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Planar Layout (Prisa) of Freeform Models
By
Gershon Elber
Department of Computer Science
Technion, Israel Institute of Technology
Haifa, 32000 Israel
+972.4.829.4338
+972.4.829.4353 fax
gershon@cs.technion.ac.il
Clilck on the images to see full-size pictures.
In 1, an algorithm has been derived to decompose a
freeform model into piecewise developable surfaces. Developable
surfaces are unique in their ability to be laid down flat with no
stretching and without any tearing. The cone and the cylinder are two
examples of developable surfaces, while a sphere is not.
Nevertheless, one can approximate arbitrary freeform surfaces using
developable surfaces. The figure on the left shows a unit sphere,
represented using a parametric NURBs surface and its piecewise
developable surface decomposition (far left), all in 3-space. The
figure to the right presents the piecewise developable surface
decomposition after it was laid down on the plane (the cross sections
are portrayed in green).
Once the decomposition of developable surfaces has been formed, the
flattening process of developable surfaces is trivial. Here, on the
left, we see the same decomposition flattened out on the plane (the
white disks are the cross sections of the respective pieces, in
3-space). On the right, we see a realized model from this
decomposition of surfaces assembled from heavy paper and the same
texture map of the earth.
This same technique may be applied to arbitrary freeform surfaces. On
the left we see an F16 airplane assembled from thick paper. The
decomposition of the F16 was computed from the NURBs model
rendered on the right.
Trimmed surfaces can also be supported. The teapot on the left,
assembled on from thick paper, was computed using this developable
surface decomposition algorithm exploiting a decomposition similar to
the following one:
The developable surface decomposition in 3D (left) laid down on
the plane (right) of the spout of the Utah Teapot. For the 2D layout,
the untrimmed surfaces are shown in magenta , while the final
trimmed version is portrayed in white.
The developable surface decomposition in 3D (left) laid down on
the plane (right) of the handle of the Utah Teapot. For the 2D layout,
the untrimmed surfaces are shown in magenta , while the final
trimmed version is portrayed in white.
The developable surface decomposition in 3D (left) laid down on
the plane (right) of the body of the Utah Teapot. For the 2D layout,
the untrimmed surfaces are shown in magenta , while the final
trimmed version is portrayed in white.
The developable surface decomposition is implemented in the freely
available solid modeling system IRIT, developed at the Technion, Israel. The decomposition of the Utah teapot was computed with the aid of a variation of this this IRIT script .
Finally, this mpeg movie (click on the image) shows an animated sequence
of this assembly process of flat sheets, for a model of a wine glass.
1. Gershon Elber. Model Fabrication using Surface Layout Projection. CAD, Vol 27, No 4, pp 283-291, April 1995.
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