Applied Mathematics, 2011, 2, 1019-1026
doi:10.4236/am.2011.28141 Published Online August 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Intersection Curves of Implicit and Parametric
Surfaces in 3
Mohamed Abdel-Latif Soliman, Nassar Hassan Abdel-All,
Soad Ali Hassan, Sayed Abdel-Naeim Badr
Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
E-mail: sayed_badr@ymail.com
Received June 9, 2011; revised June 30, 2011; accepted July 7, 2011
Abstract
We present algorithms for computing the differential geometry properties of Frenet apparatus
and higher-order derivatives of intersection curves of implicit and parametric surfaces in 3 for transversal
and tangential intersection. This work is considered as a continuation to Ye and Maekawa [1]. We obtain a
classification of the singularities on the intersection curve. Some examples are given and plotted.

t,n,b,κ,τ
Keywords: Geometric Properties, Frenet Frame, Frenet Apparatus, Frenet-Serret Formulas, Surface-Surface
Intersection, Transversal Intersection, Tangential Intersection, Dupin Indicatrices
1. Introduction
The intersection problem is a fundamental process needed
in modeling complex shapes in CAD/CAM system. It is
useful in the representation of the design of complex ob-
jects, in computer animation and in NC machining for
trimming off the region bounded by the self-intersection
curves of offset surfaces. It is also essential to Boolean
operations necessary in the creation of boundary repre-
sentation in solid modeling [1]. The numerical marching
method is the most widely used method for computing
intersection curves in . The Marching method in-
volves generation of sequences of points of an intersec-
tion curve in the direction prescribed by the local differ-
ential geometry [2,3]. Willmore [4] described how to ob-
tain the unit tangent, the unit principal normal, the unit
binormal, the curvature and the torsion of the transversal
intersection curve of two implicit surfaces [5]. Kruppa [6]
explained that the tangential direction of the intersection
curve at a tangential intersection point corresponds to the
direction from the intersection point towards the intersec-
tion of the Dupin indicatrices of the two surfaces. Hart-
mann [7] provided formulas for computing the curvature
of the transversal intersection curves for all types of in-
tersection problems in Euclidean 2-space. Kriezis et al. [8]
determined the marching direction for tangential intersec-
tion curves based on the fact that the determinant of the
Hessian matrix of the oriented distance function is zero.
Luo et al. [9] presented a method to trace such tangential
intersection curves for parametric-parametric surfaces
employing the marching method. The marching direction
is obtained by solving an undetermined system based on
the equilibrium of the differentiation of the two normal
vectors and the projection of the Taylor expansion of the
two surfaces onto the normal vector at the intersection
point. Ye and Maekawa [1] presented algorithms for
computing all the differential geometry properties of both
transversal and tangentially intersection curves of two
parametric surfaces. They described how to obtain these
properties for two implicit surfaces or parametric-implicit
surfaces. They also gave algorithms to evaluate the
higher-order derivative of the intersection curves. Aléssio
[10] studied the differential geometry properties of inter-
section curves of three implicit surfaces in for trans-
versal intersection, using the implicit function theorem.
3
4
In this study, we present algorithms for computing the
deferential geometry properties of both transversal and
tangentially intersection curves of implicit and Paramet-
ric surfaces in as an extension to the works of [1].
3
This paper is organized as follows: Section 2 briefly
introduces some notations, definitions and reviews of
differential geometry properties of curves and surfaces in
. Section 3 derives the formulas to compute the prop-
erties for the transversal intersection. Section 4 derives
the formulas to compute the properties for the tangential
intersection. Some examples of transversal and tangen-
tially intersection are given and plotted in Section 5. Fi-
nally, conclusion is given in Section 6.
3
1020 M. A.-L. SOLIMAN ET AL.
2. Geometric Preliminaries [1, 11-13]
Let us first introduce some notation and definitions. The
scalar product and cross product of two vectors and
are expressed as
a c
,ac and respectively. The
,ac
length of the vector is a,.aaa
2.1. Differential Geometry of the Curves in
3
Let be a regular curve in with arc-length
parameterization,
3
:Iα3
 

