SURF

ACE-SURFACE INTERSECTION BY

HERMITE INTERPOLATION

Eng-Wee Chionh

School of Computing, National University of Singapore, Republic of Singapore 117590

Keywords:

Hermite interpolation, surface-surface intersection.

Abstract:

A fast heuristic to approximate the intersection curve of two surface patches was originally proposed by

Sederberg and Nishita. The patches are rationally parametrized and they cut each other transversely. This

paper reports a simple generalization that greatly improves the accuracy of the original heuristic. The general-

ization either avoids or attenuates the approximation error. Avoidance is achieved when the improved heuristic

produces the exact intersection curve; attenuation is accomplished with an aggregate square distance formula

guiding the selection of a generalized constraint for a better ﬁt.

1 INTRODUCTION

Finding the intersection curve of two surfaces is a fun-

damental operation in computer graphics and com-

puter geometric design and processing (Song et al.,

2004). It is an intrinsically difﬁcult problem and ex-

pectedly solutions depend on the surface description.

In the realm of rationally parametrized surfaces, this

difﬁculty is apparent because the intersection curve

degree can be very high even when the surface de-

grees are modest. Recall that the degree of a curve in

3-space is the number of times it intersects a general

plane. By Bezout’s theorem (Cox et al., 1998), the

degree of the intersection curve of two general tensor

product surfaces of bi-degrees m

× m

and n

× n

is 4m

, and that of two general triangular sur-

faces of total degrees m and n is m

. For the popu-

lar bi-cubic surface patches in computer geometry, the

intersection curve degree works out to be 324! What

is worst, the intersection curve of two rational sur-

faces need not be rational. It is well known that an

algebraic curve is rational if and only if its genus is

zero (Fulton, 1969). Katz and Sederberg (Katz and

Sederberg, 1988) show that the genus g of the inter-

section curve of two general tensor product surfaces

of parametric bi-degrees m

× m

and n

× n

g = 8m

− 2m

+ n

)

−2n

+ m

) + 1; (1)

and that of two general triangular surfaces of paramet-

ric total degrees m and n is

g = 2m

−

n −

m +1. (2)

Thus

in general the intersection curve of two bi-

degree surfaces is non-rational and that of two trian-

gular surfaces is rational if and only if one is a plane

and the other is either a plane or a quadric surface.

Currently, the many existing methods for ﬁnding

the intersection curve of two surfaces may be broadly

classiﬁed under either geometric subdivision or topo-

logical marching (de Figueiredo, 1996). Both ap-

proaches are computationally expensive if the solu-

tion is to be within certain given tolerance. With this

state of affairs, perhaps the research on ﬁnding the

intersection curve of two rationally parametrized sur-

faces should be to improve either the speed of “ex-

act” solutions or the accuracy of approximate solu-

tions. This paper aims the latter: its sole goal is

to boost the accuracy of an approximate intersection

curve as much as possible while incurring little addi-

tional computing costs.

The remainder of the paper comprises ﬁve sec-

tions. Section 2 reviews the original Sederberg-

Nishita surface-surface intersection heuristic which

relies on a constraint to solve for end-point tangents.

Section 3 describes in detail the process and outcome

of generalizing the original constraint deployed in the

Chionh E. (2008).

SURFACE-SURFACE INTERSECTION BY HERMITE INTERPOLATION.

In Proceedings of the Third International Conference on Computer Graphics Theory and Applications, pages 23-30

DOI: 10.5220/0001095100230030

 SciTePress

heuristic for better accuracy. Section 4 derives a for-

mula to calculate some aggregate square distance and

illustrates its use in ﬁnding a better approximate in-

tersection curve. Section 5 discusses how the gen-

eralized constraint and the aggregate square distance

can be used in tandem to enhance accuracy. Some is-

sues that arise from this strategy are then discussed.

Section 6 summarizes the paper.

