Computer graphics processing and selective visual display system – Computer graphics processing – Graph generating
Reexamination Certificate
1998-10-07
2001-08-28
Zimmerman, Mark (Department: 2671)
Computer graphics processing and selective visual display system
Computer graphics processing
Graph generating
C345S426000, C345S442000, C345S440000, C345S473000, C345S474000
Reexamination Certificate
active
06281905
ABSTRACT:
BACKGROUND OF THE INVENTION
The invention relates to a method for modifying a display of a shape, suitable for enabling a user of a CAD system to modify a shape and for providing a display of the modified shape.
In the following, the word << surface >> will frequently be used. It should be understood in a broad sense and designating a surface properly speaking as well as surface-like piece-wise G
k
-continuous defined shape or a << skin >> of a volume, with k being an integer at least equal to zero, and even presenting some rips (in which case k=−1).
Engineering or styling (aesthetic) requirements may render advisable modifications of a part, or zone, of a pre-existing surface, or even of a whole surface, defined by a topology thereof for making the modified surface passing by locations indicated by a user, such as points or curves with or without tangent planes and normal curvatures fits. Such requirements constitute inner constraints. The modifications should simultaneously respect external constraint: the modified zone must have, within prescribed tolerances, a continuity or smooth transition with the unmodified part of the original surface.
Another problem which may occur in styling or engineering is to generate a display of a curved surface filling a predetermined zone, respecting external constraint along the boundary of the zone and respecting inner constraint. EP-A-0 576 218 discloses a method for solving that problem, which is however limited to obtaining a continuity not exceeding G
1
at the boundaries. That prior art method comprises generating a partial differential equation representing the surface as a physical membrane. That method is not used for modifying a pre-existing shape and does not enable to achieve a higher continuity order at the boundaries.
SUMMARY OF THE INVENTION
A first object of the invention is to provide an improved method for modifying an existing zone while respecting a continuity G
k
(with k being at least −1) at boundaries and respecting inner constraints.
For that purpose, there is provided a method comprising the steps of:
(a) defining outer constraints of said zone as being continuities G
k
along at least part of said boundaries, with k being an integer at least equal to minus one;
(b) defining inner constraints G
n
on predetermined points or curves to be respected by said zone when modified, with n being an integer at least equal to zero;
(c) selecting a deformation direction law for all points of said zone;
(d) defining a deformation in each point of said zone in a direction fulfilling said law as a result of a resilient deformation of a generalized infinite thin planar plate in equilibrium, respecting said outer and inner constraints; and
(e) generating a modified zone as a surface zone resulting from an application of said resilient deformation to said zone of the original shape.
A continuity G
0
is only a location constraint; a continuity or order
1
is a tangent plane nappe constraint. A continuity of order
2
respects the normal curvatures in all directions at a point or along a curve. Higher values of k and n are possible and have no simple geometrical counterpart. Sometimes a rip is to be obtained between the modified zone and the remainder of the surface and is designated as a continuity G
−1
.
The word “generalized” should be construed as designating a behaviour which may go beyond that of an actual thin plate subjected to mechanical actions (forces or torques) which tend to deform it. The generalization is in respect of mechanical actions and the expression of the “energy” to be minimized. Beyond G
2
, that may be considered as ultra-reality.
A continuity G
0
+G
1
along a side of the boundary requires that the “generalized” plate representing the deformation is planar along that side. An outer constraint G
0
at the boundary may be regarded, by analogy with mechanical considerations, as the requirement that there is a determined distribution of forces along the boundary. For constraints G
1
, the analogy with a plate corresponds to the application, along the boundary, of torques having a determined distribution in addition to the distribution of forces. Beyond G
1
, there is no analogy with deformation of material. But it has been found that an increase of k results in a deformation shape which has stronger slopes.
The above-defined method may be used whatever the number of curved sides of the zone and whatever the number of vertices of the boundary, the boundary being open or not, as seen on
FIGS. 1A
,
1
B,
1
C,
1
D.
Continuity will be strictly fulfilled in certain points of the boundaries, where it is actually defined. As a rule, a maximum value of the discrepancy between the edge of the modified zone and the original surface will be prescribed. For fulfilling the later condition, a trial and error method may be used: the number of points located on the boundaries where the continuity requirement is strictly respected (however subject to the quantification accuracy) until the sought precision degree be attempted for all intermediary points.
When modifying a zone,
the constraints are those to be fulfilled along sides of the boundary and similar constraints at locations which are contained within the deformed zone,
the target is a minimization of the generalized energy of the deformation.
Frequently, the inner constraints may be geometrical. For instance (and this is not exhaustive):
the surface-like model may be required to pass by at least one determined point or by determined curve;
an inner constraint may be that the deformed shape has a tangent plane of determined direction at a predetermined point.
More generally, an inner constraint may be constituted by the definition of a point and/or a curve:
by which the model should pass, and/or
at which the tangent plane and/or an osculatory plane nappe are predetermined, and/or
at which the normal curvatures and the osculatory second order surface is predetermined (it requires the tangent nappe constraint).
According to another aspect of the invention, a method of modifying a zone of a surface while respecting outer constraints as continuities G
k
with k at least −1 and inner constraints G
n
with n at least equal to zero implies defining a deformation which corresponds to a minimum over the whole parametric space of:
E
(
φ
)
=
∫
R
2
(
▽
m
φ
)
2
(
1
)
where R
2
is a two-dimensional parametric space on which said boundaries are projected,
∇ is the gradient operator,
m is a power at least equal to Sup{k,n}+2 and
&phgr; is a deformation function representing the deformation.
The search for a minimum should be made while respecting the continuity constraints along each of the sides of the zone to be deformed. The boundary may be closed or open.
In the above formula, m=1 has a physical meaning: the function E(&phgr;) defines the linear energy of an infinite horizontal membrane subjected to a uniform strain, when the vertical displacement field is &phgr;.
For m=2, E(&phgr;) defines the linear resilient flexure energy of an infinite horizontal thin plate having a Poisson coefficient equal to zero, whose vertical displacement field is &phgr;.
m=3 has no determined dynamics meaning, but represents a solution which causes smooth variations, of particular interest as regards aesthetic aspect, notably regarding the evolution of light reflection on the modified surface.
Another object of the invention is to fill a hole in a surface or to generate a patch while respecting outer constraints G
k
with k at least equal to 2 and inner constraints G
n
with n at least equal to zero. As indicated above, the method disclosed by European patent 0 576 218 is only apt to respect outer constraints G
0
or G
1
.
For that purpose, there is provided a method of modifying a display of the shape having a source surface by generating a zone connected to the source surface along boundaries, comprising the steps of:
(a) defining boundaries o
Durand Gerard
Goussard Jacques
Lieutier André
Massabo Alain
Berkowitz Marvin C.
Matra Datavision
Nath Gary M.
Nath & Associates PLLC
Nguyen Kimbinh T.
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