Multiresolution Curve Editing With Linear Constraints*

[+] Author and Article Information
Gershon Elber

Computer Science Department, Technion, Haifa 32000, Israele-mail: gershon@cs.technion.ac.il

J. Comput. Inf. Sci. Eng 1(4), 347-355 (Oct 01, 2001) (9 pages) doi:10.1115/1.1430679 History: Received August 01, 2001; Revised October 01, 2001
Copyright © 2001 by ASME
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Grahic Jump Location
Symmetries of curves: (a) X-symmetry, (b) Y-symmetry, (c) Circular-symmetry
Grahic Jump Location
The area of a closed parametric curve. The differential area in gray equals to 12|C(t)×C(t)|dt and the enclosed area by the curve is the result of integrating this differential area over the entire parametric domain of the curve. See Eq. (10)
Grahic Jump Location
In thick gray, a periodic planar B-spline cross section is shown that is constrained to be Y-symmetric as well as to present a fixed area. All the other cross sections were derived from it in few seconds via direct manipulation while the area as well as the Y-symmetry are preserved. The curve is a cubic periodic curve with twelve control points.
Grahic Jump Location
An outline of a hand is manipulated from below while the finger-tips are anchored and the total area is preserved. As a result, the width of the finger is adapted to the changes from below while the fingertips are stationary. Both linear (a) and cubic (b) curves are shown. Starting with the original curve that is shown on the left, the modifications are shown in the middle and in the right side in black with the original curve shown in thick gray color.
Grahic Jump Location
A periodic planar cubic B-spline curve in gray is directly manipulated and dragged at the selected point to the right along the (solid curved) path while preserving the enclosed area. Several snapshots are shown. In (a), and in addition to the area constraint, a tangential constraint is preserved at the bottom of the shape. In (b), a third, additional, positional constraint is added to the top left side of the curve anchoring the shape to interpolate that location throughout this direct manipulation stage.
Grahic Jump Location
A curve in the shape of a butterfly (in gray) is directly manipulated. A point on the bottom right side of the shape is moved in the bottom right direction, following the arrows. A positional as well as a tangential constraint are both placed at the top right of the butterfly, that is also constrained to be Y-symmetric. The top row shows the result of multiresolution editing in several resolutions without any constraints whereas the bottom row shows the same sequence of multiresolution operations with the constraints activated.
Grahic Jump Location
A Y-symmetric curve with a constant area has a positional constraint on the left and a tangential constraint on the right. Several direct manipulation operations are performed while, effectively, these two constraints keep the curve at a constant width, throughout. The previous operation is shown in gray and the new one is shown in black, from left to right. These examples were created in a few seconds.
Grahic Jump Location
Several examples of the illusion of two faces versus a vase. These examples were created in few minutes using a quadratic B-spline curve with 32 control points that is constrained to be Y-symmetric.
Grahic Jump Location
Multiresolution with combination with constraints could also be applied to non planar curves on surfaces. Here, two views of a curve in the shape of the letter “S” are shown on the body of the Utah teapot, potentially serving as a trimming curve for the surface.
Grahic Jump Location
Multiresolution editing without constraints is shown for the curve on the body of the Utah teapot in Fig. 9. Three resolution levels of a single select-and-drag operation and no constraints are shown.
Grahic Jump Location
Multiresolution editing with constraints is show for the curve on the body of the Utah teapot in Fig. 9. Three resolution levels of a single select-and-drag operation are shown along with two tangent constraint (black points) and one positional constraint (gray point).



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