2 THE ORIGINAL

SEDERBERG-NISHITA

HEURISTIC AND CONSTRAINT

The Sederberg-Nishita heuristic (Sederberg and

Nishita, 1991) quickly approximates the intersection

curve of two rationally parametrized surface patches

that cut transversely between two known points. This

situation may seem contrived but actually happens

naturally when subdivision is employed to ﬁnd the in-

tersection of two surfaces. Indeed by subdividing, we

transform the problem of ﬁnding the intersection of

two larger surface patches to the problem of ﬁnding

the intersections of a set of pairs of much smaller sur-

face patches such that each pair intersect transversely.

The intersection of the original surface patches is then

the concatenation of these smaller intersections with

continuity (Sederberg and Nishita, 1991).

2.1 The Problem Statement

The problem solved by the Sederberg-Nishita heuris-

tic can be formulated precisely as follows. Two sur-

face patches in the real Euclidean space R

are given

rational parametrically as

P : [0,1] × [0, 1] → R

, (3)

Q : [0,1] × [0, 1] → R

. (4)

Surfaces P(s,t) and Q(u,v) intersect transversely at

two diagonally opposite points V

and V

where

= P(0,0) = Q(0, 0), V

= P(1,1) = Q(1, 1). (5)

The task is to ﬁnd the intersection curve

C = P(s,t) ∩ Q(u,v), 0 ≤ s,t,u,v ≤ 1. (6)

2.2 Cubic Hermite Interpolation

The heuristic approximates the intersection curve us-

ing cubic Hermite interpolation. That is, given two

end-point positions V

and V

and the corresponding

tangents T

and T

of a curve, the cubic Hermite inter-

polation of the given curve is the cubic Bezier curve:

H(a) =

∑

i=0

i/3

(a), 0 ≤ a ≤ 1, (7)

where

1/3

= V

, V

2/3

= V

−

; (8)

and B

(a), i = 0,1,2,3, are the cubic Bernstein basis

functions:

(a) =

(1 − a)

3−i

. (9)

Since V

and V

are known, the heuristic amounts to

ﬁnding the end-point tangents T

and T

2.3 The Algorithm

The heuristic determines the end-point tangents T

and T

with the following steps.

1. Along the intersection curve, the parameters s, t,

u, and v are parametrized with another parameter

a such that

s = s(a), t = t(a), u = u(a), v = v(a), (10)

and at a = 0,1, we have

s(a) = t(a) = u(a) = v(a) = a. (11)

2. Since P(s(a),t(a)) = Q(u(a),v(a)), 0 ≤ a ≤ 1, we

have

dP(s(a),t(a))

dQ(u(a),v(a))

,0 ≤ a ≤ 1.

(12)

3. The multivariate chain rule gives

+ P

= Q

+ Q

, (13)

where P

, P

, Q

are the respective partial

derivatives and s

, t

, u

, v

are derivatives with re-

spect to a.

4. Linear algebra solves the above as ratios:

−t

−u

, (14)

where we treat P

as 3 × 1 column vec-

tors and the denominators are then 3 × 3 determi-

nants. Note that we are interested in the two sets

of ratios corresponding to a = 0,1.

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

5. To obtain deﬁnitive values for s

, t

, u

, and v

when a = 0,1, Sederberg and Nishita heuristically

propose the constraint

(0) + t

(0) = s

(1) + t

(1) = 2. (15)

6. With this Sederberg-Nishita constraint, we can

solve for the parametric tangents s

(a), ·· · , v

(a)

at a = 0,1, and thus obtain the tangents T

at po-

sitions V

= P

s(a)

(a) + P

t(a)

(a), (16)

= Q

u(a)

(a) + Q

v(a)

(a), (17)

for a = 0,1.

