Differential geometry of self-intersection curves of a parametric surface in R3

Available online at http://scik.org

J. Math. Comput. Sci. 6 (2016), No. 6, 1108-1132 ISSN: 1927-5307

DIFFERENTIAL GEOMETRY OF SELF-INTERSECTION CURVES OF A

PARAMETRIC SURFACE IN _R3

SOAD A. HASSAN1,∗_{, SAYED ABDEL-NAEIM BADR}1,2

1_{Department of Mathematics, Assiut University, Assiut 71516, Egypt}

2_{Department of Mathematics, University College, Umm Al-Qura University, Al-Qunfudah, KSA}

Copyright c2016 Hassan and Badr. This is an open access article distributed under the Creative Commons Attribution License, which

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.This paper presents algorithms for computing all the differential geometry properties of Frenet apparatus

of ”self-intersection curves of a parametric surface and the intersection curves of two parametric surfaces” inR3,

for transversal and tangential intersection. Some examples are given and plotted.

Keywords: geometric properties; Frenet frame; Frenet apparatus; self-intersection; surface-surface intersection;

transversal intersection; tangential intersection.

2010 AMS Subject Classification:53A25.

1.

Introduction

The intersection (also the self-intersection) problem is a fundamental process needed in model-ing complex shapes in CAD/CAM system. It is useful in the representation of the design of com-plex objects, 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 nec-essary in the creation of boundary representation in solid modeling [18]. Self-intersections

∗_{Corresponding author}

Received June 19, 2016

of a rational polynomial parametric surface are defined by finding pairs of distinct parameter values (u,v)6=S(p,q) such that S(u,v) =S(p,q) [21]. Once the shape’s offset goes beyond the local radius-of-curvature, a self-intersection is bound to occur in the offset. Other appli-cations could benefit from proper self-intersection detection and elimination as well. Another example, demonstrated herein, is the problem of creating a self-intersection-free metamorpho-sis between freeform curves [31]. The numerical marching method is the most widely used method for computing intersection curves inR3. The Marching method involves generation of

provide an algorithm for the evaluation of geometry properties for tangential intersections of two implicit surfaces in_R3.

In this paper, we study the differential geometry properties of the intersection curves of two parametric surfaces and the Self-intersection curves of a parametric surface in_R3.The intersec-tion can be transversally or tangentially. The type of intersecintersec-tion may vary point to point along the intersection curve. Finally some examples are given and plotted.

2.

Geometric preliminaries

Let us first introduce some notations and definitions. Bold letters such asa,Rwill be used for vectors and vector functions, respectively. The scalar product and cross product of two vectorsa

andcare expressed asha,cianda×c,respectively. The length of the vectoraiskak=pha,ai.

2.1. Differential geometry of the curves in _R3. Let α: I ⊂_R−→_R3 be a regular curve in

R3with arc-length parametrization,

(2.1) α(s) = (x1(s),x2(s),x3(s))

The notations for differentiation of the curveαwith respect to the arc lengthsareα0(s) = d_dsα, α00(s) = d

2 α

ds2,α000(s) = d 3

α

ds3. From the elementary differential geometry, we have

(2.2) α0(s) =t

(2.3) α00(s) =κn

(2.4) κ2(s) =α00,α00

wheret is the unit tangent vector andα00 is the curvature vector. The factor κ is the curvature andnis the unit principal normal vector. The unit binormal vectorbis defined as

(2.5) b(s) =t×n

The Frenet-Serret formulas alongα are given by

whereτ is the torsion which is given by

(2.7) τ = hb,α

000_i

κ

provided that the curvature does not vanish.

2.2. Differential geometry of the parametric surfaces in_R3. Assume thatR(u₁,u₂)is a reg-ular parametric surface withR1×R26=0,whereRr=

∂R

∂ur

(r=1,2)denote to the partial deriva-tives of the surfaceR. The unit normal vector field on the surfaceRis given by

(2.8) N= R1×R2

kR₁×R₂k

The coefficients of first fundamental form are given by

(2.9) g_pq=R_p,R_q; p,q=1,2

The coefficients of second fundamental form are given by

(2.10) L11=hR11,Ni, L12=hR12,Ni, L22=hR22,Ni

Letu_r =u_r(s),r=1,2 be functions in theu₁u₂-plane which defines a curve on the surfaceR

as

(2.11) α(s) =R(u1(s),u2(s)).

Then the fourth derivatives of the curveα are given by

(2.12) α0=R1u01+R2u02,

(2.13) α00=R11(u01)2+2R12u01u02+R22(u02)2+R1u001+R2u002,

(2.14) α000= R111(u0₁)3+3R112(u0₁)2u0₂+3R122u0₁(u₂0)2+R222(u0₂)3

(2.15) α(4)(s) = (u0₁)4R1111+ (u0₂)4R2222+4(u0₁)3u0₂R1112+6(u0₁)2(u0₂)2R1122

+4u0₁(u0₂)3R1222+6(u0₁)2u00₁R111+6(u0₂)2u00₂R222

+6(2u0₁u0₂u₁00+ (u0₁)2u00₂)R₁₁₂+6(u00₁(u0₂)2+2u0₁u0₂u00₂)R₁₂₂

+(3(u00₁)2+4u0₁u000₁)R11+ (3(u00₂)2+4u0₂u000₂)R22

+2(2u000₁u0₂+3u00₁u00₂+2u₁0u000₂)R₁₂+u₁(4)R₁+u(₂4)R₂

The projection of the curvature vector α00, the third order derivative vectorα000 and the fourth order derivative vectorα(4) onto the unit normal vector of the surfaceR, respectively are given by

