Table Of ContentBEZIER CURVES INTERSECTION USING
RELIEF PERSPECTIVE
Radoslav Hlú²ek
Department of Geometry
0 Fa
ulty of Mathemati
s and Physi
s
0
Comenius University, Bratislava
0
2 Slovak Republik
n e-mail: hlusekfmph.uniba.sk
a
J
1
2
ABSTRACT
]
G
Presentedpaperdes
ribesthemethodfor(cid:28)ndingtheinterse
tionof
lassspa
e
C rational Bezier
urves. The problem
urve/
urveinterse
tion belongs among basi
s. geometri
problems and the aim of this arti
le is to des
ribe the new te
hnique to
c
solve the problem using relief perspe
tive and Bezier
lipping.
[
1
Keywords: Bezier
urve, perspe
tive
ollineation, Bezier
lipping
v
7
1
0 1 Introdu
tion
1
0
Curve/
urve interse
tionisoneof thefundamentalproblemsof
omputational
0
0 geometry. At the present timethere exist several di(cid:27)erent approa
hes tothis prob-
/
s lembutthe endeavor istoavoiddi(cid:30)
ultiesin
al
ulationswhi
h aremainlyresults
c
: of polynomial representation rational higher degree
urves. It
an be done in
er-
v
tain
ases by using the relief perspe
tive and Bezier
lipping.
i
X
r
a Bezier
lipping, in the
ontext of plane
urve interse
tion in this paper, see e.g.
[Nishi90℄ [Nishi98℄, is an intera
tive method whi
h takes advantages of the
onvex
hull property of Bezier
urves. Regions of one
urve whi
h are guaranteed do not
interse
t a se
ond
urve
an be identi(cid:28)ed and subdivided away.
Relief perspe
tive is a mapping of spa
e into spa
e in order to
orrespond
onditions of the human seeing. Images of all obje
ts under the relief perspe
tive
are lo
ated in spa
e between two parallel planes.
2 Relief perspe
tive
The relief perspe
tivE¯e is a spe
ial
ase of perspe
tive
ollineation of the ex-
3
ϕten:dE¯ed→Eu
E¯lidean spa
e . We remind that a
ollineation is a bije
tive mωapping
3 3
that preserves
ollinearity of points. If there exists a plane su
h
ϕ(X) = X ∀X ∈ ω ϕ
that , is a perspe
tive
ollineation (for more details, see e.g.
O
[Bus53℄ [Cizm84℄). Then there exists a point (
alled the
entre of perspe
tive
1
ϕ(O) = O ϕ(α) = α ∀α; O ∈ α
ollineation) su
h that and .
The relief perspe
tive is a perspe
tive
ollineation in
luding some additional
onditions:
•
in order to produ
e
orre
t images of 3D obje
ts, it is important to respe
t
ne
essary
onditions of the human seeing. As in linear perspe
tive, see e.g.
[Cenek59℄, also in the relief perspe
tive we suppose, that obje
ts are lo
ated
inside viewing
ir
ular
one. The
one has a vertex in the eye (the
entre of
proje
tion) and the distan
e from the eye to the obje
ts has to be at least
25
m.
• ω
the image plane (the set of invariant points ) does not
ontain the
entre
O ϕ ϕ
(i.e. is a homology) and determine elements of the mapping are: the
O σ
entre , the image plane (the set of invariant points) and the vanishing
ωr
plane (the image of the plane at in(cid:28)nity).
•
no obje
ts are ideal, i.e. obje
ts andEtheir images in the relief perspe
tive
3
have not ideal points. In notions of the image plane has to be pla
ed
O ωr
between the point and the plane . The mapped obje
t, that image we
want to
on7−s→tru
t, is lo
ated in the semi-spa
e, that it is opposite to the
σO σ
semi-spa
e ("behind the image plane ").
ϕ E¯ E¯
3 3
E¯rLet be a relief persE¯pe
tive of spa
e . Let denote bϕe:aE¯pr→eimE¯arge spa
e and
3 be an imagUes∈paE¯
e ( 3 with upper index "r") suϕ
h that 3 3. The image
3
of the obje
t under the relief perspe
tive is
alled the relief of the obje
t
U Ur O
and is denoted analogi
ally . The relief perspe
tive is given by the
entre ,
σ A,Ar = ϕ(A) (A,ϕ(A) 6= O)
the imageplane and ordered pairof di(cid:27)erent points
A,Ar 6∈ σ O,A,Ar
su
h that and are
ollinear.