2.4 An Illustration

We illustrate the Sederberg-Nishita heuristic and con-

straint with the simplest example: P(s,t) and Q(u,v)

are planes intersecting transversely from V

to V

. The

intersection curve is plainly

C(a) = (1 − a)V

+ aV

, 0 ≤ a ≤ 1. (18)

Very pleasantly the cubic Hermite interpolation curve

(7) constructed by the Sederberg-Nishita heuristic

gracefully degenerates to become (18). That is,

H(a) = C(a). To see this, let

P(s,t) = V

+ Ss + Tt, 0 ≤ s,t ≤ 1, (19)

Q(u,v) = V

+Uu +V v, 0 ≤ u,v ≤ 1, (20)

with

= V

+ S + T = V

+U +V. (21)

Since the partial derivatives

= S,T,U,V, (22)

are constants, ratios (14) at a = 0,1 are the same:

|TUV |

−t

|SUV |

−u

|STV |

|STU|

. (23)

By (21) we have

|TUV | = −|SUV | (24)

Consequently the parametric tangents are

(0) = t

(0) = 1, (25)

(1) = t

(1) = 1, (26)

and the end-point tangents are

= T

= S + T = U +V = V

−V

. (27)

It is an exciting surprise that despite the use of a cu-

bic curve (7) to approximate a line (18), not only

the approximation turns out to be exact, but also the

parametrization is proper rather than improper with

index 3 (Sederberg, 1986).

Figure 1 shows two planes intersecting trans-

versely and their intersection line.

Figure 1: Two planar patches intersecting transversely and

their intersection line.

3 GENERALIZING THE

ORIGINAL HEURISTIC

CONSTRAINT

Immediately we see that instead of imposing con-

straint (15) on P(s,t), we could have imposed it on

Q(u,v). That is, we demand instead

(0) + v

(0) = u

(1) + v

(1) = 2. (28)

After all, it is completely arbitrary which of the two

given surface patches is called P(s,t) and the other

called Q(u,v).

This observation motivates the search for a more

symmetric constraint for determining the parametric

tangents s

(a), t

(a), u

(a), and v

(a) at a = 0, 1.

3.1 Other Probable Linear Constraints

To explore the possibilities, we examine condition

(11) carefully and see that it leads to

s(0) + t(0)+ u(0) + v(0) = 0, (29)

s(1) + t(1)+ u(1) + v(1) = 4. (30)

These relations suggest we consider the constraint

σs(a) + τt(a)+ µu(a) + νv(a)

= (σ + τ + µ + ν)a, (31)

or equivalently,

σs

(a) + τt

(a) + µu

(a) + νv

(a)

= σ + τ + µ + ν. (32)

Clearly constraint (32) is a generalization of the orig-

inal constraint (15) because the latter is a special case

of the former with

σ = τ = 1, µ = ν = 0. (33)

SURFACE-SURFACE INTERSECTION BY HERMITE INTERPOLATION

Consequently, a constraint completely symmetric to

the patches P, Q and the parameters s, t, u, v would

σ = τ = µ = ν = 1. (34)

It would be most desirable if being symmetric would

guarantee the best approximation. Unfortunately, this

is not to be so as we shall present a counter example

in Section 4.

The preceding sad remark and the seemingly triv-

ial derivation may cast doubts on the signiﬁcance of

the generalization. We shall restore conﬁdence by

demonstrating the usefulness of the generalized con-

straint in the following illustration and later discus-

sions. Most signiﬁcantly, we show in some situa-

tions the generalization produces the exact intersec-

tion with appropriate σ, τ, µ, and ν.

3.2 Illustrating the Generalized

Constraint

The following shows that a twisted cubic curve, which

is the intersection of a quadric cylinder and a cubic

cylinder, can be obtained exactly using the general-

ized constraint (32) with appropriate σ, τ, µ, ν.

Let the quadric cylinder α

y = βx

and the cubic

cylinder α

z = γx

be parametrized respectively as

P(s,t) = (αs,βs

,γt), 0 ≤ s,t ≤ 1; (35)

Q(u,v) = (αu, βv, γu

), 0 ≤ u,v ≤ 1. (36)

Equivalently,

P(s,t) = As + Bs

+Ct, (37)

Q(u,v) = Au + Bv +Cu

. (38)

where

A = (α,0, 0), B = (0, β, 0),C = (0,0,γ). (39)

The exact intersection C(a) from V

= 0 to V

A + B + C is easily solved analytically and can be

parametrized as

C(a) = (αa, βa

,γa

), 0 ≤ a ≤ 1. (40)

