US5361386A
Expired / Lapsed
US
⏱️ 0.0 years remaining
System for polygon interpolation using instantaneous values in a variable
Assignee: Nvidia Corp
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Filing: 1993-08-17
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Granted: 1994-11-01
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Expires: 2013-08-17
Abstract
Abstract Barycentric coordinates are determined using data readily available in computer graphics systems and defining polygons in world space so as to interpolate such polygons with regard to pixels selected by scan conversion. The barycentric coordinates are employed to interpolate the polygons, for example to accomplish smooth shading or texture mapping. The process of the system involves defining primitive data (e.g. polygons) in a space not subject to nonlinear distortion (e.g. world space), transforming select primitive data to a space that is subject to nonlinear distortion, calculating in the space subject to nonlinear distortion (e.g. screen space) a set of areas (e.g. pixels) within a defined primitive (e.g. polygon) using data transformed to space subject to nonlinear distortion, transforming the set of areas to the space not subject to nonlinear distortion, determining at least one differential value (rate of change) of a variable for the primitive, calculating display data that is not subject to nonlinear distortion and using the differential to provide display data. For example, at a pixel, the rate of change in coordinate value (differential) is determined and utilized in texture mapping to facilitate the selection of levels of fuzziness with regard to picture detail for the pixel. Specifically, the differentials are in the coordinates of the texture du land dv.
Inventors
- 👤 Gary S. Watkins
- 👤 Russell A. Brown
Bibliographic Details
Application Number
—
Jurisdiction
US
Data Source
google_patents
Last Synchronized
2026-11-01 19:11
CPC Classifications
G; G06; G06T; G06T15/00; G06T15/50; G06T15/80
Claims Analysis
Total Claims: 58
Claim 1
Independent Claim
185 words
A process for producing a computer graphics image in accordance with a defined characteristic in screen space based on primitive data defining primitives in space coordinates not subject to nonlinear distortion, said process comprising the steps of: processing select of said primitive data to provide processed data representing certain specific of said primitives for said graphics image in space coordinates subject to nonlinear distortion; from said processed data determining pixel image areas of said certain specific primitives for said graphics image, said pixel image areas being represented by pixel image area data; transforming said pixel image area data to representative area data in said space coordinates not subject to nonlinear distortion; determining at least one differential value for at least one variable specified in said space coordinates not subject to nonlinear distortion with respect to said primitive data, said differential value for selecting a scale for said defined characteristic from storage; generating display data using said selected scale for said graphics image based on said representative area data and said differential value; and displaying said graphics images in said space coordinates subject to nonlinear distortion.
Claim 2
Dependent Claim
185 words
A process for producing a computer graphics image in accordance with a defined characteristic in screen space based on primitive data defining primitives in space coordinates not subject to nonlinear distortion, said process comprising the steps of: processing select of said primitive data to provide processed data representing certain specific of said primitives for said graphics image in space coordinates subject to nonlinear distortion; from said processed data determining pixel image areas of said certain specific primitives for said graphics image, said pixel image areas being represented by pixel image area data; transforming said pixel image area data to representative area data in said space coordinates not subject to nonlinear distortion; determining at least one differential value for at least one variable specified in said space coordinates not subject to nonlinear distortion with respect to said primitive data, said differential value for selecting a scale for said defined characteristic from storage; generating display data using said selected scale for said graphics image based on said representative area data and said differential value; and displaying said graphics images in said space coordinates subject to nonlinear distortion.
Claim 3
Dependent Claim
18 words
A process according to claim 1 wherein said space coordinates not subject to nonlinear distortion comprise world space.
Claim 4
Dependent Claim
18 words
A process according to claim 1 wherein said space coordinates not subject to nonlinear distortion comprise eye space.
Claim 5
Dependent Claim
17 words
A process according to claim 1 wherein said space coordinates subject to nonlinear distortion comprise screen space.
Claim 6
Dependent Claim
28 words
A process according to claim 1 wherein said primitive data further includes texture data and wherein said differential value comprises a rate of change in said texture data.
Claim 7
Dependent Claim
39 words
A process according to claim 1 wherein said step of processing select of said primitive data includes identifying a primitive in the form of a polygon for display in said space coordinates subject to nonlinear distortion from a viewpoint.
Claim 8
Dependent Claim
24 words
A process according to claim 1 wherein said step of determining pixel image areas comprises scan converting of said primitives to specify pixel areas.