123
,,
s
xsxsxsα (2.1)
The notation for the differentiation of the curve in
relation to the arc length s is
α

d,
d
α
s
s
α

2
d
d2,
α
s
s
 α

3
3
d
d
α
s
s
 α. Then from elementary differential geometry,
we have

s
αt (2.2)

sκ
 αn (2.3)

2,κs 
αα (2.4)
where is the unit tangent vector field and
t
α is the
curvature vector. The factor is the curvature and is
the unit principal normal vector. The unit binormal vector
is defined as
κn
b

s
btn
b
(2.5)
The vectors are called collectively the Frenet-
Serret frame. The Frenet–Serret formulas along α are
given by
,,,tnb



,
,
.
sκ
sκτ
sτ
 

tn
nt
bn
(2.6)
where is the torsion which is given by
τ
,
τκ

bα (2.7)
provided that the curvature does not vanish.
2.2. Differential Geometry of the Parametric
Surfaces in
3
Assume that is a regular parametric surface. In
other words where
12
,uvR
12
RR
0, (1,2
r
r
r
u

R
R)
de-
note to partial derivatives of the surface . The unit sur-
face normal vector field of the surface is given by
R
R
12
12
RR
NRR
(2.8)
The first fundamental form coefficients of the surface
are given by
R
,;,1,2
pqp q
gpqRR (2.9)
The second fundamental form coefficients of the surface
R
are given by
11 1112122222
,, ,,,LLLRNRNR N (2.10)
Let
,
r
us 1, 2r
in the 12
uu -plane defines a curve
on the surface which can be written as
R

21
,
s
usuαRs
2
u
(2.11)
Then the three derivatives of the curve are given by
α
11 2
u
RαR (2.12)
 
11 112122221 122
2uuu u
22
u u
 

 RRRRαR
222
uu
(2.13)
 


 
33
12111 12
11 1 11212122222
22
112121221
12
2
3
33
uuuu uuu
uu uu
uu
u
 
 
 




αRR R
RR R
RR
R
(2.14)
The projection of the curvature vector onto the unit
normal vector field of the surface is given by

α
R
 
22
12
11112 1 2
uu

22 2
12
,2Lu LLu
 

RR
RR
α
(2.15)
2.3. Differential Geometry of the Implicit
Surfaces in
3
Assume that
123
,, 0fxxx
is a regular implicit sur-
face. In other words 0
f, where

12 3
,,
f
fff is
the gradient vector of the surface
f
, p
p
f
f
x
, then the
unit surface normal vector field of the surface
f
is given
by
f
Nf (2.16)
Let
 
123
,,
s
xsxs sax (2.17)
be a curve on the surface
f
with constraint
123
,, 0fxxx
then we have
Copyright © 2011 SciRes. AM
M. A.-L. SOLIMAN ET AL.
1021


123
123
123
,, ,
,, ,
,, .
x
xx
x
xx
x
xx



 
 
α
α
α
(2.18)
112233
d0
d
ffxf xfx
s
   (2.19)
 

22
22
11 122233 3
2
12 1 213 1 3232 3
112233
d
d
2
0
f
f
xfx fx
s
f
xxfxxfxx
fxf xfx

 
 

 

(2.20)
The projection of the curvature vector onto the unit
normal vector field of the surface

α
f
is given by
222
123
,η
f
ff


f
αf (2.21)
where
 


2
22
11 1222333
12 1213 1 32323
2
η
f
xfx fx
f
xxfxxfxx

 
 

3. Transversal Intersection Curves
Consider the intersecting implicit and parametric surfaces
and

123
,, 0fxxx
12
,;Ru uR
0,fRR
324
such that, 12 . Then the
intersection curve of these surfaces can be viewed as a
curve on both surfaces as
112
c,uc
0cuc
 


