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Developing the Developable Surfaces in a Space to the Plane using
Some Triangle Pieces
Kusno and Nur Hardiani
Jember University, Jl. Kalimantan No. 37, Jember, Indonesia
Department of Tadris Matematika, State Islam University, Mataram, Indonesia
Keywords: Developing, Developable Surfaces, Space, Plane, Triangle.
Abstract: This paper deals with the development in the space to the plane of the polygons, the cone, the cylindrical
surfaces and the developable quartic Bézier patches in which its boundary curves are respectively parallel,
and the normal vectors of the surfaces must be in the same orientation. The method is as follows, we
approximate the surfaces into some triangle pieces then we transform consecutively these pieces in the plane.
The result of the study shows that the use of the triangle approximation method can develop effectively these
surfaces in the space to the plane. In addition, it can be applied to detect all surface measures of an object that
are defined by those surface types.
1 INTRODUCTION
Some methods related to the development of surface
to the plane have been presented. The development of
the pipeline surfaces can be carried out by
enumerating of two boundary curves in some
approximation polygons (Weiss and Furtner, 1998).
We can develop a surface to the plane by using the
techniques of interactive piecewise flattening of
parametric 3-D surfaces, leading to a non-distorted
(Bennis and Gagalowicz, 1991). After that,
developing an arbitrary developable surface into a
flattened pattern is based on the geodesic curve length
preservation and linear mapping principles
(Clements, 1991; Gan et al., 1996). We can simulate
the physical model of transitional pipeline parts
whose cross sections are plane curve and polygon and
are made of unwrinkled or unstretched materials. It is
based on the approximation of the boundary surface
triangulation (Obradović et al., 2014). Different from
the previous methods, we are interested in the
discussion about the development of the convex
polygons, the conic/cylindrical surfaces and the
developable quartic Bezier patches in a space to the
plane using the triangle pieces.
This paper is organized in the following steps. In
the first, we talk about the development of the triangle
and polygon plane surfaces in space to the plane. In
the second, we evaluate the development of the cone
and the cylinder defined by the linear interpolation of
two parallel circles. In the third, the construction and
the development of developable quartic Bezier
patches in a space to the plane are introduced. Finally,
the results will be summarized in the conclusion
section.
2 DEVELOPING THE TRIANGLE
AND THE POLYGON PLANE
SURFACE IN A SPACE TO THE
PLANE
Let a triangle plane . The vector
and
form an angle in the space orthonormal coordinate
. The problem is how to develop the plane
in the plane orthonormal coordinate in
which the vector
of the side development of
triangle is align to the determined unit vector
(Figure 1a).
To develop the triangle in space to the
plane can be undertaken as follows (Gan et
al., 1996).
1). Determine the vector′′
and
calculate the unit vector
.
2). Evaluate the measure of angle
Kusno, . and Hardiani, N.
Developing the Developable Surfaces in a Space to the Plane using Some Triangle Pieces.
DOI: 10.5220/0008523104270431
In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 427-431
ISBN: 978-989-758-407-7
Copyright
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2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
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