(2.16)

α00, R1×R2

kR₁×R₂k

=L₁₁(u0₁)2+2L₁₂u0₁u0₂+L₂₂(u0₂)2,

(2.17) α000,N= (u0₁)3hR111,Ni+3(u0₁)2u0₂hR112,Ni+3u₁0(u0₂)2hR122,Ni

+(u0₂)3hR₂₂₂,Ni+3(u0₁L₁₁+u0₂L₁₂)u00₁+3(u0₁L₁₂+u0₂L₂₂)u00₂,

(2.18)

D α(4),N

E

= (u0₁)4hR₁₁₁₁,Ni+4(u0₁)3u0₂hR₁₁₁₂,Ni+6(u0₁)2(u0₂)2hR₁₁₂₂,Ni

+ (u0₂)4hR₂₂₂₂,Ni+4u0₁(u0₂)3hR₁₂₂₂,Ni+6(u0₁)2u00₁hR₁₁₁,Ni

+6(u0₂)2u00₂hR₂₂₂,Ni+6(2u₁0u0₂u00₁+ (u0₁)2u00₂)hR₁₁₂,Ni

+6(u00₁(u0₂)2+2u0₁u0₂u00₂)hR₁₂₂,Ni+3(u00₁)2L₁₁+6u00₁u00₂L₁₂

+3(u00₂)2L22+4(u0₁L11+u0₂L12)u000₁ +4(u0₁L12+u0₂L22)u000₂.

2.3. Self-intersection of a parametric surface. Self-intersection point pof a parametric sur-faceR=R(u₁,u₂);c₁<u₁<c₂,c₃<u₂<c₄is defined by finding two pairs of distinct param-eter values(γ₁,γ₂)6= (ν1,ν2)in theu1u2-plane, such that p=R(γ1,γ2) =R(ν1,ν2).[21].

Consider a surfaceR=R(u₁,u₂);c₁<u₁<c₂,c₃<u₂<c₄, which intersect it self at a curve (with arc length parametrization)α(s). Assume that(v1(s),v2(s)) and(w1(s),w2(s)) are two distinct paths in theu₁u₂-plane, defines the curveα(s)(see Fig. 2.1), then we can write

(2.19) α(s) =R(v1(s),v2(s)) =R(w1(s),w2(s)); (v1(s),v2(s))6= (w1(s),w2(s))

We can consider the surfaceR(u1,u2)as two distinct regular surfacesP(v1,v2)and Q(w1,w2) which intersect at the curveα(s),where

Thus the curveα(s)can be viewed as a curve on both surfaces as

(2.21) α(s) =P(v1(s),v2(s)) = (P

1_,P2_,P3_),

α(s) =Q(w1(s),w2(s)) = (Q1,Q2,Q3).

According to (2.21), we can write

(2.22) Pj(v₁(s),v₂(s)) =Qj(w₁(s),w₂(s)); j=1,2,3.

FIGURE 1. Fig 2.1

3.

Transversal intersection

Assume that the surfaces (2.20) are intersecting transversally at the curve (2.21). 3.1. The unit tangent vector field. Differentiation (2.22) with respect tosyields

(3.1)

P₁1v0₁+P₂1v0₂=Q1₁w0₁+Q1₂w0₂,

P₁2v0₁+P₂2v0₂=Q2₁w0₁+Q2₂w0₂, P₁3v0₁+P₂3v0₂=Q3₁w0₁+Q3₂w0₂

Since the surfaceQ(w₁(s),w₂(s))is regulare, thene without loss of generality, we have

Ql₁ Ql₂ Qm₁ Qm₂

6

=0, {l,m} ⊂ {1,2,3}

The system (3.1) can be written as

(3.2) P

l

1v01+P2lv02=Ql1w01+Ql2w02,

(3.3) P₁qv0₁+P₂qv0₂=Qq₁w0₁+Qq₂w0₂, {l,m,q}={1,2,3}.

whereP_ij=P_ij(v₁(s),v₂(s)) = ∂P

j

∂ui

,Q_ij=Q_ij(w₁(s),w₂(s)) =∂Q

j

∂ui

, i=1,2. Solving the coef-ficientsw0₁andw0₂from linear system (3.2) and substituting into (3.3)yields

(3.4) v0₁=v0₁, v0₂=−η

ζ

v0₁.

where

(3.5)

η=Qq₁A12−Q₂qA11−B12P₁q, ζ =Qq₁A22−Q₂qA21−B12P₂q,

A_{i j} =

P_il P_im

Ql_j Qm_j

, B₁₂=

Ql₁ Qm₁

Ql₂ Qm₂

, i,j=1,2.