Claim 9
Dependent Claim
33 words
A process according to claim 1 wherein said step of transforming said pixel image area data to representative area data includes generating values in barycentric coordinates for said certain specific of said primitives.
Claim 10
Dependent Claim
27 words
A process according to claim 8 wherein said step of transforming said pixel image area data to representative area data includes determining texture values u and v.
Claim 11
Dependent Claim
24 words
A process according to claim 9 wherein said step of determining at least one differential value includes determining texture differential values du and dv.
Claim 12
Dependent Claim
27 words
A process according to claim 10 wherein said step of generating display data includes a table of look-up operation indexed by said differential values du and dv.
Claim 13
Dependent Claim
20 words
A process according to claim 11 wherein said table look-up operation comprises three table look-ups for three color component images.
Claim 14
Dependent Claim
35 words
A process according to claim 10 wherein said step of generating display data includes a table look-up operation indexed by said differential values du and dv and addressed by said values of u and v.
Claim 15
Dependent Claim
27 words
A process according to claim 1 wherein said step of transforming said pixel image area data to representative area data includes determining texture values u and v.
Claim 16
Dependent Claim
23 words
A process according to claim 1 wherein said step of generating display data includes a table look-up operation indexed by said differential value.
Claim 17
Dependent Claim
29 words
A process according to claim 15 wherein said one differential value is instantaneous value of texture value u and another differential value is instantaneous value of texture value v.
Claim 18
Dependent Claim
189 words
A component of a computer graphics system for producing dynamic computer graphics images using a display unit based on primitive data defining primitives in space coordinates not subject to nonlinear distortion, said computer graphics system providing processed data representing certain specific of said primitives for said graphics images in space coordinates subject to nonlinear distortion, said component comprising: scan means for receiving said processed data for determining pixel image areas of said certain specific primitives for said graphics images, said pixel image areas being represented by pixel image area data; means for transforming said pixel image area data to representative area data in said space coordinates not subject to nonlinear distortion; means for determining at least one differential value for at least one variable specified in said space coordinates not subject to nonlinear distortion with respect to said primitive data; means for providing a selected a texture scale in accordance with said differential value; means for generating display data using said selected texture scale for said graphics images based on said representative area data and said differential value; and means for providing said display data to said display unit.
Claim 19
Dependent Claim
189 words
A component of a computer graphics system for producing dynamic computer graphics images using a display unit based on primitive data defining primitives in space coordinates not subject to nonlinear distortion, said computer graphics system providing processed data representing certain specific of said primitives for said graphics images in space coordinates subject to nonlinear distortion, said component comprising: scan means for receiving said processed data for determining pixel image areas of said certain specific primitives for said graphics images, said pixel image areas being represented by pixel image area data; means for transforming said pixel image area data to representative area data in said space coordinates not subject to nonlinear distortion; means for determining at least one differential value for at least one variable specified in said space coordinates not subject to nonlinear distortion with respect to said primitive data; means for providing a selected a texture scale in accordance with said differential value; means for generating display data using said selected texture scale for said graphics images based on said representative area data and said differential value; and means for providing said display data to said display unit.
Claim 20
Dependent Claim
27 words
A component according to claim 17 wherein said means for generating display data includes a barycentric coordinate computing network for computing barycentric coordinates for said certain primitives.
Claim 21
Dependent Claim
20 words
A component according to claim 18 wherein said means for transforming includes means for determining texture values u and v.
Claim 22
Dependent Claim
21 words
A component according to claim 19 wherein said means for determining includes means for determining texture differential values du and dr.
Claim 23
Dependent Claim
23 words
A component according to claim 20 wherein said means for generating includes a look-up table indexed by said differential values du and dr.
Claim 24
Dependent Claim
198 words
A computer graphics process of displaying a graphics image by interpolating a graphics object as for shading or texturing pixels in accordance with a defined characteristic, comprising the steps of: mapping, in space coordinates not subject to nonlinear distortion, at least a part of said object as a planar polygon; selecting vertices of a triangular polygon lying within said planar polygon, said vertices of said triangular polygon designated P 1 , P 2 and P 3 ; mapping an eyepoint O in relation to said planar polygon; mapping a perspective plane in relation to said eyepoint and specifying pixel locations in said perspective plane; generating signal-represented barycentric coordinates for said planar polygon at said vertices, in said space coordinates not subject to nonlinear distortion; scan converting said planar polygon to identify points P thereof coinciding to said pixels, said points P being in said space coordinates not subject to nonlinear distortion; interpolating said points P using said barycentric coordinates with respect to said defined characteristic to provide representative signal values; storing said representative signal values for scan presentation in a display; and driving a scan display structure with said representative signal values for scan presentation of an image.