To ﬁnd an approximation intersection curve H(a) by

the heuristic we ﬁrst compute

= A + 2Bs, P

= C, (41)

= A + 3Cu

, Q

= B. (42)

The parametric tangent ratio is

(a) : t

(a) : u

(a) : v

(a)

= |C,A + 3Cu

,B|

: −|A + 2Bs,A + 3Cu

,B|

: −|A + 2Bs,C,B|

: |A + 2Bs,C,A + 3Cu

= 1 : 3u

: 1 : 2s (43)

To apply the original constraint on P(s,t), we set

σ = τ = 1, µ = ν = 0 and obtain parametric tangents

(0),t

(0),u

(0),v

(0) = 2,0, 2, 0; (44)

(1),t

(1),u

(1),v

(1) =

,1; (45)

and end-point tangents

= 2A, T

A + 2B + 3C

. (46)

The resulting approximation cubic Hermite curve is

− 3a + 4

A+a

(2 − a)B+

(3 − a)C. (47)

To apply the original constraint on Q(u, v), we set

σ = τ = 0, µ = ν = 1 and obtain parametric tangents

(0),t

(0),u

(0),v

(0) = 2,0, 2, 0; (48)

(1),t

(1),u

(1),v

(1) =

,2,

; (49)

and end-point tangents

= 2A, T

2A + 4B + 6C

. (50)

The resulting approximation cubic Hermite curve is

− 5a + 6

A +

(5 − 2a)B + a

C. (51)

In both the above cases the original constraint did

not produce the exact intersection. But observe that

the intersection is found by setting

s = a,t = a

,u = a,v = a

. (52)

Thus we may simply set either σ or µ to 1 and the rest

to zero to produce the exact intersection (40). Either

way, the parametric tangents are

(0),t

(0),u

(0),v

(0) = 1,0, 1, 0; (53)

(1),t

(1),u

(1),v

(1) = 1,3, 1, 2; (54)

and the end-point tangents are

= A, T

= A + 2B + 3C. (55)

This example reveals that the original constraint may

not produce the exact intersection even when it is a

cubic curve. In contrast, the generalized constraint is

able to produce the exact intersection.

It is worth mentioning that this example does not

contradict Bezout’s theorem despite that the degree

of the intersection curve of a degree two and a degree

three surfaces is three but not six. This is because

here the quadric cylinder and the cubic cylinder has

another intersection line x = w = 0 at inﬁnity of multi-

plicity three. (Here w is the homogenizing coordinate

to enlarge the afﬁne 3-space to a projective 3-space.)

Figure 2 shows the intersection curve, a twisted

cubic curve, and the transversely intersecting quadric

cylinder patch and cubic cylinder patch. Figure 3

shows the exact intersection curve (with σ = 1, τ =

µ = ν = 0 or µ = 1, σ = τ = ν = 0) and the two approx-

imation intersection curves obtained with σ = τ = 1,

µ = ν = 0 and σ = τ = 0, µ = ν = 1.

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

Figure 2: The intersection curve of a quadric cylinder patch

and a cubic cylinder patch.

Figure 3: The two approximation intersection curves (in

thin line) and the exact intersection curve (in thick line) of

a quadric cylinder patch and a cubic cylinder patch.

4 ESTIMATING THE CLOSENESS

OF FIT

By now we see that there are inﬁnitely many gener-

alized constraints for determining the parametric tan-

gents s

(a), t

(a), u

(a), and v

(a) at a = 0,1. This

calls for some criteria to assess among constraints

κ which gives a better or best Hermite interpolation

(a) to approximate the exact intersection P(s,t) ∩

Q(u,v), ideally parametrized as C(a), 0 ≤ a ≤ 1. For

this purpose we propose a closeness of ﬁt measure-

ment called aggregate square distance.