Since

(3.6)

2

∑

g_{i j}v0_iv0_j=1, g_{i j}=P_i,P_j.

Substituting (3.4) into (3.6) yields

(3.7) v01=

ζ q

g11ζ2−2g12η ζ+g22η2

, v0₂= _q −η

g11ζ2−2g12η ζ+g22η2

.

Differentiation (2.21) with respect tosyields

(3.8) t=α0(s) =P1v01+P2v02=Q1w01+Q2w02

The unit tangent vectorα0(s)can be obtain by substituting (3.7) into (3.8), as follows

(3.9) t= ζP1−ηP2

kζP1−ηP2k

.

Using (3.7), (3.8) and (3.9), we obtain

(3.10) v0₁= ζ

kζP1−ηP2k

, v0₂= −η

kζP1−ηP2k ; Using (3.8), (3.9) and (3.10), we obtain

(3.11) w0₁= ζA12−ηA22

B₁₂kζP1−ηP2k

, w0₂= ηA21−ζA11

B₁₂kζP1−ηP2k

The tangent vector field of the self-intersection curve of the surfaceR=R(u₁,u₂)is given by (3.12) t= ζR1(v1,v2)−ηR2(v1,v2)

kζR1(v1,v2)−ηR2(v1,v2)k

.

Using (3.10) and (3.11) we obtain

(3.13)

v0₁= ζ

kζR1(v1,v2)−ηR2(v1,v2)k

, v0₂= −η

kζR1(v1,v2)−ηR2(v1,v2)k

,

w0₁= ζA12−ηA22

B₁₂kζR1(v1,v2)−ηR2(v1,v2)k

, w0₂= ηA21−ζA11

B₁₂kζR1(v1,v2)−ηR2(v1,v2)k

,

where

(3.14)

η =A12Rq₁(w1,w2)−A11Rq₂(w1,w2)−B12Rq₁(v1,v2), ζ =A22Rq₁(w1,w2)−A21Rq₂(w1,w2)−B12Rq₂(v1,v2),

Ai j =

Rl_i(v₁,v₂) Rm_i (v₁,v₂)

Rl_j(w1,w2) Rm_j(w1,w2)

, B12=

Rl₁(w₁,w₂) Rm₁(w₁,w₂)

Rl₂(w1,w2) Rm₂(w1,w2) .

3.2. Curvature and curvature vector. Assume that the intersection curveα(s)is given by (3.15) α(s) = (x1(s),x2(s),x3(s))

Then we have

(3.16) α

0_{(s) = (x}0

1(s),x02(s),x03(s)), α000(s) = (x0001(s),x0002(s),x0003(s)), α00(s) = (x00₁(s),x00₂(s),x00₃(s)), α(4)(s) = (x(₁4)(s),x₂(4)(s),x(₃4)(s)).

Since the curvature vector is perpendicular on the tangent vector, then we have (3.17)

h

x0₁ x0₂ x0₃

i h

x00₁ x00₂ x00₃

iT

=0

The projection of curvature vector fieldα00(s) of the curve (2.20) onto the unit normal vector fieldsN1= (N₁1,N₁2,N₁3)andN2= (N₂1,N₂2,N₂3)of both surfaces (2.21) are given by

(3.18)

h

N₁1 N₁2 N₁3

i h

x00₁ x00₂ x00₃

iT

= (v0₁)2L1₁₁+2v0₁v0₂L1₁₂+ (v0₂)2L1₂₂,

h

N₂1 N₂2 N₂3

i h

x00₁ x00₂ x00₃

iT

Solving the system (3.17)and (3.18) forx00₁,x₂00andx₃00,we obtain (3.19)     

x00₁ x00₂

x00₃

     =     

x0₁ x0₂ x0₃ N₁1 N₁2 N₁3

N₂1 N₂2 N₂3



   

−1

   

0

(v0₁)2L1₁₁+2v0₁v0₂L1₁₂+ (v0₂)2L₂₂1

(w0₁)2L2₁₁+2w0₁w0₂L2₁₂+ (w0₂)2L2₂₂



   

The curvature vector field α00(s) can be computed by using (3.12), (3.13) and (3.19). The curvatureκ is given by

κ2=α00,α00

The curvature and curvature vector of self-intersection curves of the surfaceR(u1,u2)are given by replacing the surfacesP(v₁,v₂)andQ(w₁,w₂)withR(v₁,v2)andR(w1,w2),respectively. 3.3. Torsion and third order derivative. Since the intersection curve α(s) views as a curve on both surfaces (2.20), then the Eq. (2.13) satisfies on both surfaces thus

(3.20) α00= (v0₁)2P11+2v01v02P12+ (v02)2P22+v100P1+v002P2,

(3.21) α00= (w0₁)2Q₁₁+2w0₁w0₂Q₁₂+ (w0₂)2Q₂₂+w₁00Q₁+w00₂Q₂.