Claim 25
Dependent Claim
198 words
A computer graphics process of displaying a graphics image by interpolating a graphics object as for shading or texturing pixels in accordance with a defined characteristic, comprising the steps of: mapping, in space coordinates not subject to nonlinear distortion, at least a part of said object as a planar polygon; selecting vertices of a triangular polygon lying within said planar polygon, said vertices of said triangular polygon designated P 1 , P 2 and P 3 ; mapping an eyepoint O in relation to said planar polygon; mapping a perspective plane in relation to said eyepoint and specifying pixel locations in said perspective plane; generating signal-represented barycentric coordinates for said planar polygon at said vertices, in said space coordinates not subject to nonlinear distortion; scan converting said planar polygon to identify points P thereof coinciding to said pixels, said points P being in said space coordinates not subject to nonlinear distortion; interpolating said points P using said barycentric coordinates with respect to said defined characteristic to provide representative signal values; storing said representative signal values for scan presentation in a display; and driving a scan display structure with said representative signal values for scan presentation of an image.
Claim 26
Dependent Claim
99 words
A process according to claim 22 wherein said step of generating signal-represented barycentric coordinates includes the steps of: mapping areas A 1 , A 2 and A 3 of triangles defined by one of said points P and said vertices P 1 , P 2 and P 3 , as P 1 PP 3 , P 1 PP 2 and P 2 PP 3 ; specifying depth dimensions of said vertices as z' 1 , z' 2 and z' 3 in said space coordinates not subject to nonlinear distortion; and generating barycentric coordinates for said planar polygon as, ##EQU30##
Claim 27
Dependent Claim
24 words
A process according to claim 22 wherein said interpolating step comprises interpolation of intensity i of said pixels in accordance with said barycentric coordinates.
Claim 28
Dependent Claim
56 words
A process according to claim 60 wherein modulation of intensity i is accomplished by a form of the equation, i=b' 1 i 1 +b' 2 i 2 +b' 3 i 3 where i 1 , i 2 , and i 3 specify intensities at said vertices P 1 , P 2 , and P 3 .
Claim 29
Dependent Claim
23 words
A process according to claim 22 wherein said interpolating step comprises interpolation of multiple components of surface normal vectors at said points P.
Claim 30
Dependent Claim
17 words
A process according to claim 22 wherein said interpolating step comprises interpolation of texture for said pixels.
Claim 31
Dependent Claim
43 words
A process according to claim 27 wherein texture coordinates for said vertices are designated as vectors (tx 1 ,tx 2 ,tx 3 ) and (ty 1 ,ty 2 ,ty 3 ) and said texture of said points P is interpolated as: tx=(b'.sub.1,b'.sub.2,b'.sub.3)·(tx.sub.1,tx.sub.2,tx.sub.3) ty=(b'.sub.1,b'.sub.2,b'.sub.3)·(ty.sub.1,ty.sub.2,ty.sub.3).
Claim 32
Dependent Claim
184 words
A computer graphics system for displaying a graphics image by interpolating a graphics object as for shading or texturing pixels in accordance with a defined characteristic, wherein said object is represented in space coordinates not subject to nonlinear distortion, at least in part by a planar polygon with defined vertices P 1 , P 2 , P 3 and further, wherein an eyepoint O is specified along with a perspective plane, said system comprising: means for generating signal-represented barycentric coordinates for said planar polygon at said vertices, in said space coordinates not subject to nonlinear distortion; means for scan converting said planar polygon to identify points P thereof coinciding to said pixels, said points P being in said space coordinates not subject to nonlinear distortion; means for interpolating said points P using said barycentric coordinates with respect to said defined characteristic to provide representative signal values; means for storing said representative signal values for scan presentation in a display; and means for receiving said representative signal values from said means for storing to display a representation of said planar polygon as a graphics image.
