Optimum Steiner Ratio for Gradient-Constrained NetworksConnecting Three Points in 3-Space, Part II:The Gradient-Constraint m Satisfies 1 ≤ m ≤ √

3

Doreen ThomasDepartment of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia

Kevin PrendergastDepartment of Electrical and Electronic Engineering, The University of Melbourne, Victoria 3010, Australia

Jia WengDepartment of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia

The authors have proved in a previous article that theSteiner ratio (the minimum ratio of the length of a Steinerminimal tree to the length of a minimal spanning tree)for a gradient-constrained network on three points is1/2+(

√4 + 3m2−1)/(4

√1 + m2) if the maximum gradient

m of an edge in the network satisfies m ≤ 1. In this arti-cle, we continue this study and use the same strategy toshow that this result still holds when the maximum valueof m satisfies 1 ≤ m ≤ √

3. However the ratio becomes(3m +

√9m2 + 12)/(4

√3m2 + 3) if m ≥ √

3. © 2010 WileyPeriodicals, Inc. NETWORKS, Vol. 57(4), 354–361 2011

Keywords: minimum networks; gradient-constrained networks;Steiner trees; locally minimal.

1. INTRODUCTION

A gradient-constrained network is a network interconnect-ing a given set N of points in 3D-space, where each edge inthe network is embedded in Euclidean space so as to havegradient no greater than a given upper bound m. Here by thegradient g(pq) of an edge pq we mean the absolute value ofthe slope from p = (px, py, pz) to q = (qx, qy, qz), that is,

g(pq)def==

|pz − qz|√(px − qx)2 + (py − qy)2

. (1)

A gradient-constrained Steiner Minimum Tree (SMT) S is agradient-constrained network whose total length is minimum

Received July 2008; accepted June 2010Correspondence to: D. Thomas; e-mail: doreen.thomas@unimelb.edu.auDOI 10.1002/net.20406Published online 1 October 2010 in Wiley Online Library(wileyonlinelibrary.com).© 2010 Wiley Periodicals, Inc.

for the given point set N . The nodes of the network other thanthose in N are referred to as Steiner points, which are addedto reduce the length of the tree. Where no Steiner pointsare added, the network, denoted by T , is called a MinimalSpanning Tree (MST). Clearly, the length of the SMT on N ,denoted by LS(N), is no greater than the length of the MST onN , denoted by LT (N). The ratio ρ(N) = LS(N)/LT (N) (≤ 1)

measures the improvement of the SMT over the MST. Let ρ

be the infimum of ρ(N):

ρ = infN

{ρ(N)} = infN

{LS(N)

LT (N)

}< 1.

ρ is called the Steiner ratio for the gradient-constrainedSMTs.

A gradient-constrained SMT can be regarded as a shortestnetwork in a gradient metric space where the gradient metricfor an edge pq is defined in terms of the Euclidean and verticalmetrics, denoted by | · | and | · |v respectively:

|pq|g =

|pq| =√

(px − qx)2 + (py − qy)2 + (pz − qz)2

if g(pq) ≤ m,

|pq|v = |pz − qz|/µ if g(pq) ≥ m,

where

µ = m√1 + m2

= sin φ ≤ 1, (2)

and φ is the angle at which pq meets the horizontal plane.From this definition the following statement is trivial.

Theorem 1.1. For any two points p and q in space |pq| ≤|pg|g.

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It is well-known that for Euclidean (unconstrained) SMTs,the Steiner ratio is achieved by an equilateral triangle andρ = infN {ρ(N)} = √

3/2 [3]. In our previous article,Optimum Steiner Ratio for Gradient-Constrained NetworksConnecting Three Points in 3-Space, Part I [6] (referred to as‘Part I’ hereafter) the Steiner ratio for gradient-constrainedSMTs spanning three points and with a maximum gradientconstraint m ≤ 1 was discussed as well as in [5]. In that arti-cle, we proved that in a gradient metric space the Steiner ratiois also achieved by an equilateral triangle. More precisely, weproved the following result:

Theorem 1.2 (Theorem 5.9 in [6]). For the SMT on a set Nof three points (i.e., N = �) in a gradient metric space theSteiner ratio is

ρ� = inf� ρ(�) = 1

2+

√4 + 3m2 − 1

4√

1 + m2if 0 ≤ m ≤ 1.

This article is a continuation of Part I for the case m > 1.We prove that, in this case, the Steiner ratio is still achievedby an equilateral triangle and that

Theorem 1.3 (Main theorem). For the SMT on a set N ofthree points (i.e., N = �) in a gradient metric space theSteiner ratio, as illustrated in Figure 1, is

ρ� = inf� {ρ(�)}

=

ρ1 = 1

2+

√4 + 3m2 − 1

4√

1 + m2if 0 ≤ m ≤ √

3;

ρ2 = 3m + √9m2 + 12

4√

3m2 + 3if m ≥ √

3.

Remark 1.1. Note that the expression for the Steiner ratio ρ

is the same when the domain of m is extended from 0 ≤ m ≤ 1to 1 ≤ m ≤ √

3 but the expression changes when m ≥ √3.

