaluating any point on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices, the computer apparatus comprising:

• an input device to receive input specifying coordinates of the control vertices defining a patch of the surface;
• a processor programmed to evaluate any desired point on the patch without the need to modify the patch by subdividing by determining whether one of the control vertices comprises a vertex having an extraordinary valence and performing an exact surface evaluation of the point on the patch using predefined eigenbasis functions;
• and
• a storage device for storing the evaluated location.

A computer apparatus for evaluating any location on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices having a corresponding set of subdivision rules, the method comprising:

• an input device to receive input specifying coordinates of the control vertices defining a patch of the surface;
• a processor programmed to evaluate any location on the patch without subdividing the patch by determining whether one of the control vertices comprises a vertex having an extraordinary valence and performing an exact surface evaluation of the point on the patch using predefined eigenbasis functions;
• and
• a storage device for storing the evaluated location.

A computer-based method for evaluating a property of any location on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices having a corresponding set of subdivision rules and admitting a regular parameterization for regular patches corresponding to regular sets of control vertices, the method comprising:

• receiving input specifying coordinates of the control vertices defining a patch of the surface;
• determining whether one of the control vertices comprises a vertex having an extraordinary valence;
• projecting the specified control vertices into a space of predefined eigenbasis functions corresponding to eigenvectors derived from a matrix representation of the subdivision rules;
• performing an exact surface evaluation of any desired location on the patch, without the need to modify the patch by subdividing, as a function of a valence of one of the control vertices, the location in the parameter space of the patch, and the set of projected control vertices;
• storing the evaluated location in a computer memory;
• and
• wherein the evaluating comprises evaluating according substantially to s ⁡ ( u , v ) = ∑ i = 1 K ⁢ ϕ i ⁡ ( u , v ) ⁢ ρ i .

A computer-based method for evaluating a property of any location on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices having a corresponding set of subdivision rules and admitting a regular parameterization for regular patches corresponding to regular sets of control vertices, the method comprising:

• receiving input specifying coordinates of the control vertices defining a patch of the surface;
• determining whether one of the control vertices comprises a vertex having an extraordinary valence;
• projecting the specified control vertices into a space of predefined eigenbasis functions corresponding to eigenvectors derived from a matrix representation of the subdivision rules;
• performing an exact surface evaluation of any desired location on the patch, without the need to modify the patch by subdividing, as a function of a valence of one of the control vertices, the location in the parameter space of the patch, and the set of projected control vertices;
• storing the evaluated location in a computer memory;
• and
• wherein the evaluating comprises evaluating according substantially to
• s(u,w)=Ĉ0TΦ(u, w).

A computer program product, disposed on a computer readable medium, for evaluating a property of any location on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices having a corresponding set of subdivision rules and admitting a regular parameterization for regular patches corresponding to regular sets of control vertices, the computer program product comprising instructions for causing a computer to:

• receive input specifying coordinates of the control vertices defining a patch of the surface;
• project the specified control vertices into a space of predefined eigenbasis functions corresponding to eigenvectors derived from a matrix representation of the subdivision rules;
• evaluate any location on the patch, without the need to modify the patch, as a function of a valence of one of the control vertices, the location in the parameter space of the patch, and the set of projected control vertices, wherein one of the control vertices comprises a vertex having an extraordinary valence;
• and
• store the evaluated location in a computer memory;
• and
• wherein the instruction that cause the computer to evaluate comprise instructions that cause the computer to evaluate according substantially to:
• s ⁡ ( u , v ) = ∑ i = 1 K ⁢ ϕ i ⁡ ( u , v ) ⁢ ρ i .

A computer program product, disposed on a computer readable medium, for evaluating a property of any location on a computer generated surface, the surface being specified by a set of patches, each patch being defined by a set of control vertices having a corresponding set of subdivision rules and admitting a regular parameterization for regular patches corresponding to regular sets of control vertices, the computer program product comprising instructions for causing a computer to:

• receive input specifying coordinates of the control vertices defining a patch of the surface;
• determine whether one of the control vertices comprises a vertex having an extraordinary valence;
• project the specified control vertices into a space of predefined eigenbasis functions corresponding to eigenvectors derived from a matrix representation of the subdivision rules;
• exactly evaluate any location on the patch, without the need to modify the patch by subdividing, as a function of a valence of one of the control vertices, the location in the parameter space of the patch, and the set of projected control vertices;
• store the evaluated location in a computer memory;
• and
• wherein the instruction that cause the computer to evaluate comprise instructions that cause the computer to evaluate according substantially to:
• s(v,w)=Ĉ0TΦ(v,w).