123 123
s,s,s ;,,0,sx xxfxxxα
 

121123 24
s,s ;,.
s
uu cuccucαR
Then we have
 

12
s,s ,1,2,3
i
isxRuui
where Then the surface
 

123
12
s,s,,.uu RRRR
f
can be expressed as


123
12
,,,huufRRR0 (3.1)
Thus the intersection curve is given by



12 12
1123 24
s,s ;,0,
,
suu huu
cuccu c

 
Rα (3.2)
3.1. Tangential Direction
Differentiation (3.1) yields
1212
0hhuu
(3.3)
where ,
i
i
h
hu
then we have
1
2
2
21
,uu
hh
h
0 (3.4)
Since
α is the unit tangent vector field of the curve
(3.2), then we have
22211112
,uuuu
 
RR RRα1
1
(3.5)
which can be written as
 
11
22
11 122222
g2gguuuu


(3.6)
Substituting (3.4) into (3.6) yields
 

 

1
22
2
221112121221
2
1
22
2
12111 212122
g2 ,
g2
hhhhg hg
hhhhgh
u
ug

 .
(3.7)
The unit tangent vector field of the intersection curve is
given by substituting (3.7) into (2.12) as follows
21 12
;
ζhh
ζ
tζRR
(3.8)
3.2. Curvature and Curvature Vector
The curvature vector is given by differentiation (3.8)
with respect to s as follows
 

2
3
122
211122 12
12
21212211 122112
,
2
hh hh
h hhhhhh h


 
 ζζ ζζζ
ζ
ζζ RR
R
α
R
R
(3.9)
The unit principal normal vector field, the curvature
and the unit binormal vector are given by using (2.3) (2.4)
and (2.5) as follows
2
2
2
3
2
2
,,
,
,
,
,.
,
κ




 
ζζ ζζζ
n
ζζ ζζζ
ζζ ζζζ
ζ
ζζ ζζζ
ζ
bζζζζζζ
(3.10)
3.3. Torsion and Higher-Order Derivatives
Equation (3.7) can be written as
Copyright © 2011 SciRes. AM
1022 M. A.-L. SOLIMAN ET AL.
2
12
1
,
h
uh
u


ζζ
(3.11)
Differentiation (3.13) we obtain

12 22
12
11 12
2
1212111121 12
221 212122 1
12
21 2
1
222
,,
,,
.
ζ
hh
u
hh
hhhh
hhh
uu
uu
u
uh
u

 





 





 
ζζ
ζζ
ζ
ζζ
ζζ
ζRR RR
RR RR
(3.12)
Differentiation (3.12) we obtain


 
2
1222 112
11
2
22
12
4322
222
21
122 222
3
11 12
21 2
22
11
111 122
1
12 22
2
3
12
,
,, ,
2
,
2,
,
,
uu
u
uu u
hhh
uu
h
h
hh
h
u
u
uu
h
u
h
hh
u
u

 





 










 
 


ζζ
ζζζ
ζ
ζζζζζ ζζ
ζζζζ
ζζ
ζζ ζ
ζζ
ζζ
ζ
ζζ
ζζ ζ




2
12
4322
1121211 112112
22 12121221 22
2112 112 112 111
111211 121 112
2222 122 122 122
1
1
2
22 212 22
2
1
2
,, ,
2,
2
2
2
2
h
uh hhh
hhhh
hhh
hhh
hhh
h
uh
u
u
u
2
 




 





 



ζζζζζ ζζ
ζζζζ
ζRR RR
RR RR
RRR
RRR
RRR
RR
1 222
122 1112222 11
112221121 12
12
2
2
2.
2
h
hhh
hh
uu h







 

R
RRR
RRR

(3.13)
Substituting 11122
and 2 into (2.14) we
obtain the third-order derivative vector of the intersection
curve. Hence the torsion can be obtained by (2.7).
,,,,uuuuu   u
We can compute all higher-order derivatives of the in-
tersection curve by a similar way.
4. Tangentially Intersection Curves
Assume that the surfaces and

123
,, 0fxxx
12
,RuuR; 1123 24
,cuccu c

P
are intersecting
tangentially at a point on the curve (3.2) then the unit
surface normal vector field of both surfaces are parallel to
each other. In other words
12
12


RR
f
fRR
which can be written as

12
12
,AA

f
fRR RR
(4.1)
Then we can write

23 32
11212
31 13
21212
12 21
31212
,
,
.



