Method and system for computing a transverse derivative field along a curve on a surface
View Patent ↗According to one embodiment of the invention, a computerized method for computing a transversal derivative field along a curve on a surface includes receiving data defining the surface, the surface represented in terms of parameters u and v as S(u,v), and receiving data defining the curve, the curve represented in terms of a parameter t as c(t). The method also includes receiving a number equal to N constraint conditions for the derivative field, wherein N is a positive integer and determining the transversal derivative L(t) field according to the formula L ( t ) = ∂ S ( u ( t ) , v ( t ) ) ∂ w , where i is an index identifying a particular one of N numbers of constraint conditions and L(t i ) is the value of the transversal derivative field at the ith constraint condition and w is a scalar function of u and v that satisfies ∂ S ( u ( t i ) , v ( t i ) ) ∂ w = L ( t i ) .
1. A computerized method for computing a transversal derivative field along a curve on a surface comprising:
receiving data defining the surface, the surface represented in terms of parameters u and v as S(u,v);
receiving data defining the curve, the curve represented in terms of a parameter t as c(t);
receiving a number equal to N constraint conditions for the derivative field, wherein N is a positive integer;
determining a scalar function w(u,v) according to the constraint formula
L
(
t
i
)
=
∂
S
(
u
(
t
i
)
,
v
(
t
i
)
)
∂
w
,
wherein i is an index identifying a particular one of N numbers of constraint conditions and L(t i ) is the value of the transversal derivative field at the ith constraint condition; and
determining by a computer the transversal derivative L(t) field according to the formula
∂
S
(
u
(
t
)
,
v
(
t
)
)
∂
w
=
L
(
t
)
.
wherein:
u,v, and t are real numbers:
S(u,v), w(u,v), c(t), and L(t) are real value functions; and
u(t) and v(t) are real value functions of the curve parameter t that relate the
parameterization of the curve t and the parameterization of the surface S(u,v).
2. The method of claim 1 , and further comprising displaying the transversal derivative field on a computer display.
3. The method of claim 1 , and further comprising using the determined transversal derivative field to generate a computer image.
4. The method of claim 1 , wherein the scalar function is a NURBS function.
5. The method of claim 4 , and further comprising selecting a proper degree and knot vector for the NURBS function.
6. The method of claim 1 , wherein the scalar function is a spline.
7. The method of claim 6 , and further comprising selecting a proper degree and knot vector for the spline.
8. The method of claim 1 , and further comprising computing a derivative of the transversal derivative field with respect to the scalar function for use in generating a computer image.
9. A computerized method for computing a transversal derivative field along a curve on a surface comprising:
receiving data defining the surface, the surface represented in terms of parameters u and v as S(u,v);
receiving data defining the curve, the curve represented in terms of a parameter t as c(t);
receiving a number equal to N constraint conditions for the derivative field, wherein N is a positive integer;
determining by a computer the transversal derivative L(t) field according to the formula
L
(
t
)
=
∂
S
(
u
(
t
)
,
v
(
t
)
)
∂
w
,
wherein w is any scalar function of either u and v, or t, that satisfies at least N constraint equations:
∂
S
(
u
(
t
i
)
,
v
(
t
i
)
)
∂
w
=
L
(
t
i
)
where i is an index identifying a particular one of N numbers of the constraint conditions and L(t i ) is the value of the transversal derivative field at the ith constraint; and
wherein:
u,v, and t are real numbers;
S(u,v), w(u,v), c(t), and L(t) are real value functions; and
u(t) and v(t) are real value functions of the curve parameter t that relate the
parameterization of the curve t and the parameterization of the surface S(u,v).
10. The method of claim 9 , and further comprising displaying the transversal derivative field on a computer display.
11. The method of claim 9 , and further comprising using the determined transversal derivative field to generate a computer image.
