Outline
The target of this work is to generate smooth curve/surface shapes
in a three dimensional space. The smooth curve/surface shapes are
automatically generated in efficient performance from some given
positional and normal constraints. The representation of the surface
shape is a mesh. However, if the representation of parametric Bezier
surfaces are requested, the mesh can be
converted to a set of Bezier surfaces by applying an aditional
technique. This technique can be applied to the areas of
- surface design in mechanical CAD
- 3D-contents creation for
- e-commerce
- game
- animation
Research contents
Figure 1 and Figure 2 show examples of smooth surface generation.
In Figure 1, starting from a noisy shape of Figure 1(a), a smooth shape
of Figure 1(b) is generated under the constraints on boundaries.
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| (a): Initial shape having noisy portion. |
(b): Smoothed surface with overlaid constraints.
Red points show positional constraints, and green arrows show normal
constraints. |
| Figure 1: Smooth surface generation.
Positions and normals are fixed on the boundaries of the three holes.
Noisey portions on Figure (a) are removed on Figure (b).
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Figure 2 shows a deformation process from a flat plain to a hand shape.
Only giving some positional constraints on finger joints generates
such a realistic hand shape.
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| (a): A flat plain as an initial shape. |
(b): Green points show the positions where positional
constraints are given.
Four constraints are given for each joints, four for a wrist,
and one for each finger tip.
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| (c): Constraints are given so that the green points move to
the red points. Each yellow line shows a correspondence between a green point and a red point.
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(d):
For each joint and a wrist,
generate a curve (red curve)
smoothly interpolating four red points.
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| (e): A smooth surface shape interpolating
the red curves. In this process, the curves are treated as
positional constraints.
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(f): Another view of Figure (e).
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| Figure 2: Smooth surface generation.
Starting from a flat plain, a hand shape is generated.
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Table 1 summarizes the input and output data of the algorithm. The
curves and surfaces treated in this algorithm are polygonal curves and
surfaces, not parametric curves and surfaces such as NURBS or Bezier.
However, if the representation of parametric Bezier surfaces are
requested, the polygonal surface can be
converted to a set of Bezier surfaces by applying an aditional
technique. As for constraints, two types of constraints in the table
is acceptable. It will be possible to extend the types of constraints
to other variations.
| Table 1: Input and output of the algorithm |
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In case of curve |
In case of surface |
| Input |
- Initial polyline
- Constraints
- Positional constraints
- Normal constraints
- ....
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- Initial polygon (mesh)
- Constraints
- Positional constraints
- Normal constraints
- ....
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| Output |
Smooth polyline |
Smooth polygon (mesh) |
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Table 2 lists the advantages of the algorithm.
| Table 2: Advantages of the algorithm |
- Minimization of curvature variation
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The algorithm approximately estimates curvature from a discrete
representation of a polyline and a polygon and minimizes variation
of the curvature over the entire shape. Curvature is a natural
indicator for the smoothness; therefore the algorithm generates fairer
shape than the other discrete approaches that do not estimate curvature.
Bent physical objects like beams and plates are in the situation
that the curvature is minimized. The word "Physically-based" on the page
title is coming from the fact that the algorithm simulates the physical
behavior.
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- Efficient performance for interactive design
- The computational order of the algorithm is linear to the number
of polygon nodes. The efficient performance is attained by the algorithmic device
that a polygon is internally represented by a hierarchical
representation.
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- Iterative approach suitable for interactive design
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The algorithm iteratively updates the positions of polygonal nodes;
therefore the accuracy of the solution increases stepwise. In case
of designing a rough shape, users can design efficiently by ending the
iterations in an early stage.
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- Shape control with a few constraints
- The
shape is automatically determined by giving a few constraints.
Typical constraints are positional ones and normal ones. Moreover,
scattered points in 3D space with no linkage with polygonal nodes can
be constraints to generate a shape roughly approximating the scattered
points.
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- Interpolation to specified points
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A well-known smoothing method called "Subdivision Surface" does not
generate a surface interpolating specified points, whereas our
algorithm can generate an interpolating shape.
The Interpolation makes the design process intuitively easy.
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*Conceptually to say, "Subdivision Surface"
rounds a polygonal object by cutting its corners.
Due to the cutting operation, the generated
surface does not interpolate the initial corners.
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Publications
- Atsushi Yamada, Kenji Shimada, Tomotake Furuhata, and Ko-Hsiu Hou,
A Discrete Spring Model to Generate Fair Curves and Surfaces,
Pacific Graphics '99, 1999 Oct.
- Atsushi Yamada, Kenji Shimada, Tomotake Furuhata, Ko-Hsiu Hou,
Fair Curve and Surface Generation Based on a Discrete Spring Model,
IPSJ Graphics and CAD Symposium '99,
pp. 43-48, 1999, Jun, (in Japanese).
- Atsushi Yamada, Tomotake Furuhata, Takayuki Itoh, Kenji
Shimada, Curve And Surface Fitting to Scattered Points Based on a
Discrete Spring Model, ISPJ Graphics & CAD workshop, 99-CG-96,
pp. 31-36, 1999, (in Japanese).
- Atsushi Yamada, Tomotake Furuhata,
Fair Curve and Surface Generation Based on a Discrete Spring Model -
resolving of folding problem of Laplacian smoothing-,
Proceedings of JSPE annual conference, p. 146, 1999, Sep, (in Japanese).
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