fRRRR
fRRRR
fRRRR
A
A
A
(4.2)
Since

12
,,1,2,sususi
i
xR 3,
u
i
i
then we have
11 22
ii
u
xR R
(4.3)
4.1. Tangential Direction
Projecting the curvature vector onto the two unit nor-
mal vectors of both surfaces yields

α
12
12
,,



α
RR
fRR
α
f (4.4)
Using (2.15) (2.21) and (4.4) we obtain
 


 

2
22
11 1222333
12 1 213 1 32323
2'
12111 1212
2
22 2
2ff
2u
fxf xf x
fxx xxxx
ALuLuLu


  




 
RR
(4.5)
Substituting (4.3) into (4.5) yields
2
1112 22
11
2
22
2aa 0,0
uu
uu
au

 
 


 (4.6)
where

 
 
 

22
12
1112 11111221
2
312231
3311211231 11311
22
12
1 21122
2
312231
3312231
222222
2222223
112 2
1 21122
3312 2
3
2
12121212
1
12 1231221
2,
2,
aA Lff
ffff
aA Lff
ffff
aA Lff
ff
f
 
 
 
 
 
 
RRR R
RRRRRR
RRR R
RRRRRR
RRRR RR
RRRRRR

23 321331
23121 213121 2
f. RR RRRR RR
3
3
R
R
Copyright © 2011 SciRes. AM
M. A.-L. SOLIMAN ET AL.
1023
R
],
R
This can be written in a matrix form as follows
T
ijij ij
 afRRH (4.7)
where
and
123T
123
[],[
ijij ij ij
fff RRR fR
123T
[]
iiii
RRR
11 1213
12 22 23
13 2333
f
ff
f
ff
f
ff





H
.
is the
Hessian matrix of the surface
f
Solving (4.6) for 1
2
u
u
yields
2
121211 22
12
11
(a )
,aa
uBu Ba
 

 a
(4.8)
Substituting (3.7) and (4.7) into (4.8) we obtain


1
22
1111222
1
22
2111222
2
2.
uBBgBgg
uBgBgg


(4.9)
Then the unit tangent vector field of the intersection
curve is given by
12
12
B
B
RR
tRR
(4.10)
From the previous formulas, it is easy to see that, there
are four distinct cases for the solution of (4.6) depending
upon the discriminant these cases are
as the following [1]

2
121122
Δ,aaa
Lemma 1. The point is a branch point of the inter-
section curve (3.2) if and there is another intersec-
tion branch crossing the curve (3.2) at that point.
P
0Δ
Lemma 2. The surfaces and intersect at the
point and at its neighborhood, if and
fh
ΔP0

22 2
111222 0.aaa(Tangential intersection curve).
Lemma 3. The point is an isolated contact point of
the surfaces
P
f
and , if hΔ0
.
Lemma 4. The surfaces
f
and have contact of at
least second order at the point , if .
(Higher-order contact point).
h
P1112220aaa
4.2. Curvature and Curvature Vector
Differentiation (4.6) and using (4.9) we obtain


121
2
1112 22
1211 12
11 12
,
2;0
uBu a
aBaB a
au aBa
aBa
 



 
,
.
(4.11)
where


TTT
11 111
TT T
22 22
11 1213
123 122223
13 23 33
()
,
HH H,
ijijiji jj
ijijijij
iii
iiii
iii
au
u
fff
fff
fff
 
 






tHRfRRHR RHR
fRRHRRHRR QR
QtH
(4.11)
Since the curvature vector is perpendicular to the tangent
vector, then we have ,

αα 0. Using (2.12) (2.13) and
(4.9) we obtain
21 324
au aua

(4.13)
where

2111231222
232
42 111121
12 2221222112
,,
,2B,
2,, ,,
aBg gaBgg
auB
BB

 

RR RR
RRRRRRRR
Solving the linear system (4.11) and (4.13) yields
34 4
1
32
412
2
32
,
B
aa aB
uaaB
aaa
uaa


