by Using Subdivision Surfaces

Ryutarou Ohbuchi

Yamanashi University, Yamanashi, Japan.

Shigeo Takahashi

Gunma University, Gunma, Japan.

Yoshiyuki Kokojima

Gunma University, Gunma, Japan.

We have been working on a new geometric
morphing algorithm for polygons and polyhedrons.
The algorithm is capable of morphing between
shapes having different genera, e.g., between
a sphere and a torus, with little intervention.
At the same time, the algorithm offers powerful
control over the morphing process; for example,
user-specified evolution of topology, vertex-to-vertex
feature correspondence, representation of
both smooth and sharp features, user-specified
deformation of shapes during the morph, and
spatially non-uniform progress of the morphing.

Our algorithm directly interpolates vertices
of polyhedral source shapes by using subdivision
surfaces. The algorithm employs variational
optimization to produce (N+1)-dimensional
subdivision surfaces by treating vertices
of the N-dimensional polyhedrons as geometric
constraints. For example, three-dimensional
(3D) polyhedrons are interpolated by four-dimensional
(4D) subdivision (hyper-)surfaces. Morphed
shape can be extracted by intersecting the
subdivision surfaces with another (N+1)-dimensional
surface. Smoothness of subdivision surface
gives us smooth shape transitions while manipulability
of the subdivision surfaces gives us powerful
control over the morphing, which include
feature correspondence and shape transition
control

- - Two to one contour correspondence: two contours are merged

- - Two to one contour correspondence: one contour disappears

- - Pulling out the "nose" of the letter "3" in the 2 -> 3 transition

- - Non-uniform progress in the 8 -> 7 -> 1 transition
- Original morphing
- Increasing the transition rate at the bottom
- Interpolating mesh and transparent surface
- Morphing (MPEG movie, Animation GIF)

- Increasing the transition rate at the top
- Interpolating mesh and transparent surface
- Morphing (MPEG movie, Animation GIF)

- Increasing the transition rate on the left side
- Interpolating mesh and transparent surface
- Morphing (MPEG movie, Animation GIF)

- Increasing the transition rate on the right side
- Interpolating mesh and transparent surface
- Morphing (MPEG movie, Animation GIF)

- - Trans-genus morphing
- Initial mesh (in 4D)
- Meshes after first subdivision and after smoothing

- Morphing (MPEG movie, animation GIF)

- Controlling the transition with constraints (MPEG movie, animation GIF)

- - Spatial transition control for 3D
- Original morphing (MPEG movie, animation GIF)

- Increasing the transition rate on the right (MPEG movie, animation GIF)

- Original morphing (MPEG movie, animation GIF)