Taking the cross product of both hand sides of (3.20) withP₁andP₂and projecting the results vector onto the surface normal vectorN1,we obtain

(3.22)

v00₁= |α

00_,P

2,N1|

kP1×P2k

−(v0₁)2|P11,P2,N1|

kP1×P2k

−2v0₁v0₂|P12,P2,N1|

kP1×P2k

−(v0₂)2|P22,P2,N1|

kP1×P2k

,

v00₂= |P1,α

00_,N

1|

kP₁×P₂k −(v

0

1)2

|P₁,P11,N1|

kP₁×P₂k −2v

0

1v02

|P₁,P₁₂,N₁| kP₁×P₂k −(v

0

2)2

|P₁,P₂₂,N₁| kP₁×P₂k .

Taking the cross product of both hand sides of (3.21) withQ₁andQ₂and projecting the results vector onto the surface normal vectorN₂,we obtain

w00₁ = |α

00_,Q

2,N2|

kQ₁×Q₂k −(w

0

1)2

|Q₁₁,Q₂,N₂| kQ₁×Q₂k −2w

0

1w02

|Q₁₂,Q₂,N₂| kQ₁×Q₂k −(w

0

2)2

|Q₂₂,Q₂,N₂| kQ₁×Q₂k ,

(3.23)

w00₂ = |Q1,α

00_,N

2|

kQ₁×Q₂k −(w

0

1)2

|Q₁,Q₁₁,N₂| kQ₁×Q₂k −2w

0

1w02

|Q₁,Q₁₂,N₂| kQ₁×Q₂k −(w

0

2)2

|Q₁,Q₂₂,N₂| kQ₁×Q₂k .

then we have

α0,α000=−κ2

Which can be written in the matrix form as

(3.24)

h

x0₁ x0₂ x0₃

i h

x000₁ x₂000 x000₃

iT

=−κ2

The projection of the third order derivative α000(s)of the intersection curve α(s)onto the unit normal vector fields N₁= (N₁1,N₁2,N₁3) and N₂ = (N₂1,N₂2,N₂3) of both surfaces P(v₁,v₂)and

Q(w₁,w₂)are given by

(3.25)

h

N₁1 N₁2 N₁3

i h

x000₁ x000₂ x000₃

iT

=ψ₁,

h

N₂1 N₂2 N₂3

i h

x000₁ x000₂ x000₃

iT

=ψ₂,

where

(3.26)

ψ₁= (v0₁)3hP111,N1i+3(v0₁)2v₂0 hP112,N1i+3v0₁(v0₂)2hP122,N1i

+(v0₂)3hP₂₂₂,N₁i+3v0₁v00₁L1₁₁+3(v00₁v0₂+v0₁v00₂)L1₁₂+3v0₂v00₂L1₂₂,

ψ₂= (w0₁)3hQ₁₁₁,N2i+3(w01)2w20 hQ112,N2i+3w01(w02)2hQ122,N2i

+(w0₂)3hQ₂₂₂,N₂i+3w0₁w00₁L2₁₁+3(w00₁w0₂+w0₁w00₂)L2₁₂+3w0₂w00₂L2₂₂.

Solving the system (3.25) and (3.25), we obtain

(3.27)     

x000₁ x000₂

x000₃

     =     

x0₁ x₂0 x0₃ N₁1 N₁2 N₁3

N₂1 N₂2 N₂3



   

−1

   

−κ2

ψ₁ ψ₂      .

The third order derivative α000(s) can be computed by using (3.12), (3.18), (3.26) and (3.27). The torsionτis given by (2.7). The torsion and third order derivative of self-intersection curves of the surfaceR(u₁,u₂)are given by replacing the surfacesP(v₁,v₂)andQ(w₁,w₂)toR(v₁,v₂)

4.

Tangentially Intersection curves

Assume that the surfaces (2.20) are intersecting tangentially at a point p on the curve (2.21), then the unit surface normal vector fields of both surfaces are parallel to each other. in other words

(4.1) P1×P2

kP₁×P₂k =δ

Q₁×Q₂

kQ₁×Q₂k, δ =±1.

4.1. Tangential direction. Differentiation (2.22), with respect tosyields (4.2) P₁jv0₁+P₂jv0₂=Q₁jw0₁+Q₂jw₂0, j=1,2,3 whereP_ij=P_ij(v1(s),v2(s)) =

∂Pj

∂ui

,Q_ij=Q_ij(w1(s),w2(s)) = ∂Qj

∂ui

, i=1,2

since the unit surface normal vector fields of both surfaces are parallel to each other, then the system (4.2) reduced to only two Eqs..Projecting the curvature vectorα00(s)onto the two unit normal vector fields of both surfaces and using (4.1), we obtain

α00, Q1×Q2

kQ₁×Q₂k

=δ

α00, P1×P2

kP₁×P₂k

Using (2.16) and (4.1), we obtain (4.3)

2

∑

L2_{i j}w0_iw0_j=δ 2

∑

L1_{i j}v0_iv0_j

where

L1_{i j} =

P_{i j},N₁, L2_{i j}=

Q_{i j},N₂,

Assume that the system (4.2) reduced to

(4.4) P

l

1v01+Pl2v02=Ql1w01+Ql2w02,

Pm₁v0₁+P₂mv0₂=Qm₁w₁0 +Qm₂w0₂; {l,m} ⊂ {1,2,3},

then we have

(4.5)





w0₁ w0₂

 = 1 B12  

A12 A22

−A₁₁ −A₂₁



 



v0₁ v0₂



, B126=0 where

(4.6) A_{i j} =

Pl_i Pm_i

Ql_j Qm_j

, B₁₂=

Ql₁ Qm₁

Ql₂ Qm₂

Substituting (4.5) into (4.3) yields

(4.7) a₁₁(v

0

2

v0₁)