(4.14)
The curvature vector of the intersection curve is obtained
by substituting 11
,,uuu
2
,
and into (2.13).
2
u
4.3. Torsion
If we have a branch point, then we can compute the torsion
by taking the limit of the torsion of transversal intersection
curve at this point. If we have tangential intersection curve,
then we can compute 1
u
 and by differentiation 12
u u
and 2.u
Substituting 11122
,,,,uuuuu
 
, and 2into (2.14)
we obtain the third-order derivative vector of the intersec-
tion curve. Then we can obtain the torsion by using (2.7).
u
5. Examples
Example 1. Consider the intersection of the implicit and
the parametric surfaces

22
12
12 22
90,
,3sin,3cos; 02
fxx
uu uu

R
(5.1)
as shown in Figure 1.
Transversal intersection: Using (3.1) yields
22
12
9cos 0huu
 (5.2)
Copyright © 2011 SciRes. AM
1024 M. A.-L. SOLIMAN ET AL.
The intersection curves
P
(0, 1, 0)
Figure 1. Transversal and tangential intersection.
Differentiation (5.1) and (5.2) we obtain




122 21
222 1112111
222 2
122
222 2
2222 2
2, 9sin2,1,0,0,
18cos 2,0,
36sin2,
1, 0, 0,30, cos,sin,
30,sinsinu,cos ,
30,coscosu, sin.
huhu
huhhh
hu
uu
u
u
 




 
R
RR
R
R
2
(5.3)
Using (3.8) and (5.2), we obtain
2112
222
22
2
sin tan
,,
1sinu31sin31sin
cos 0.
uuuu
uu
u





t
2
,
(5.4)
Using (3.12) and (5.2), hence

221212
1222
222
2222
2
22
3
12
2
2
18sincos ,6cos ,6sin,
2cos 26sin26cos 2
,,
cos1 sin1 sin1 sin
18cos1 sin,
72 sin
,.
1sin
uuuuuu
uuu u
uuuu
uu
uu
u








ζ
ζ
ζ
ζζ
,
(5.5)
Using (2.4), (2.5), (3.12), (3.13) and (5.4) then we have




12
22
222
22
12
22 2
22
3
22
2
12 22
2
2
2sin cos
,,
91 sin31sin31sin
2sin cos
,,
321sin1sin21sin
21sin,
3
cos2sin tan1
,0,.
2
321sin
uu
uuu
uuu
uu u
κu
uu uu
u
2
2
2
2
2
,
,
u

 




 


 






α
n
b
(5.6)
Using (3.15) and (3.16) hence
21
12
22
22
12
12
22 22
22
sin ,,
1sinu9cos 1sin
sin cos
,.
9(1 sin)9(1 sin)
uu
uu
u
uu
uu
uu




 


2
2
u
u
(5.7)
Using (3.17) and (5.7) hence


2
22
17
22
2
122 22
27
22
2
sin2 3cos,
91 sin
(2sintancoscos 2).
81 1sin
uu
u
u
uuu uu
u
u




(5.8)
Using (2.7) and (2.14) yields
 

22
22112
77
22
22
22
22
222 22
7
22
2
3(23cos)sin2u6usin
,,
27(1 sin)27(1 sin)
4sincossin1 sinsin
27 1sin
uu u
uu
uuu uu
u
 



α
(5.9)
12 22
2
2
234
222222
53
221 22122
24
122122
24tan4sin10 cossin
42cosu
4cos sin7cossincos sin
cossin2 cossin3 cossin
2 costan6 costan
uu uuu
uuuu uu
uuuuuu uu
uuuuuu


 

2
(5.10)
Tangentially intersection: The surfaces are intersect-
ing tangentially at the points . Consider the
0,1, 0P
point
10,1, 0,P using (4.7) (4.8) (4.9) and (5.3), then we
have
12 3
12
2, 0,18,
1
3,, .
23
aa a
Bu u
1
2


  (5.11)
Then this means that the point is a branch
point (Figure 1). From (4.10) and (5.11), we obtain
Δ0,1
P
11
,0,
22


t
(5.12)
Using (2.13) and (4.14) hence
Copyright © 2011 SciRes. AM
M. A.-L. SOLIMAN ET AL.
1025