U Ur
Note therefore all mapped obje
ts and their imagEes uE¯nder the relief per-
3 3
spe
tive have no ideal points, we
an study them in or using a(cid:30)ne and
ϕ/U
proje
tive methods as well. It is also
lear that the restri
tion of the relief
ϕ
perspe
tive , whi
h is a bije
tive mapping, is again a bije
tion.
3 Representation of the relief perspe
tive in an analyti
al
form
X X′ E ⊆ E¯ [x,y,z]
3 3
[Sxu′,pyp′,ozs′e] and aretwϕodi(cid:27)erentpointsof wE¯ith
oordinates
3
and respe
tively. If isa
ollineationof thespa
e we use thefollowing
equations:
a x+a y +a z +a
′ 11 12 13 14
x =
a x+a y +a z +a
41 42 43 44
a x+a y +a z +a
′ 21 22 23 24
y =
a x+a y +a z +a (1)
41 42 43 44
a x+a y +a z +a
′ 31 32 33 34
z = ,
a x+a y +a z +a
41 42 43 44
2
a (i,j ∈ {1..4}) det(a ) 6= 0
i,j ij
where are real numbers and . In equaϕ¯ti=onϕs/({1E)¯ar−e
3
used non-homogeni
oordinates, whi
h
orrespond to a restri
tion
α} α a x +a x +a x +a x = 0
41 1 42 2 43 3 44 0
, where is a plane .
In order to obtain simple mapping equations for relief perspe
tive from (1) we
O [0,0,0] σ
assume, that the
entre is a point with
oordinates and is a plane
z−1 = 0
. In this spe
ial
ase we obtain the
onvenient representation of the relief
perspe
tive in the following form
(1+k)x
′
x =
z +k
(1+k)y
′
y =
z +k (2)
(1+k)z
′
z =
z +k
k ∈ R+ − {1} k
where the span (the geometri
representation of the parameter
will be explained in the next lines).
4 Some properties of the relief perspe
tive
• ωr ≡ 1+k ν ≡ −k
the vanishing plane and the neutral plane (the preimage
O,ν ωr,σ
of the plane at in(cid:28)nity) are parallel and for distan
es and we have
| Oν |=| ωrσ |= k
k
where the parameter represents the span
• α ≡ z −c (c 6= 0) (α 6= σ,ν)
if a plane is parallel to the image plane , then
α αr ≡ (1+k)c
the relief of the plane is the plane c+k and these planes are parallel
α k αr k σ
(it is obvious that )
•
the rel7−ie→f of a point pla
ed in the semi-spa
e, that is opposite to the semi-
σO z = 1
spa
e , is a point in a part of spa
e with boundary planes and
z = 1+k z > 1 z < 1+k
(in an interse
tion of spa
es and ) (Fig. 1)
• Ur U
the orthographi
proje
tion of the obje
t (the relief of the obje
t ) onto
σ U S
the image plane is the
entral proje
tion of the obje
t from the point ,
S ∈ l∩ν (l;O ∈ l∧l ⊥ σ)
where
5 Rational Bezier
urves and relief perspe
tive
E¯
3
V , iT∈h{e0r,e1l,ie.f..p,enr}spe
tive is a mapp[xin,gyo]f the extended Eu
lidean spa
e E¯. Let
i i i 2
begivenpoints, betheir
oordinatesofthespa
e and
w , i ∈ {0,1,...,n}
i
positive real numbers be their weights. By these elements and
3
Figure 1
Bn(t)
i
Bernstein polynomials , see e.g. [Farin93℄, is de(cid:28)ned a n-th planar rational
Bezier
urve
n n
x w Bn(t) y w Bn(t)
i i i i i i
P(t) = [i=0 , i=0 ], t ∈ h0,1i
Pn Pn (3)
w Bn(t) w Bn(t)
i i i i
i=0 i=0
P P
and also the spa
e nonrational Bezier
urve
n n n
P¯(t) = [ x w Bn(t), y w Bn(t), w Bn(t)], t ∈ h0,1i
i i i i i i i i
(4)
i=0 i=0 i=0
X X X
with
ontrol points
V = [x w , y w , w ].
i i i i i i
(5)
ϕ : E¯ → E¯r E¯ Xr[x,y,z](Xr 6∈ σ)
Let 3 3 betheXre[laie,fbp,ce]rs∈peE¯
tive ofthespa
e 3 and
3
be the relief of the point . From the equations (2) we get
(1+k)a (1+k)b (1+k)c
x = , y = , z =
c+k c+k c+k (6)
z = (1+k)c
and a
ording to the relation c+k we obtain
k(1+k)
c+k =
1+k −z
[a,b,c] X
From theXabro[xv,eyr,ezs]ults
oordinates of theap=ointkx , thba=t isktyhe pcre=imagkze of
the relief , are
omputed a
ording to 1+k−z, 1+k−z, 1+k−z.