2

+2a₁₂(v

0

2

v0₁) +a22 =0; v

0

16=0, where

(4.8)

a₁₁= (A₁₂)2L2₁₁−2A₁₂A₁₁L₁₂2 + (A₁₁)2L₂₂2 −δ(B12)2L111

a₁₂=A₁₂A₂₂L2₁₁−(A₁₁A₁₂+A₁₂A₂₁)L2₁₂+A₁₁A₂₁L₂₂2 −δ(B12)2L112

a₂₂= (A₂₂)2L2₁₁−2A₂₁A₂₂L₁₂2 + (A₂₁)2L₂₂2 −δ(B12)2L122 Solving (4.7) yield

(4.9) v

0

2

v0₁ =

−a12± p

(a12)2−a11a22

a₁₁ .

In other words

(4.10) v0₂=λv0₁; λ = −a12± p

(a₁₂)2_−a 11a22

a11

Sinceα0(s)is the unit tangent vector of the curveα(s)on the surfaceP(v1,v2),then we have

(4.11)

2

∑

g_{i j}v0_iv0_j=1, g_{i j} =P_i,P_j

Substituting (4.10) into (4.11) yields

(4.12)

v0₁=_q 1

g₁₁+2g₁₂λ+g22λ2

,

v0₂=_q λ

g₁₁+2g₁₂λ+g22λ2

.

The unit tangent vector of the tangential intersection curves of the parametric surfacesP(v1,v2) andQ(w₁,w₂)can be obtained by

(4.13) t= P1+λP2

kP₁+λP2k

.

Lemma 1. The point p is a branch point of the intersection curve,if∆>0and there is another intersection branch crossing the intersection curve at that point.

Lemma 2. The surfaces P(v1,v2) and Q(w1,w2) intersect at the point p and at its

neighbor-hood, if∆=0and(a11)2+ (a12)2+ (a22)26=0. (Tangential intersection curve).

Lemma 3. The point p is an isolated contact point of the surfaces P(v1,v2) and Q(w1,w2),if

∆<0.

Lemma 4. The surfaces P(v1,v2) and Q(w1,w2) have contact of at least second order at the

point p,if a₁₁=a₁₂=a₂₂=0. (Higher-order contact point).

Using (4.5),(4.12) and (4.13), we obtain

(4.14) v0₁= 1

kP₁+λP2k

, v0₂= λ

kP₁+λP2k ;

Using(4.34)and(4.43), we obtain

(4.15) w0₁= A12+λA22

B12kP1+λP2k

, w0₂= −A11−λA21

B12kζP1−ηP2k

.

Then the tangent vector field of the tangential self-intersection curves of the surfaceR=R(u₁,u₂)

is given by

(4.16) t= R1(v1,v2) +λR2(v1,v2)

kR₁(v₁,v₂) +λR2(v1,v2)k

,

(4.17)

v0₁= 1

kR1(v1,v2) +λR2(v1,v2)k

, v0₂= λ

kR1(v1,v2) +λR2(v1,v2)k

,

w0₁= A12+λA22

B₁₂kR₁(v₁,v₂) +λR2(v1,v2)k

, w0₂= −A11−λA21

B₁₂kR₁(v₁,v₂) +λR2(v1,v2)k

where

(4.18)

λ = −a12± p

(a₁₂)2_−4a 11a22

2a₁₁ ,

a₁₁= (A₁₂)2L₁₁2 −2A₁₂A₁₁L2₁₂+ (A₁₁)2L2₂₂−δ(B12)2L111,

a₁₂=A₁₂A₂₂L2₁₁−(A₁₁A₁₂+A₁₂A₂₁)L2₁₂+A₁₁A₂₁L2₂₂−δ(B12)2L112,

a₂₂= (A₂₂)2L₁₁2 −2A₂₁A₂₂L2₁₂+ (A₂₁)2L2₂₂−δ(B12)2L122,

Ai j=

Rl_i(v₁,v₂) Rm_i (v₁,v2)

Rl_j(w₁,w₂) Rm_j(w₁,w2)

, L1_{i j}=Ri j(v1,v2),N1

,

B₁₂=

Rl₁(w₁,w2) Rm₁(w1,w2)

Rl₂(w₁,w₂) Rm₂(w₁,w₂)

, L2_{i j}=

R_{i j}(w₁,w₂),N2,

4.2. Curvature and curvature vector. Since the intersection curve views as a curve on both surfaces, then Eq. (2.13) satisfies on both surfaces thus

(4.19) (v0₁)2P₁₁+2v0₁v0₂P₁₂+ (v0₂)2P₂₂+v00₁P₁+v00₂P₂

= (w0₁)2Q₁₁+2w0₁w0₂Q₁₂+ (w0₂)2Q₂₂+w₁00Q₁+w00₂Q₂

Taking the cross product of both hand sides of (4.19) withQ₁andQ₂and projecting the resulting equations onto the surface normal vectorN₂,we obtain