4.1 The Aggregate Square Distance

Let (s(a),t(a)), (u(a), v(a)) be the pre-images of the

intersection curve C(a), 0 ≤ a ≤ 1, in the parameter

spaces (s,t), (u,v) respectively. That is,

(s(a),t(a)) = P

−1

(C(a)), (56)

(u(a),v(a)) = Q

−1

(C(a)), (57)

where 0 ≤ a ≤ 1. Let |V | be the usual Euclidean norm

for the 1 × 3 vector V ∈ R

; that is,

|(x,y, z)| =

+ y

+ z

. (58)

Consider the integral

I =

|P(s(a),t(a))− Q(u(a),v(a))|

da. (59)

It is obvious that I = 0 if and only if P(s(a),t(a)) =

Q(u(a),v(a)). That is, the integral vanishes if and

only if P(s(a),t(a)) and Q(u(a),v(a)) give the exact

intersection curve C(a).

However, if (s(a),t(a)) and (u(a), v(a)) are only

approximations of the pre-images of the actual inter-

section curve C(a), the integral will give the aggre-

gate square distance of the two approximation curves

(a) = P(s(a),t(a)) and C

(a) = Q(u(a),v(a)). In

other words, the integral is a measurement of how

close the two approximation curves C

(a) and C

(a)

are. It is thus reasonable to extrapolate it to be a mea-

surement of how well H(a) ﬁts C(a). Admittedly this

is a heuristic estimation of the closeness of ﬁt, but em-

pirically it seems to reﬂect the closeness of ﬁt quite

well as shall be illustrated later.

After adopting the aggregate square distance as a

measurement for the closeness of ﬁt of H(a) to C(a),

our remaining task is to ﬁnd some approximation pre-

images (s(a),t(a)) and (u(a), v(a)) of C(a) with as

little computing costs as possible.

4.2 Approximating the Pre-Images

Recall that we have already found the parametric tan-

gents s

(a), t

(a), u

(a), and v

(a) for a = 0,1. It is

thus with no additional computing costs if we approx-

imate the pre-image of the intersection curve C(a) in

the parameter space (s,t) also with a Hermite cubic

interpolation H

(a), by setting the control vertices in

(7) as

= (0,0), (60)

1/3

= V

(0),t

(0))

, (61)

2/3

= V

−

(1),t

(1))

, (62)

= (1,1). (63)

The approximation pre-image H

(a) of the intersec-

tion curve C(a) in the (u,v) space is constructed sim-

ilarly with (s

(a),t

(a)) replaced by (u

(a),v

(a)) for

a = 0,1.

To summarize, the intersection curve C(a) and its

pre-images are approximated by three approximation

SURFACE-SURFACE INTERSECTION BY HERMITE INTERPOLATION

curves, one in the Euclidean 3-space (x,y,z) and two

in the parameter 2-spaces (s,t) and (u,v):

C(a) = P(s,t) ∩ Q(u,v) ≈ H(a), (64)

−1

(C(a)) ≈ H

(a), (65)

−1

(C(a)) ≈ H

(a). (66)

Furthermore, we estimate

|H(a)−C(a)|

da (67)

using

|P(H

(a)) − Q(H

(a))|

da. (68)

The following theorem afﬁrms the relevance of in-

tegral (68) as a measurement of closeness of ﬁt.

Theorem 1 If integral (68) vanishes and C(a) is cu-

bic in a, then H(a) = C(a).

Proof.

Integral (68) vanishes if and only if P(H

(a)) =

Q(H

(a)). But the equality means

P(H

(a)) = Q(H

(a)) = P(s,t) ∩ Q(u,v) = C(a).

Thus C(a) and H(a) have the same end-points and

end-point tangents. Consequently they give the same

curve since a parametric cubic curve over [0,1] is

completely determined by its end-points and end-

point tangents. Q.E.D.

We emphasize that H

(a) and H

(a) are by-

products from the construction of H(a). It is thus fair

to claim they are available free. Integral (68) can be

integrated numerically with high efﬁciency because

the integrands are polynomials of low degrees.

4.3 Illustrations

We repeat the tutorial example in (Sederberg and

Nishita, 1991) to illustrate the role of the aggregate

square distance in identifying a constraint for a closer-

ﬁtting approximation H(a).