ϕ : E¯ → E¯r E¯ [x,y,z]
Lemma Let 3 [3 bkex reliekfyperspkze
t]ive of the spa
e 3. The point
is the relief of the point 1+k−z,1+k−z,1+k−z .
4
V w 1+ k > w ∀i
i i i
By givE¯en planar points and their weights , where , we
an
3
de(cid:28)ne in points
kx w ky w kw
i i i i i
W = [ , , ] i ∈ {0,1,...,n}
i
1+k −w 1+k −w 1+k −w (7)
i i i
W
i
With respe
t to the lemma above it is obvious, that the reliefs of the points are
V
i
represeP¯n(tte)d by the points (5). Using the points , the sP¯p(at
)e nonrational Beϕzier
urve of the form (4) is de(cid:28)ned and we noti
e, that under mapping is
an image of the
urve
n n n
kx w Bn(t) ky w Bn(t) kw Bn(t)
i i i i i i i i
Q(t) = [ i=0 , i=0 , i=0 ]
n P n P n P (8)
(1+k −w )Bn(t) (1+k −w )Bn(t) (1+k −w )Bn(t)
i i i i i i
i=0 i=0 i=0
P P P
This
urve Q(t) is the spa
e rational Bezier
urve de(cid:28)ned by
ontrol points of the
Ω Ω = 1 + k − w
i i i
form (7) and their weights , that are
omputed a
ording to
be
ause of
kx w ky w kw
i i i i i
Ω = kx w , Ω = ky w , Ω = kw
i i i i i i i i
1+k −w 1+k −w 1+k −w
i i i
Applying all these results we obtain the following theorem
P¯
Theorem The
urve (t) is a relief of the
urve Q(t).
In order to get more information about Bezier
urves and the relief per-
O[0,0,0]
spe
tive, let us assume having a
entral proje
tion with the
entre and
z −1 = 0 P(t)
the plane . In this
ase the planar rational
urve of the form (3) is
Q(t)
the image of the spa
e rational Bezier
urve de(cid:28)ned by (8).
P¯(t)
Let the spa
e nonrational Bezier
urve n expressed by (4) be given. What
z +k = (w + k)Bn(t)
i i
is the relief of this
urve? We know and now the relief
i=0
P¯r(t) Pr(t) P
of the given
urve
an be written as
n n n
(1+k)x w Bn(t) (1+k)y w Bn(t) (1+k)w Bn(t)
i i i i i i i i
P¯r(t) = [i=0 , i=0 , i=0 ]
Pn Pn Pn (9)
(w +k)Bn(t) (w +k)Bn(t) (w +k)Bn(t)
i i i i i i
i=0 i=0 i=0
P P P
t ∈ h0,1i,
where
ontrol verti
es are points expressed as follow
(1+k)x w (1+k)y w (1+k)w
V¯r = [ i i, i i, i]
i w +k w +k w +k
i i i
¯ ¯
Ω Ω = w +k
i i i
and weights are
omputed by .
5
6 Interse
tion of spa
e rational Bezier
urves
P(t) Q(u)
Let , be spa
e rational Bezier
urves of the form (8). The aim is to
(cid:28)nd their interse
tion.
The solution to this problem
an be found using already known fa
ts. A
-
ording to the theorem, whi
h was formed in the previous se
tion, reliefs of spa
e
rational Bezier
urves of the form (8) are spa
e nonrational Bezier
urves of the
k = 2
form (4) (in Fig. 2 Bezier
urves of the form (8) and their reliefs for are
[0,0,0]
shown). The
entral proje
tions of these nonrational
urves from the point
Figure 2
z −1 = 0
onto the plane are planar rational Bezier
urves de(cid:28)ned by (3). Their
interse
tion is possible to (cid:28)nd spe
i(cid:28)
ally by utsi∈ngh0t,h1eimethod of BePzise(rt)
lipping.
This method spe
i(cid:28)esuv∈alhu0e,s1oif the parametQers(u) of the
urve , respe
-
tivelyRthse parameter of the
urve , that
orrespond to the
ommon
point (the upper indexes "r" and "s" denote the reliefs and
entral proje
tions
of
urves or pRosints). This point is the interse
tion of both
urves . The preimage
of the point in the
entral proje
tion is the spa
e point, whi
h is the relief of
P(t) Q(u)
the interse
tion of the spa
e rational Bezier
urves and .