(4.20)

w00₂= |Q1,P1,N2|

kQ₁×Q₂kv

00

1+

|Q₁,P₂,N₂| kQ₁×Q₂kv

00

2+

c₁₁

kQ₁×Q₂k,

w00₁= |P1,Q2,N2|

kQ₁×Q₂kv

00

1+

|P₂,Q₂,N₂| kQ₁×Q₂kv

00

2+

c₁₂

where

(4.21)

c₁₁= (v0₁)2|Q₁,P₁₁,N₂|+2v0₁v₂0|Q₁,P₁₂,N₂|+ (v0₂)2|Q₁,P₂₂,N2|

−(w0₁)2|Q₁,Q₁₁,N2| −2w0₁w0₂|Q1,Q12,N2| −(w0₂)2|Q1,Q22,N2|,

c12= (v0₁)2|P11,Q2,N2|+2v0₁v₂0|P12,Q2,N2|+ (v0₂)2|P22,Q2,N2|

−(w0₁)2|Q₁₁,Q₂,N₂| −2w0₁w0₂|Q₁₂,Q₂,N₂| −(w0₂)2|Q₂₂,Q₂,N₂|.

Projecting the vectorα000(s)ontoN1andN2then using (4.1), we obtain

(4.22) δ(v0₁L1₁₁+v0₂L1₁₂)v00₁+δ(v0₁L1₁₂+3v0₂L1₂₂)v00₂

= (w0₁L2₁₁+w₂0L2₁₂)w00₁+ (w0₁L2₁₂+3w0₂L2₂₂)w00₂+c13

3 , where

(4.23) c₁₃ = (w0₁)3hQ₁₁₁,N₂i+3(w0₁)2w0₂hQ₁₁₂,N₂i+ (w0₂)3hQ₂₂₂,N₂i

+3w0₁(w0₂)2hQ₁₂₂,N₂i −(v0₁)3hP₁₁₁,N₂i −(v0₂)3hP₂₂₂,N₂i −3v0₁(v0₂)2hP₁₂₂,N₂i −3(v0₁)2v₂0 hP₁₁₂,N₂i.

Since the curvature vector is perpendicular on the tangent vector, then we have

(4.24) (v0₁g₁₁+v0₂g₁₂)v00₁+ (v0₁g₁₂+v₂0g₂₂)v00₂=−



   

hP₁₁,P₁i(v0₁)3+hP₂₂,P₂i(v0₂)3

+(2hP12,P1i+hP11,P2i)(v0₁)2v0₂

+(2hP₁₂,P₂i+hP₂₂,P₁i)v0₁(v0₂)2),



   

We can computev00₁,v00₂,w00₁andw00₂by solving (4.20), (4.22) and (4.24).

The curvature vector and the curvature of the tangential intersection curves of the parametric surfacesP(v₁,v₂)andQ(w₁,w₂)can be computed by using (2.13) and (2.4) respectively.

The curvature and curvature vector of the tangential self-intersection curves of the surface

R(u₁,u₂)are given by replacing the surfacesP(v₁,v₂)andQ(w₁,w₂)toR(v₁,v₂)andR(w₁,w₂),

4.3. Torsion and third order derivative. The intersection curve views as a curve on both surfaces, then Eq. (2.14) satisfies on both surfaces thus

(4.25)        

(v0₁)3P111+3(v0₁)2v0₂P112

+3v0₁(v0₂)2P₁₂₂+ (v0₂)3P₂₂₂

+3v0₁v00₁P11+3(v00₁v0₂+v0₁v00₂)P12

+3v0₂v00₂P₂₂+v000₁P₁+v000₂P₂,

        =        

(w0₁)3Q₁₁₁+3(w0₁)2w0₂Q₁₁₂

+3w0₁(w0₂)2Q₁₂₂+ (w0₂)3Q₂₂₂

+3w0₁w00₁Q₁₁+3(w00₁w0₂+w0₁w00₂)Q₁₂

+3w0₂w00₂Q₂₂+w000₁Q₁+w000₂Q₂,

       

Taking the cross product of both hand sides of (4.25) withQ₁andQ₂and projecting the resulting equations onto the surface normal vectorN₂,we obtain

(4.26)

w000₂ = |Q1,P1,N2|

kQ₁×Q₂kv

000

1 +

|Q₁,P2,N2|

kQ₁×Q₂kv

000

2 +

c14

kQ₁×Q₂k,

w000₁ = |P1,Q2,N2|

kQ₁×Q₂kv

000

1 +

|P₂,Q₂,N₂| kQ₁×Q₂kv

000

2 +

c₁₅

kQ₁×Q₂k.

where

(4.27)

c₁₄= (v0₁)3|Q₁,P₁₁₁,N₂|+3(v0₁)2v0₂|Q₁,P₁₁₂,N₂|+3v0₁(v0₂)2|Q₁,P₁₂₂,N₂| −3w0₁(w0₂)2|Q₁,Q₁₂₂,N2|+3v0₁v₁00|Q1,P11,N2|+3v0₂v00₂|Q1,P22,N2|