Claim 33
Dependent Claim
184 words
A computer graphics system for displaying a graphics image by interpolating a graphics object as for shading or texturing pixels in accordance with a defined characteristic, wherein said object is represented in space coordinates not subject to nonlinear distortion, at least in part by a planar polygon with defined vertices P 1 , P 2 , P 3 and further, wherein an eyepoint O is specified along with a perspective plane, said system comprising: means for generating signal-represented barycentric coordinates for said planar polygon at said vertices, in said space coordinates not subject to nonlinear distortion; means for scan converting said planar polygon to identify points P thereof coinciding to said pixels, said points P being in said space coordinates not subject to nonlinear distortion; means for interpolating said points P using said barycentric coordinates with respect to said defined characteristic to provide representative signal values; means for storing said representative signal values for scan presentation in a display; and means for receiving said representative signal values from said means for storing to display a representation of said planar polygon as a graphics image.
Claim 34
Dependent Claim
101 words
A system according to claim 29 wherein said means for generating signal-represented barycentric coordinates includes, means for mapping areas A 1 , A 2 and A 3 of triangles defined by one of said points P and said vertices P 1 , P 2 and P 3 as, P 1 PP 3 , P 1 PP 2 and P 2 PP 3 ; means for specifying depth dimensions of said vertices as z' 1 , z' 2 and z' 3 in said space coordinates not subject to linear distortion; and means for generating barycentric coordinates for said planar polygon as, ##EQU31##
Claim 35
Dependent Claim
56 words
A process according to claim 30 wherein modulation of intensity i is accomplished by a form of the equation, i=b' 1 i 1 +b' 2 i 2 +b' 3 i 3 where i 1 , i 2 , and i 3 specify intensities at said vertices P 1 , P 2 , and P 3 .
Claim 36
Dependent Claim
40 words
A process according to claim 30 wherein said texture signals manifest tx 1 , tx 2 , tx 3 , ty 1 , ty 2 , ty 3 and said means for interpolating functions to accomplish the formula, tx=(b'.sub.1,b'.sub.2,b'.sub.3)·(tx.sub.1,tx.sub.2,tx.sub.3) ty=(b'.sub.1,b'.sub.2,b'.sub.3)·(ty.sub.1,ty.sub.2,ty.sub.3).
Claim 37
Dependent Claim
31 words
A system according to claim 29 further including means for providing intensity signals and means for interpolation of said intensity signals in accordance with said barycentric coordinates to shade said pixels.
Claim 38
Dependent Claim
46 words
A system according to claim 29 wherein said means for interpolating comprises means for providing surface normal signals representative of multiple components of surface normal vectors for said planar polygon and means for interpolation of said surface normal signals in accordance with said signal-represented barycentric coordinates.
Claim 39
Dependent Claim
32 words
A system according to claim 29 wherein said means for interpolating includes means for providing texture signals and means for interpolation of said texture signals in accordance with said signal-represented barycentric coordinates.
Claim 40
Dependent Claim
204 words
A computer graphics process of displaying a graphics image by interpolating a graphics object as for shading or texturing in accordance with a defined characteristic, comprising the steps of: mapping, in space coordinates not subject to nonlinear distortion, at least a part of said object as in form of a planar triangular polygon having at least three vertices; selecting said vertices of said planar triangular polygon, said vertices of said triangular polygon being designated P 1 , P 2 and P 3 ; mapping an eyepoint O in relation to said planar triangular polygon; mapping a perspective plane in relation to said eyepoint and specifying pixel locations in said perspective plane; generating signal-represented barycentric coordinates for said planar triangular polygon at said vertices, in said space coordinates not subject to nonlinear distortion; scan converting said planar triangular polygon to identify points P thereof coinciding to said pixels, said points P being in said space coordinates not subject to nonlinear distortion; interpolating said points P using said barycentric coordinates with respect to said defined characteristic to provide representative signal values; storing said representative signal values for scan presentation; and driving a scan display structure with said representative signal values for scan presentation of an image.
Claim 41
Dependent Claim
204 words
A computer graphics process of displaying a graphics image by interpolating a graphics object as for shading or texturing in accordance with a defined characteristic, comprising the steps of: mapping, in space coordinates not subject to nonlinear distortion, at least a part of said object as in form of a planar triangular polygon having at least three vertices; selecting said vertices of said planar triangular polygon, said vertices of said triangular polygon being designated P 1 , P 2 and P 3 ; mapping an eyepoint O in relation to said planar triangular polygon; mapping a perspective plane in relation to said eyepoint and specifying pixel locations in said perspective plane; generating signal-represented barycentric coordinates for said planar triangular polygon at said vertices, in said space coordinates not subject to nonlinear distortion; scan converting said planar triangular polygon to identify points P thereof coinciding to said pixels, said points P being in said space coordinates not subject to nonlinear distortion; interpolating said points P using said barycentric coordinates with respect to said defined characteristic to provide representative signal values; storing said representative signal values for scan presentation; and driving a scan display structure with said representative signal values for scan presentation of an image.