12. The method of claim 9 , wherein the scalar function is a NURBS function.
13. The method of claim 12 , and further comprising selecting a proper degree and knot vector for the NURBS function.
14. The method of claim 9 , wherein the scalar function is a spline.
15. The method of claim 14 , and further comprising selecting a proper degree and knot vector for the spline.
16. The method of claim 9 , and further comprising computing a derivative of the transversal derivative field with respect to the scalar function for use in generating a computer image.
17. An apparatus comprising a logic encoded in a computer-readable medium operable, when executed on a processor, to:
receive data defining the surface, the surface represented in terms of parameters u and v as S(u,v);
receive data defining the curve, the curve represented in terms of a parameter t as c(t);
receive a number equal to N constraint conditions for the derivative field, wherein N is a positive integer;
determine the transversal derivative L(t) field according to the formula
L
(
t
)
=
∂
S
(
u
(
t
)
,
v
(
t
)
)
∂
w
,
wherein w is any scalar function of either u and v, or t, that satisfies at least N constraint equations:
∂
S
(
u
(
t
i
)
,
v
(
t
i
)
)
∂
w
=
L
(
t
i
)
where i is an index identifying a particular one of N numbers of constraint conditions and L(t i ) is the value of the transversal derivative field at the ith constraint; and
wherein:
u,v, and t are real numbers;
S(u,v), w(u,v), c(t), and L(t) are real value functions; and
u(t) and v(t) are real value functions of the curve parameter t that relate the
parameterization of the curve t and the parameterization of the surface S(u,v).
18. The logic of claim 17 , wherein the logic is further operable to display the transversal derivative field on a computer display.
19. The logic of claim 17 , wherein the logic is further operable to use the determined transversal derivative field to generate a computer image.
20. The logic of claim 17 , wherein the scalar function is a NURBS function.
21. The logic of claim 20 , wherein the logic is further operable to select a proper degree and knot vector for the NURBS function.
22. The logic of claim 17 , wherein the scalar function is a spline.
23. The logic of claim 22 , wherein the logic is further operable to select a proper degree and knot vector for the spline.
24. The logic of claim 17 , wherein the logic is further operable to compute a derivative of the transversal derivative field with respect to the scalar function for use in generating a computer image.
25. A computer system comprising:
a processor;
a computer-readable medium accessible by the processor;
a computer program encoded on the computer-readable medium operable, when executed on the processor, to:
receive data defining the surface, the surface represented in terms of parameters u and v as S(u,v);
receive data defining the curve, the curve represented in terms of a parameter t as c(t);
receive a number equal to N constraint conditions for the derivative field wherein N is a positive integer;
determine the transversal derivative L(t) field according to the formula
L
(
t
)
=
∂
S
(
u
(
t
)
,
v
(
t
)
)
∂
w
,
wherein w is any scalar function of either u and v, or t, that satisfies at least N constraint equations:
∂
S
(
u
(
t
i
)
,
v
(
t
i
)
)
∂
w
=
L
(
t
i
)
where i is an index identifying a particular one of N numbers of constraint conditions and L(t i ) is the value of the transversal derivative field at the ith constraint; and
wherein:
u,v, and t are real numbers;
S(u,v), w(u,v), c(t), and L(t) are real value functions; and
u(t) and v(t) are real value functions of the curve parameter t that relate the
parameterization of the curve t and the parameterization of the surface S(u,v).
26. The computer system of claim 25 , wherein the computer program is further operable to display the transversal derivative field on a computer display.
27. The computer system of claim 25 , wherein the computer program is further operable to use the determined transversal derivative field to generate a computer image.
28. The computer system of claim 25 , wherein the scalar function is a NURBS function.
29. The computer system of claim 25 , wherein the scalar function is a spline.
30. The computer system of claim 29 , wherein the computer program is further operable to select a proper degree and knot vector for the spline.
31. The computer system of claim 25 , wherein the computer program is further operable to compute a derivative of the transversal derivative field with respect to the scalar function for use in generating a computer image.