+(v0₂)3|Q₁,P₂₂₂,N₂| −3w0₁w00₁|Q₁,Q₁₁,N₂| −3(w0₁)2w0₂|Q₁,Q₁₁₂,N₂| −(w0₁)3|Q₁,Q₁₁₁,N2| −(w0₂)3|Q1,Q222,N2| −3w0₂w00₂|Q1,Q22,N2|

+3(v00₁v0₂+v0₁v00₂)|Q₁,P₁₂,N₂| −3(w00₁w0₂+w0₁w₂00)|Q₁,Q₁₂,N₂|,

c₁₅= (v0₁)3|P₁₁₁,Q₂,N₂|+3(v0₁)2v0₂|P₁₁₂,Q₂,N₂|+3v0₁(v0₂)2|P₁₂₂,Q₂,N₂|

+(v0₂)3|P222,Q2,N2|+3v0₁v00₁|P11,Q2,N2| −3(w0₁)2w0₂|Q112,Q2,N2|

+3v0₂v00₂|P₂₂,Q₂,N₂| −3w0₁(w0₂)2|Q₁₂₂,Q₂,N₂| −3w₂0w00₂|Q₂₂,Q₂,N₂| −(w0₁)3|Q₁₁₁,Q₂,N2| −(w0₂)3|Q222,Q2,N2| −3w0₁w00₁|Q11,Q2,N2|

+3(v00₁v0₂+v0₁v00₂)|P₁₂,Q₂,N₂| −3(w00₁w0₂+w0₁w₂00)|Q₁₂,Q₂,N₂|,

Projecting the vectorα(4)(s)onto the two unit normal vector fields of both surfaces and using (2.18) and (4.1), we obtain

(4.28) (

w0₁L2₁₁+w0₂L2₁₂)w000₁ + (w0₁L2₁₂+w0₂L₂₂2 )w000₂

=δ(v0₁L1₁₁+v0₂L1₁₂)v000₁ +δ(v0₁L1₁₂+v₂0L1₂₂)v000₂ +c16

where

(4.29) c₁₆= (v0₁)4hP₁₁₁₁,N₂i+4(v0₁)3v0₂hP₁₁₁₂,N₂i+6(v0₁)2(v0₂)2hP₁₁₂₂,N₂i

+(v0₂)4hP₂₂₂₂,N₂i+4v0₁(v0₂)3hP₁₂₂₂,N₂i+6(v0₁)2v00₁hP₁₁₁,N₂i

+6(v0₂)2v00₂hP₂₂₂,N₂i+6(2v₁0v0₂v00₁+ (v0₁)2v00₂)hP₁₁₂,N₂i

+6(v00₁(v0₂)2+2v0₁v0₂v00₂)hP₁₂₂,N₂i+3δ(v00₁)2L1₁₁+6δv00₁v00₂L1₁₂

+3δ(v00₂)2L₂₂1 −(w0₁)4hQ₁₁₁₁,N₂i −4(w0₁)3w0₂hQ₁₁₁₂,N₂i −6(w0₁)2(w₂0)2hQ₁₁₂₂,N₂i −(w0₂)4hQ₂₂₂₂,N₂i −3(w00₁)2L2₁₁

−4w0₁(w0₂)3hQ₁₂₂₂,N2i −6(w₁0)2w00₁hQ111,N2i −6w00₁w00₂L2₁₂

−6(w0₂)2w00₂hQ₂₂₂,N₂i −6(2w0₁w0₂w00₁−(w0₁)2w00₂)hQ₁₁₂,N₂i −6(w00₁(w0₂)2−2w0₁w0₂w00₂)hQ₁₂₂,N2i −3(w00₂)2L2₂₂

Since

α0,α000=−κ2

which can be written as (4.30)

(v0₁g11+v02g12)v0001 +(v01g12+v02g22)v0002 =−( κ2+ (v01)4hP111,P1i+3(v01)3v02hP112,P1i

+3(v0₁)2(v0₂)2hP₁₂₂,P₁i+v0₁(v0₂)3hP₂₂₂,P₁i

3(v0₁v0₂v00₁+ (v0₁)2v00₂)hP12,P1i+3(v0₁)2v00₁hP11,P1i

+3v0₁v0₂v00₂hP₂₂,P₁i+ (v0₁)3v0₂hP₁₁₁,P₂i

+3(v0₁)2(v0₂)2hP112,P2i+3v0₁(v₂0)3hP122,P2i

+(v0₂)4hP₂₂₂,P₂i+3v0₁v0₂v00₁hP₁₁,P₂i

+3(v00₁(v0₂)2+v0₁v0₂v00₂)hP₁₂,P₂i+3(v0₂)2v00₂hP₂₂,P₂i)

We can computev000₁,v000₂,w000₁ andw000₂ by solving (4.26), (4.28) and (4.30).

The third derivative vector and the torsion of the tangential intersection curves of two para-metric surfacesP(v₁,v₂)andQ(w₁,w₂)can be computed by using (2.14) and (2.7), respectively. The third derivative vector and the torsion of the tangential self-intersection curves of the surfaceR(u₁,u₂) are given by replacing the surfacesP(v₁,v₂)and Q(w₁,w₂)to R(v₁,v₂)and

5.