Consider two bilinear patches

P(s,t) =

∑

i, j=0

i j

(1 − s)

1−i

(1 −t)

1− j

, (69)

and

Q(u,v) =

∑

i, j=0

i j

(1 − u)

1−i

(1 − v)

1− j

, (70)

where

[0,0, 0] , [0, 1, 4],[3,3,0],[4,0, 4] ; (71)

and

[0,0, 0] , [0, 4, 4],[4,2,0],[4,0, 4] . (72)

The example in (Sederberg and Nishita, 1991)

used the Sederberg-Nishita constraint on P(s,t) by

setting

σ = τ = 1, µ = ν = 0, (73)

and obtained the parametric tangents

(0),t

(0),u

(0),v

(0) =

,1; (74)

(1),t

(1),u

(1),v

(1) = 2,0, 2,

. (75)

The resulting end-point tangents were

= [2,−6, 8] , (76)

and thus

1100

(a) =







−2a(a

− a − 2)

−

(2a

+ 5a − 7)

(2a

− a + 2)







. (77)

The aggregate square distance for this approximation

1100

= 0.0053561. (78)

Now we try using the Sederberg-Nishita con-

straint on Q(u,v) instead of P(s,t); that is, we set

σ = τ = 0, µ = ν = 1. (79)

The parametric tangents are

(0),t

(0),u

(0),v

(0) =

; (80)

(1),t

(1),u

(1),v

(1) =

,0,

. (81)

The resulting end-point tangents are

,−

, (82)

and thus

0011

(a) =







−(2a

− a − 6)

− 8a + 7)

(2a

− a + 4)







. (83)

The aggregate square distance for this approximation

0011

= 0.00032996. (84)

Since ρ

0011

is much smaller than ρ

1100

, we expect

0011

(a) to be a closer ﬁt than H

1100

(a). This is

found to be so by visually inspecting a plot of P(s,t),

Q(u,v), H

1100

(a), and H

0011

(a).

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

Finally we assess the performance of the symmetric

constraint

σ = τ = µ = ν = 1. (85)

The parametric tangents are

(0),t

(0),u

(0),v

(0) =

;(86)

(1),t

(1),u

(1),v

(1) =

,0,

. (87)

The resulting end-point tangents are

,−

, (88)

and thus

1111

(a) =







−(46a

− 37a − 108)

−6(a

+ 20a − 21)

(50a

− 23a + 72)







. (89)

The aggregate square distance for this approximation

1111

= 0.0016125. (90)

We observe that

0011

< ρ

1111

< ρ

1100

. (91)

With a visual inspection of a plot of P(s,t), Q(u,v),

0011

(a), H

1111

(a), and H

1100

(a), it is satisfying to

see that the aggregate square distance does faithfully

reﬂect the closeness of ﬁt.

Figure 4 shows the bilinear patches and their

intersection curve. Figure 5 and ﬁgure 6 shows

the three approximation intersection curves H

1100

(a),

0011

(a), H

1111

(a) and the actual intersection curve

C(a). Visually we ascertain that H

0011

(a) ﬁts best

while H

1100

(a) ﬁts worst and H

1111

(a) is somewhere

in between. This is consistent with the prediction of

their aggregate square distances.

Figure 4: The bilinear patches P(s,t), Q(u, v), and their in-

tersection C(a).

Figure 5: The cubic Hermite curves H

1100

(a), H

1111

(a),

0011

(a) and the exact intersection curve C(a), 0 ≤ a ≤ 1,

for the bilinear patches.

Figure 6: The top curve is the exact intersection curve. In

increasing distance from the top curve are the curves ob-

tained with constraints σ = τ = 0,µ = ν = 1; σ = τ = µ =

ν = 1; σ = τ = 1, µ = ν = 0 respectively. All curves are

plotted from 0.3 to 0.7 to magnify their separation.

5 OBSERVATIONS AND

SPECULATIONS

From the above discussions, we realize that the

approximation cubic Hermite curves H

1100

(a) and

0011

(a) ﬁt the intersection curve C(a) with differ-

ent accuracy and one can be a much better ﬁt than

the other. The winner can be determined by compar-

ing their aggregate square distances ρ

1100

and ρ

0011

which can be computed efﬁciently with little addi-

tional computing costs.

