Curve/curve intersection is one of the fundamental problems of computational geometry. At
the present time there exist several different approaches to this problem but the endeavor is to
avoid difficulties in calculations which are mainly results of polynomial representation rational
higher degree curves. It can be done in certain cases by using the relief perspective and Bezier
Bezier clipping, in the context of plane curve intersection in this paper, see e.g.
[Nishi90] [Nishi98], is an interactive method which takes advantages of the convex
hull property of Bezier curves. Regions of one curve which are guaranteed do not intersect a
second curve can be identified and subdivided away.
Relief perspective is a mapping of space into space in order to correspond conditions of the human seeing. Images of all objects under the relief perspective are located in space between two parallel planes.
2 Relief perspective
The relief perspective is a special case of perspective collineation of the extended
Euclidean space . We remind that a collineation is a bijective mapping
that preserves collinearity of
points. If there exists a plane such that ,
is a perspective collineation (for more details, see e.g. [Bus53] [Cizm84]). Then there
exists a point (called the centre of perspective collineation) such that and
The relief perspective is a perspective collineation including some additional conditions:
in order to produce correct images of 3D objects, it is important to respect necessary conditions of the human seeing. As in linear perspective, see e.g. [Cenek59], also in the relief perspective we suppose, that objects are located inside viewing circular cone. The cone has a vertex in the eye (the centre of projection) and the distance from the eye to the objects has to be at least 25 cm.
the image plane (the set of invariant points ) does not contain the centre (i.e. is a homology) and determine elements of the mapping are: the centre , the image plane (the set of invariant points) and the vanishing plane (the image of the plane at infinity).
no objects are ideal, i.e. objects and their images in the relief perspective have not ideal points. In notions of the image plane has to be placed between the point and the plane . The mapped object, that image we want to construct, is located in the semi-space, that it is opposite to the semi-space ("behind the image plane ").
Let be a relief perspective of space . Let denote be a preimage space and be an image space ( with upper index "r") such that . The image of the object under the relief perspective is called the relief of the object and is denoted analogically . The relief perspective is given by the centre , the image plane
and ordered pair of different pointssuch that and are collinear.
Note therefore all mapped objects and their images under the relief perspective have no ideal points, we can study them in or using affine and projective methods as well. It is also clear that the restriction of the relief perspective , which is a bijective mapping, is again a bijection.
3 Representation of the relief perspective in an analytical form
Suppose and are two different points of with coordinates and respectively. If is a collineation of the space we use the following equations:
where are real numbers and . In equations
(3 Representation of the relief perspective in an analytical form) are used non-homogenic coordinates, which correspond to a restriction
, where is a plane
In order to obtain simple mapping equations for relief perspective from (3 Representation of the relief perspective in an analytical form) we assume, that the centre is a point with coordinates and is a plane . In this special case we obtain the convenient representation of the relief perspective in the following form
where the span (the geometric representation of the parameter will be explained in the next lines).
4 Some properties of the relief perspective
the vanishing plane and the neutral plane (the preimage of the plane at infinity) are parallel and for distances and we have
where the parameter represents the span
if a plane is parallel to the image plane , then the relief of the plane is the plane and these planes are parallel (it is obvious that )
the relief of a point placed in the semi-space, that is opposite to the semi-space , is a point in a part of space with boundary planes and (in an intersection of spaces and ) (Fig. 4 Some properties of the relief perspective)
the orthographic projection of the object (the relief of the object ) onto the image plane is the central projection of the object from the point , where
5 Rational Bezier curves and relief perspective
The relief perspective is a mapping of the extended Euclidean space . Let be given points, be their coordinates of the space and positive real numbers be their weights. By these elements and Bernstein polynomials , see e.g. [Farin93], is defined a n-th planar rational Bezier curve
and also the space nonrational Bezier curve
with control points
Let be the relief perspective of the space and be the relief of the point . From the equations (3 Representation of the relief perspective in an analytical form) we get
and according to the relation we obtain
From the above results coordinates of the point , that is the preimage of the relief
, are computed according to , ,
Lemma Let be relief perspective of the space . The point is the relief of the point ,,.
By given planar points and their weights , where , we can define in points
With respect to the lemma above it is obvious, that the reliefs of the points are represented by the points (5). Using the points , the space nonrational Bezier curve of the form (4) is defined and we notice, that under mapping is an image of the curve
This curve Q(t) is the space rational Bezier curve defined by control points of the form (7) and their weights , that are computed according to because of
Applying all these results we obtain the following
Theorem The curve (t) is a relief of the curve Q(t).
In order to get more information about Bezier curves and the relief perspective, let us
assume having a central projection with the centre and the plane . In this case
the planar rational curve of the form (3) is the image of the space
rational Bezier curve defined by (8).
Let the space nonrational Bezier curve expressed by (4) be given. What is the relief of this curve? We know and now the relief of the given curve can be written as
where control vertices are points expressed as follow
and weights are computed by .
6 Intersection of space rational Bezier curves
Let , be space rational Bezier curves of the form (8). The aim is to find their intersection.
The solution to this problem can be found using already known facts. According to the theorem, which was formed in the previous section, reliefs of space rational Bezier curves of the form (8) are space nonrational Bezier curves of the form (4) (in Fig. 6 Intersection of space rational Bezier curves Bezier curves of the form (8) and their reliefs for are shown).
The central projections of these
nonrational curves from the point onto the plane are planar rational Bezier
curves defined by (3). Their intersection is possible to find specifically by using
the method of Bezier clipping. This method specifies values of the parameter
of the curve , respectively the parameter
of the curve , that correspond to the common point (the upper indexes "r" and "s" denote the reliefs and central projections of curves or
points). This point is the intersection of both curves . The preimage of the point in
the central projection is the space point, which is the relief of the intersection of the space
rational Bezier curves and .
The following scheme shows a whole proces of finding the intersection of the and curves:
– relief perspektive
– central projection .
The central projection is not a bijective mapping and in case that the intersection of curves and exists (in opposite case the given curves and certainly do not intersect) the intersection of curves and does not have to exist. This situation occurs when preimages of point on the curves and in the central projection are different.
7 Conclusions and future work
The relations between Bezier curves and the relief perspective have been described. The
necessary and sufficient conditions for expressing the space nonrational Bezier curve as the
relief of the space rational Bezier curve have been formed. We have shown that to the planar
rational curve of the form (3), defined by planar points
and their weights , it is possible to assign a class of the space curves by the relief
perspective. One of them is nonrational curve defined by (4) and two are
rational curves defined by (8) and (5 Rational Bezier curves and relief perspective).
The described method can be used as a direction for application how to find the intersection of
the space rational curves of the form (8). It is possible to express
every polynomial curve in Bezier’s representation and due to this representation the method can be
applied to all polynomial curves after modifications (e.g. spline curves which are considered as
curves consist of Bezier segments) for solution to the curve/curve or curve/line intersection
Despite the author’s attempt he did not succeed in finding similar comparable published methods on
website. In addition to this he is not able to compare his results with any other. Obviously
author’s knowledge is limited and he would appreciate to get information about any other similar
In the future work we want to extend possibilities of the relief perspective in geometric modeling. Our aim is to find an answer whether any polynomial 3D curve can be converted to the space curve of the form (8).
References 111some of the publications have not equivalents in English
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- [Cizm84] Čižmár, J.: Grupy geometrických transformácií. Alfa, Bratislava 1984
- [Cenek59] Čeněk, G., Medek, V.: Deskriptívna geometria I. SVTL, Bratislava 1959
- [Farin93] Farin, G.E.: Curves and Surfaces for Computer Aided Geometric Design. Academic Press, 3. Edition 1993
- [Hlus99] Hlúšek, R.: Shape modifications of Bezier curves by relief perspective, Proceedings of SCG’99, Kočovce
- [Kader25] Kadeřávek, F.: Relief. JanŠtenc, Praha 1925
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- [Nishi98] Nishita, T.: Applications of Bezier clipping method and their Java applets, Proceedings of Spring conference on computer graphics, Budmerice 1998