Claim 42
Dependent Claim
155 words
A process according to claim 36 wherein said step of generating signal-represented barycentric coordinates includes the steps of: mapping a first ray extending from said eyepoint to one of said pixels as a vector OC from said eyepoint to said one pixel; mapping a second, a third and a fourth ray extending from said eyepoint to each of said vertices P 1 , P 2 , P 3 as vectors OP 1 , OP 2 , and OP 3 from said eyepoint to said vertices; formulating digital signal representations for said vectors OC, OP 1 , OP 2 and OP 3 ; generating triple scalar products using said vectors OC, OP 1 , OP 2 and OP 3 , defined as K 1 , K 2 , K 3 ; and generating barycentric coordinates for said planar triangular polygon as, b'.sub.1 =K.sub.1 /(K.sub.1 +K.sub.2 +k.sub.3) b'.sub.2 =K.sub.2 /(K.sub.1 +K.sub.2 +K.sub.3) b'.sub.3 =K.sub.3 /(K.sub.1 +K.sub.2 +K.sub.3).
Claim 43
Dependent Claim
104 words
A process according to claim 36 wherein said step of generating signal-represented barycentric coordinates includes the steps of: mapping areas A 1 , A 2 , and A 3 of triangles defined by one of said points P and said vertices P 1 , P 2 , and P 3 , as P 1 PP 3 , P 1 PP 2 , and P 2 PP 3 ; specifying depth dimensions of said vertices as z' 1 , z' 2 , and z' 3 in said space coordinates not subject to nonlinear distortion; and generating barycentric coordinates for said planar triangular polygon as, ##EQU32##
Claim 44
Dependent Claim
57 words
A process according to claim 38 wherein modulation of intensity i is accomplished by a form of the equation, i= b' 1 i 1 +b' 2 i 2 +b' 3 i 3 where i 1 , i 2 , and i 3 specify intensities at said vertices P 1 , P 2 , and P 3 .
Claim 45
Dependent Claim
17 words
A process according to claim 38 wherein said interpolating step comprises interpolation of texture for said pixels.
Claim 46
Dependent Claim
47 words
A process according to claim 40 wherein texture coordinates for said vertices are designated as vectors (tx 1 , tx 2 , tx 3 ) and (ty 1 , ty 2 , ty 3 ) and said texture of said one point is interpolated as: tx=(b'.sub.1,b'.sub.2,b'.sub.3)·(tx.sub.1,tx.sub.2,tx.sub.3) ty=(b'.sub.1,b'.sub.2,b'.sub.3)·(ty.sub.1,ty.sub.2,ty.sub.3).
Claim 47
Dependent Claim
24 words
A process according to claim 36 wherein said interpolating step comprises interpolation of intensity i of said pixels in accordance with said barycentric coordinates.
Claim 48
Dependent Claim
24 words
A process according to claim 36 wherein said interpolating step comprises interpolation of multiple components of a surface normal vector at said one point.
Claim 49
Dependent Claim
180 words
A computer graphics system for displaying a graphics image by interpolating a graphics object as for shading or texturing pixels in accordance with a defined characteristic, wherein said object is represented in space coordinates not subject to nonlinear distortion, at least in part by a planar polygon with defined vertices P 1 , P 2 , P 3 and further, wherein an eyepoint O is specified along with a perspective plane, said system comprising: means for generating signal-represented barycentric coordinates for said planar polygon at said vertices, in said space coordinates not subject to nonlinear distortion; means for scan converting said polygon to identify points P thereof coinciding to said pixels, said points P being in said space coordinates not subject to nonlinear distortion; means for interpolating said points P using said barycentric coordinates with respect to said defined characteristic to provide representative signal values; means for storing said representative signal values for scan presentation; and means for receiving said representative signal values from said means for storing to display a representation of said planar polygon as a graphics image.