Intuitively, we would expect the approximation

cubic Hermite curve H

1111

(a), in terms of accuracy,

to be worst than the best of H

1100

(a), H

0011

(a) but

better than the worst of them. This speculation is sup-

ported by our example. It would be very helpful in

establishing the applicability of the aggregate square

SURFACE-SURFACE INTERSECTION BY HERMITE INTERPOLATION

distance measurement if this guess can be established

as a fact with some proof.

We have shown that when the surface patches are

planar, the Sederberg-Nishita heuristic produces the

exact intersection line with a proper parametrization,

despite the use of a cubic Hermite interpolation. This

has greatly enhanced the credibility of the Sederberg-

Nishita constraint used in the heuristic. It would be

interesting in theory and useful in practice if more

cases can be discovered in which the cubic Hermite

curve is the actual intersection curve rather than just

an approximation. It would be even better if some

sufﬁcient or necessary conditions can be laid down

under which the generalized constraint produces the

exact intersection curve.

In this paper we only exploit polynomial cubic

Hermite interpolation. It should be a worthwhile ef-

fort to explore the use of rational Hermite interpola-

tion (Goldman, 2003) for approximating the intersec-

tion curve of two surface patches. Note that some ra-

tional curves, such as circular arcs, do not have poly-

nomial parametrization.

6 CONCLUSIONS

We ﬁrst reviewed a quick heuristic originally pro-

posed by Sederberg and Nishita. The heuristic ﬁnds a

cubic Hermite interpolation curve that approximates

the intersection of two rationally parametrized sur-

face patches when they intersect transversely. The

power of the heuristic was established empirically in

(Sederberg and Nishita, 1991). We further enhanced

its credibility by showing that when the surfaces are

planes the heuristic actually produces the exact inter-

section line parametrized properly.

The heuristic utilizes a constraint to decide the

parametric tangents in order to compute the Hermite

interpolation. The constraint is arbitrarily applied to

one of the two surfaces in the heuristic. We proposed

that the constraint should be applied to both surfaces

individually thus producing two approximating cubic

Hermite curves. The better-ﬁtting one should then

be used. The assessment is based on their aggregate

square distances which can be easily evaluated. An

example was given to illustrate that much improve-

ment in accuracy could be achieved. All this was done

with very little additional computing costs, as it was

crucial not to sacriﬁce speed of the heuristic to attain

better accuracy.

This way of applying the constraint can be consid-

ered as special cases of deploying of what we called

the generalized constraint. We showed that there are

situations in which the generalized constraint could

produce the exact intersection curve but the original

constrain could not.

We also discussed some interesting open problems

with this new perspective on the constraint.

REFERENCES

Cox, D., John, L., and Donal, O. (1998). Using Algebraic

Geometry. Springer, New York.

de Figueiredo, L. H. (1996). Surface intersection us-

ing afﬁne arithmetic. Proceedings of Graphics Inter-

face’96.

Fulton, W. (1969). Algebraic Curves: An Introduction to Al-

gebraic Geometry. W. A. Benjamin, Inc., New York.

Goldman, R. N. (2003). Pyramid Algorithms : A dynamic

programming approach to curves and surfaces for ge-

ometric modeling. Morgan Kaufmann, San Francisco.

Katz, S. and Sederberg, T. W. (1988). Genus of the intersec-

tion curve of two rational surface patches. Computer

Aided Geometric Design, 5:253–258.

Sederberg, T. W. (1986). Improperly parametrized rational

curves. Computer Aided Geometric Design, 3:1:67–

75.

Sederberg, T. W. and Nishita, T. (1991). Geometric hermite

approximation of surface patch intersection curves.

Computer Aided Geometric Design, 8:97–114.

Song, X., Sederberg, T., Zheng, J., Farouki, R., and Hass, J.

(2004). Linear perturbation methods for topologically

consistent representations of free-form surface inter-

sections. Computer Aided Geometric Design, 21:303–

319.

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications