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Control-Line Curves and

Control-Plane Surfaces

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*A. Ardeshir Goshtasby*

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`Control-Line Curves: The formulations for parametric curves
defined by control points is revised to improve their design
capabilities. By simply replacing the control points with control lines
in the formulations of these curves, powerful tools will be obtained
that enable creation of complex geometries more precisely and in a
shorter amount of time. Curves designed by control lines will be called
control-line curves or plus curves. The plus implying that in addition
to the control points, tangent vectors at the control points are used
to design a curve. A control-line curve with Bernstein-Bézier
basis functions is called a Bézier+ curve.`

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**Examples:**

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`(a)
(b)
(c)`

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`Fig. 1. ``(a)–(c) Different control-line
Bézier curves of order 3 obtained by varying the tangent
direction at the middle point.`

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`(a)
(b)`

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`Fig. 2. ``(a) A traditional Bézier curve of
order 5. The curve does not resemble its control polygon. (b) The curve
obtained using the revised formulation.`

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`(a)
(b)
(c)
(d) `

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`Fig. 3. ``(a) A Bézier curve of order 3. (b) -
(d) Bézier+ curves obtained from the same control points and the
same tangent directions, but with different tangent magnitudes. `

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` `

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`(a)
(b) (c)
(d)`

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`Fig. 4. ``Comparison of Bézier curves (yellow)
and Bézier+ curves (blue). The control points are shown in
white, the control lines are shown in green, and the control polygons
are shown in purple. (a) Bézier and Bézier+ curves are
tangent to the control polygon at start and end points. (b) Closed
Bézier and Bézier+ curves obtained from the same control
points. (c) A Bézier+ curve may have one more inflection point
than a Bézier curve of the same order. (d) Bézier+ curves
do not always provide linear precision. `

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`Control-Plane Surfaces: ``The same idea can be
extended to parametric surfaces`** also. By replacing the control
points with control planes, it becomes possible to revise the local
shape of a surfaces by changing the orientation of the control points
(or normals to the control planes). An Example is demonstrated below.**

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`(a) (b)`

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`(c) (d)`

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`(e) (f)`

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`(g) (h)`

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`Fig. 5. ``(a) A `**Bézier **patch
defined by nine control points. Once the control points are selected,
the surface becomes fixed. Note that the patch is very far from the
center control point. (b) A **Bézier+ patch obtained from
the same control points. The normal vector magnitudes are interactively
increased until the surface gets sufficiently close to the control
point. (c) Overlaying of (a) and (b) for comparison purposes. (d)-(h)
Different surfaces obtained by changing the orientation of the center
control-plane (normal).**** **

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Use of Control-Plane Surfaces in Smooth
Parametric Representation of Polygon Meshes

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`To show the power of control-plane surfaces, suppose
a polygon mesh is given and we want to represent the mesh by a smooth
surface. If we use the polygon faces as the control planes in the
equation of rational Gaussian surfaces [1], we will obtain a smooth
parametric surface approximating the mesh. An example of this is shown
in Fig. 6. Note that the mesh is not required to be made of a
homogenous mesh elements, such as triangles. A mixture of different
polygon types may be used. In this example, a mesh made up of
triangluar and quadrilateral mesh elements is shown. Fig. 6a shows the
mesh, and Figs. 6b - f show surfaces obtained with different smoothness
levels of the rational Gaussian.`

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`Fig. 6. ``Approximation of a polygon mesh by a
control-plane surface. The polygon faces are used as the control planes
in the equation of a rational Gaussian surface [1]. (a) The given mesh.
(b) - (f) Control-plane surfaces approximating the mesh at increasing
smoothness levels.`

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`Examples of larger data sets are shown in Fig. 7.
Figs. 7a, 7c, and 7e show three polygon meshes with thousands of faces
and Figs. 7b, 7d, and 7f, show approximation of the meshes by
control-plane surfaces. These data sets are courtesy of the U.S.
Geological Survey, Stanford University, and SGI.`

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`Fig. 7. ``Smooth parametric representation of irregular
polygon meshes by control-plane surfaces. (a) A digital elevation map
courtesy of the U.S. Geological Survey and (b) its smooth
representation. (c) The Stanford bunny and (d) its smooth
representation. (e) A vase courtesy of SGI and (f) its smooth
representation.`

[1] A. Goshtasby, "Geometric Modeling Using
Rational Gaussian Curves and Surfaces," *Computer-Aided Design*,
May 1995, pp. 363-375.

For more information contact A. Goshtasby (agoshtas@cs.wright.edu).

Last modified: 1/31/03.