Examples

Example1. Consider the two parametric surfaces

(5.1) P= (1+sinv2,cosv2,v1)

Q= (2 cosw1cosw2,2 sinw1cosw2,2 sinw2)

FIGURE 2. Fig 5.1

Transversal intersection: Using (3.9) and (5.1), we obtain

(5.2) t= (cosv2sinw_q2,−sinv2sinw2,−cosw2cos(v2+w1))

1−cos2_w

2sin2(w1+v2)

.

Using (3.19) and (5.1), we obtain

(5.3) α00= (a1

a₄, a2

a₄, a3

where

(5.4)

a₁= cosv₂cos3(v₂+w₁)cos3w₂+2 sinw₁cos(v₂+w₁)cos4w₂

−cosv2cos(v2+w1)cos3w2−2 sinw1cos(v2+w1)cos2w2

+cosv₂cos(v₂+w₁)cosw₂+2 sinv₂cos4w₂−4 sinv₂cos2w₂+2 sinv₂

a₂= −sinv₂cos3(v₂+w₁)cos3w₂−2 cosw₁cos(v₂+w₁)cos4w₂

−sinv2cos(v2+w1)cosw2+2 cosv2cos4w2−4 cosv2cos2w2

+sinv₂cos(v₂+w₁)cos3w₂+2 cosw₁cos(v₂+w₁)cos2w₂+2 cosv₂

a₃= sinw₂



   

cos2(v₂+w₁)cos2w₂−2 cosv₂cosw₂sinw₁

−2 cosw₁cosw₂sinv₂+2 cosv₂cos3w₂sinw₁

+2 cosw₁cos3w₂sinv₂−cos2w₂+1



   

a₄= 1₈





cos(2v₂+2w₁−2w₂) +cos(2v₂+2w₁+2w₂)

−2 cos 2w₂+2 cos(2v₂+2w₁) +6





Using (3.21) and (5.1) we obtain

(5.5)     

x000₁

x000₂ x000₃

     =      

cosv2sinw2

√

1−cos2_w

2sin2(w1+v2)

−sinv2sinw2

√

1−cos2_w

2sin2(w1+v2)

−cosw2cos(v2+w1)

√

1−cos2_w

2sin2(w1+v2)

sinv2 cosv2 0

cosw₁cosw₂ sinw₁cosw₂ sinw₂

      −1 ×       

−(a1)

2

+ (a₂)2+ (a₃)2 (a₄)2

−3v0₂v00₂

3(w0₁)2w0₂sin 2w2−6w0₁w00₁cos2w2−6w0₂w00₂       

point,λ =±1. Using (4.13)and (5.1) at the pointp(2,0,0), we obtain

(5.6) t= (0,±√1

2, 1

√

2).

Using (4.20), (4.24), (2.4), (2.13) and (5.1) at the point p(2,0,0), we obtain

(5.7) n= (−1,0,0), κ= 1₂.

(5.8) b= (0,−1

2

√

2,±1

2

√

2)

Using (4.26), (4.28), (4.30), (2.7), (2.14) and (5.1) at the point p(2,0,0), we obtain

α000(s) = (0,±₃₂1√2,−₃₂9√2) τ =∓5₈. Example2: Consider the surface

(5.9) R(u₁,u₂) = (u₁−u

3 1 3 +u1u

2

2,−u2−u21u2+

u3₂

3 ,u 2

1−u22); −5<u1,u2<5.

FIGURE 3. Fig 5.2

Let us find the Frenet vectors, the curvature and the torsion of the transversal self-intersection curve at the transversal self-intersection pointp(28₃√2,0,−7) =R(√2,3) =R(√2,−3)∈R(u₁,u₂).

Using (3.12) and (5.2) at the point p(28₃√2,0,−7),we obtain

(5.10) t= (5

9

√

3,0,−1

9

√

6).

Using (3.19), (2.4), (2.13) and (5.2) at the pointp(28₃√2,0,−7),we obtain (5.11) _κn= (₉₇₂1 √2,0,₉₇₂5 ), κ =₃₂₄1

√

Using (3.21), (2.7), (2.14) and (5.2) at the pointp(28₃√2,0,−7),we obtain

(5.12) α000= (−_{314 928}23

√

3,0,−_{78 732}11 √6), τ=0.

6.

Conclusion

Algorithms for computing all the differential geometry properties of, self-intersection curves of a parametric surface and the intersection curves of two parametric surfaces inR3, for

transver-sal and tangential intersection. This paper is an extension to the works of Ye and Maekawa (1999). They gave an example of implicit-parametric surfaces intersection and they computed the tangent vector field and refired to how obtain the curvature vector, the curvature, the tor-sion and the higher derivatives of the intersection curves by using they method. But this paper introduce a direct formulas to compute all the properties. The types of singularity on the inter-section curve are characterized. The questions of how to exploit and extend these algorithms to compute the differential geometry properties of intersection curves between three surfaces in

R4,can be topics of future research.

Conflicts of Interests

The authors declare that there is no conflict of interests.

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