The following s
heme shows a whole pro
es of (cid:28)nding the interse
tion of the
P(t) Q(u)
and
urves:
P(t) −ϕ→ Pr(t) −ψ→ Ps(t)
s
B.−
li→pping R
Q(u) −ϕ→ Qr(u) −ψ→ Qs(u) )
s s s ψ−1 r ϕ−1
R = P (t)∩Q (u) −→ R −→ R
R = P(t)∩Q(u)
ϕ ϕ : E¯3 → E¯3
ψ (cid:21) relief perspektive ψ : E¯3 → E¯2
(cid:21)
entral proje
tion .
6
ψ
The
entral prPojse(
tt)ion Qissn(uot) a bije
tive mapping and in
ase that the inPte(tr)-
se
tion of
urves and exists (in opposite
ase the given
urves
Q(u) P(t) Q(u)
and
ertainly do not interse
t) the interse
tion of
urves Rsand does
Pnort(th)ave toQerx(ius)t. This situation o
urs when preimages of point on the
urves
and in the
entral proje
tion are di(cid:27)erent.
7 Con
lusions and future work
The relations between Bezier
urves and the relief perspe
tive have been de-
s
ribed. The ne
essary and su(cid:30)
ient
onditions for expressing the spa
e nonra-
tionalBezier
urve asthereliefofthespa
erationalBezier
urve have beenformed.
We havVe sh∈owE¯n that to the planar rwational
urve of the form (3), de(cid:28)ned by planar
i 2 i
points and their weights , it is possible to assign a
lass of the spa
e
urves by the relief perspe
tive. One of them is nonrational
urve de(cid:28)ned by (4)
and two are rational
urves de(cid:28)ned by (8) and (9).
The des
ribedmethod
anbe used asadire
tionforappli
ationhow to(cid:28)nd the
interse
tion of the spa
e rational
urves of the form (8). It is possible to express
every polynomial
urve in Bezier's representation and due to this representation
the method
an be applied to allpolynomial
urves after modi(cid:28)
ations (e.g. spline
urves whi
h are
onsidered as
urves
onsist of Bezier segments) for solution to
the
urve/
urve or
urve/line interse
tion problems.
Despite the author's attempt he did not su
eed in (cid:28)nding similar
omparable
published methods on website. In addition to this he is not able to
ompare his
results with any other. Obviously author's knowledge is limited and he would
appre
iate to get information about any other similar method.
In the future work we want to extend possibilities of the relief perspe
tive in
geometri
modeling. Our aim is to (cid:28)nd an answer whether any polynomial 3D
urve
an be
onverted to the spa
e
urve of the form (8).
1
Referen
es
[Bus53℄ Busemann, H., Kelly, P.J.: Proje
tive geometry and proje
tive metri
s.
New York, A
ademi
Press In
., Publishers 1953
[Cizm84℄ ƒiºmár, J.: Grupy geometri
ký
h transformá
ií. Alfa, Bratislava 1984
[Cenek59℄ ƒen¥k, G., Medek, V.: Deskriptívna geometria I. SVTL, Bratislava
1959
[Farin93℄ Farin,G.E.: CurvesandSurfa
esforComputerAidedGeometri
Design.
A
ademi
Press, 3. Edition 1993
1
some of the publi
ations have not equivalents in English
7
[Hlus99℄ Hlú²ek, R.: Shape modi(cid:28)
ations of Bezier
urves by relief perspe
tive,
Pro
eedings of SCG'99, Ko£ov
e
[Kader25℄ Kade°ávek, F.: Relief. Jan’ten
, Praha 1925
[Kader54℄ Kade°ávek, F., Kepr, B.: Prostorová perspektiva a reliefy. Nakladatel-
ství ƒeskoslovenské akademie v¥d, Praha 1954
[Mull23℄ M¶ller, E.: Vorlesungen ¶ber darstellende Geometrie I. Franz Deuti
ke
in Leipzig und Wien 1923
[Nishi90℄ Nishita, T., Sederberg, T.: Curve Interse
tion using Bezier
lipping.
CAD, Vol.22, No.9, November 1990
[Nishi98℄ Nishita, T.: Appli
ations of Bezier
lipping method and their Java ap-
plets, Pro
eedingsofSpring
onferen
eon
omputergraphi
s,Budmeri
e1998
8