Claim 50
Dependent Claim
180 words
A computer graphics system for displaying a graphics image by interpolating a graphics object as for shading or texturing pixels in accordance with a defined characteristic, wherein said object is represented in space coordinates not subject to nonlinear distortion, at least in part by a planar polygon with defined vertices P 1 , P 2 , P 3 and further, wherein an eyepoint O is specified along with a perspective plane, said system comprising: means for generating signal-represented barycentric coordinates for said planar polygon at said vertices, in said space coordinates not subject to nonlinear distortion; means for scan converting said polygon to identify points P thereof coinciding to said pixels, said points P being in said space coordinates not subject to nonlinear distortion; means for interpolating said points P using said barycentric coordinates with respect to said defined characteristic to provide representative signal values; means for storing said representative signal values for scan presentation; and means for receiving said representative signal values from said means for storing to display a representation of said planar polygon as a graphics image.
Claim 51
Dependent Claim
86 words
A system according to claim 44 wherein said means for generating signal-represented barycentric coordinates includes, means for generating signal-represented triple scalar products using vectors OC, OP 1 , OP 2 , OP 3 to provide representations of K 1 , K 2 , K 3 ; and means for generating signal-represented barycentric coordinates b' 1 , b' 2 and b' 3 for said planar polygon using a form of the equation, b'.sub.1 =K.sub.1 /(K.sub.1 +K.sub.2 +K.sub.3) b'.sub.2 =K.sub.2 /(K.sub.1 +K.sub.2 +K.sub.3) b'.sub.3 =k.sub.3 /(K.sub.1 +K.sub.2 +K.sub.3).
Claim 52
Dependent Claim
46 words
A system according to claim 45 wherein said means for generating multiple products comprises means for generating said multiple products of said vectors by applying equations as a dot product and a cross-product as, K.sub.1 =OC·[OP.sub.2 ×OP.sub.3 ] K.sub.2 =OC·[OP.sub.3 ×OP.sub.1 ] K.sub.3 =OC·[OP.sub.1 ×OP.sub.2 ].
Claim 53
Dependent Claim
101 words
A system according to claim 44 wherein said means for generating signal-represented barycentric coordinates includes, means for mapping areas A 1 , A 2 and A 3 of triangles defined by one of said points P and said vertices P 1 , P 2 and P 3 as, P 1 PP 3 , P 1 PP 2 and P 2 PP 3 ; means for specifying depth dimensions of said vertices as z' 1 , z' 2 and z' 3 in said space coordinates not subject to nonlinear distortion; and means for generating barycentric coordinates for said planar polygon as, ##EQU33##
Claim 54
Dependent Claim
55 words
A system according to claim 47 wherein said means for interpolation accords to a form of the equation, i=b' 1 i 1 +b' 2 i 2 +b' 3 i 3 where i 1 , i 2 , and i 3 specify intensities at said vertices P 1 , P 2 , and P 3 .
Claim 55
Dependent Claim
39 words
A process according to claim 47 wherein texture signals manifest tx 1 , tx 2 , tx 3 , ty 1 , ty 2 , ty 3 and said means for interpolating functions to accomplish the formula, tx=(b'.sub.1,b'.sub.2,b'.sub.3)·(tx.sub.1,tx.sub.2,tx.sub.3) ty=(b'.sub.1,b'.sub.2,b'.sub.3)·(ty.sub.1,ty.sub.2,ty.sub.3).
Claim 56
Dependent Claim
31 words
A system according to claim 44 further including means for providing intensity signals and means for interpolation of said intensity signals in accordance with said barycentric coordinates to shade said pixels.
Claim 57
Dependent Claim
46 words
A system according to claim 44 wherein said means for interpolation comprises means for providing surface normal signals representative of multiple components of surface normal vectors for said planar polygon and means for interpolation of said surface normals signals in accordance with said signal-represented barycentric coordinates.
Claim 58
Dependent Claim
32 words
A system according to claim 44 wherein said means for interpolating includes means for providing texture signals and means for interpolation of said texture signals in accordance with said signal-represented barycentric coordinates.
References Cited (Backward Citations)
US4821212A
Cited
USPTO Annuity & Maintenance Windows
| Annuity Stage | Window Opens | Payment Deadline | Surcharge Period Until | Estimated Fee | Status |
|---|---|---|---|---|---|
| 3.5 Years (Year 4) | 1997-10-31 | 1998-05-02 | 1998-11-01 | $2000.00 | Window Closed / Lapsed |
| 7.5 Years (Year 8) | 2001-10-31 | 2002-05-02 | 2002-11-01 | $3760.00 | Window Closed / Lapsed |
| 11.5 Years (Year 12) | 2005-10-31 | 2006-05-02 | 2006-11-01 | $7700.00 | Window Closed / Lapsed |