21
10,1, 0,
6
1
0,1, 0,,
6
11
,
0,
0, .
22
uu
κ







n
b
α
(5.13)
Using (5.10) at , we obtain
10,1, 0P
2
4
22
2
π2
2
22
122 22
33
122 22
5
222212
4
122122
1
lim42 sin2cossin
42cos
22costan42 cossin
32 cossin72cossin
10 2cossin2cossin4 2tan
2 2cossin6 2costan
0
u
uu
u
uuu uu
uuu uu
uuuuuu
uuuu uu





2
u

(5.14)
Example 2. Consider the intersection of the implicit and
the parametric surfaces

222
123
12 22
90,
R,3sin,3cos,0 2
fxxx
uu uu


(5.15)
as shown in Figure 2.
At 1, 0x
12
// .fRR
0,
Using (4.7) and (5.15),
we have Δ this means that the surfaces are intersect-
ing tangentially in a curve as (Figure 2). Then from (4.8)
and (4.9), we have
12
1
0, 0,3
uuB

(5.16)
Using (4.10) we have
2
0, cos,sinuut
2
2
(5.17)
Using (5.16) hence
112
0, 0uuuu
 
 

(5.18)
Using (2.4) and (2.13) hence the curvature vector and the
curvature are given by
10
x
Figure 2. Tangential intersection.
P (0, 3, 0)
Figure 3. Tangential intersection.


22
22
10, sin, cos,
3
1
0, sin, cos,3
uu
uuκ
 
 
α
n
(5.19)
Using (2.5) (2.7)and (2.14) hence


22
10, cos,sin,
9
1, 0, 0,0.
uu
τ
 
 
α
b
(5.20)
Example 3. Consider the intersection of the implicit and
the parametric surfaces

22
12
12
(6)90,
,3 3sin,3cos.
fx x
uu
  
R2
u
(5.21)
as shown in Figure 3.
At the point
0,3, 0P,
 
12
// .fRR
0,
Using (4.7)
and (5.21), we have Δ
this means that the point
is an isolated tangential contact point (Figu re 3 ).
P
Example 4. Consider the intersection of the implicit
and the parametric surfaces
 

222
31 2
222
122112
0,
1,1,
fxxx
uuuuuu

R3
.
(5.22)
as shown in Figure 4.
P (1, 0, 1)
Figure 4. Transversal intersection.
Copyright © 2011 SciRes. AM
M. A.-L. SOLIMAN ET AL.
Copyright © 2011 SciRes. AM
1026
At the point

1, 0,1,
f
R
PSS
on the intersection
curve (Figure 4), we have


 
111 111
22212
112122 222
11122 222
2 12122112
1, 0, 2,0, 0, 2,24,
0,2,0,2, 0, 0,0,2,0,
0,2,0,2, 0, 0,0, 0,6,
2, 10,12,
0,24, 0.
h
hh hh
hh hh

 
 
 
 
RR
RRR
RRR
(5.23)
Using (3.8) and (5.23), we obtain
0,1,0t (5.24)
Using (3.12) (3.13) and (5.23) we obtain

2, 0,3,
23
(,0,),13
1313 κ


α
n.
(5.25)
Using (2.5) (2.7) (2.14) (3.17) and (5.25) we obtain
33
,19, ,
44
32
,0,,.
52
1313 τ


 






α
b3
(5.26)
6. Conclusions
Algorithms for computing the differential geometry prop-
erties of intersection curves of implicit and parametric sur-
faces in are given for transversal and tangential inter-
section. This paper is an extension to the works of Ye and
Maekawa [1]. They gave algorithms to compute the dif-
ferential geometry properties of intersection curves be-
tween two parametric surfaces then they applied it on a
simple example for implicit and parametric surfaces inter-
section. This paper presented direct and simple formulas to
compute all differential geometry properties, which may
reduce the time it takes to calculate those properties. The
types of singularities on the intersection curve are charac-
terized. The questions of how to exploit and extend these
algorithms to compute the differential geometry properties
of intersection curves between three surfaces in , can
be topics of future research.
3
4
7. Acknowledgements
The authors would like to thank the reviewers for their
valuable comments and suggestions.
8. References
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