In a gradient metric space every edge pq can be representedas a straight line segment. However, if the gradient of pq isgreater than m, while it is straight in the gradient metric space,it will be represented as a bent edge prq in Euclidean space,where pr, rq are straight line segments with gradient m. Anedge pq is labelled as “f”, “m”, or “b”, and called an f-edge,m-edge, or b-edge if g(pq) < m, g(pq) = m or g(pq) > m,respectively.

From now on let N = {a, b, c} (or denoted by N = �abc)be a given set of three points a, b, c, and s be the Steiner pointin a network joining the three points. Let Lab, Lbc, Lca denotethe respective labels of the edges of �abc. Then the triangleabc can be classified by its labeling, denoted by LabLbcLca,and referred to as the labeling of �abc (or the triangle label-ing). Note that if s is degenerate, that is, collapses into aterminal, then LS = LT and ρ(N) = 1. Hence, we mayassume that s is nondegenerate. Moreover, let Hs be the hor-izontal plane through s. If all three edges as, bs, cs lie on oneside of Hs, then trivially the SMT is not minimal. Hence, we

FIG. 1. The Steiner ratio for three points in the gradient metric space.[Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

may assume that, without loss of generality, as is below (oron) Hs but cs is above (or on) Hs. Let Las, Lbs, Lcs denotethe respective labels of the edges in the SMT of �abc. If bs isbelow Hs or on Hs, then we say the labeling of s (or the SMTlabeling) is Lcs/LbsLas. Similarly, the labeling is LcsLbs/Lasif bs is above or on Hs. This labeling characterizes andclassifies the Steiner point s and the SMT on �abc.

In this paper to prove Theorem 1.3 we follow the samestrategy as in the case of m ≤ 1, that is, we take the followingsteps:

• Normalizing and parameterizing gradient-constrained tri-angles, that is, proving that

(a) it is sufficient to show that the Steiner ratio ρ� isachieved by the Steiner minimal tree S spanning anequilateral �abc located in the first octant of thegradient metric space;

(b) the Cartesian coordinates of the vertices of �abc canbe expressed in terms of two parameters k and β wherek is the x-coordinate of vertex c and β is the anglebetween the inclined plane containing �abc and thexy-plane;

• Partitioning the parameter domain � (bounded by 0 ≤ k ≤1/2, 0 ≤ β ≤ π/2) into

(a) regions by the labelings of �abc, and further into(b) subregions by the labelings of s;

• Proving that

(a) ρ(k, β) does not increase when the point (k, β) moveseither horizontally or vertically in subregions of �,hence ρ(k, β) achieves its minimum on the boundaryin each subregion;

(b) the Steiner ratio is achieved at k = 0, β = π/2, henceTheorem 1.3 is proved.

2. FUNDAMENTALS

There are two corner stones in the study of gradient-constrained SMTs, which we briefly review: the variationalargument and the properties of right circular cones.

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2.1. The Variation of Gradient Metric

Suppose e = ps is an edge in gradient metric space. LetLe = |ps|g be the length of e. Suppose s is perturbed to s′ indirection v with p being fixed. Let L̇e(v)g be the directionalderivative of Le. The following proposition is proved in [1].

Proposition 2.1. Suppose e = ps is an edge and s is per-turbed to s′ in direction v with p being fixed. Let L̇e(v)g bethe directional derivative of Le in the gradient metric space.

(i) If g(ps) < m then L̇e(v)g = − cos ∠pss′.(ii) If g(ps) > m then L̇e(v)g = − cos ∠zss′/µ where z is a

point on the vertical line through s such that ∠psz ≤ 90◦.(iii) If g(ps) = m then L̇e(v)g = − cos ∠pss′ or =

− cos ∠zss′/µ depending on whether g(ps′) ≤ m org(ps′) ≥ m, respectively.

Corollary 2.2. (i) If s moves horizontally on the plane Hs

and g(ps) > m, then L̇e = 0, that is, the length of ps does notchange.

(ii) Suppose s moves vertically. If g(ps) ≥ m and Le =|ps|g is increasing (or decreasing), then L̇e = 1/µ (or L̇e =−1/µ respectively).

2.2. Right Circular Cones

Let Cp denote the right circular cone whose vertex is p andthe gradient of whose generating lines is m. The followingproposition is proved in [2, 8].

Proposition 2.3. Suppose Ca, Cc are two right circularcones and s is a Steiner point joining a and c.

(i) If g(cs) = m, then s lies on (the surface of) Cc and theintersection Cc

⋂ Hs is a circle, denoted by Rc,s. If s movesalong Rc,s, then L̇cs = 0. Hence, if as, bs are f-edges andthe SMT labeling is m/ff, then s is determined by g(cs) =m, ∠(t, as) = ∠(−t, bs), cos ∠csa + cos ∠csb = −1, wheret is the vector tangent to Rc,s at s.

(ii) If g(ac) > m, then the intersection Ca⋂ Cc is an

ellipse, denoted by Ea,c, and s lies on Ea,c. If s moves alongEa,c, then L̇as +L̇cs = 0. Hence, if bs is an f-edge and the SMTlabeling is m/fm, then s is determined by g(cs) = g(as) =m, bs ⊥ Ea,c.

3. NORMALIZATION AND PARAMETRIZATION

In Part I, the authors proved a series of results for the m ≤ 1case. Note that the arguments for a number of these resultsare independent of the value of m. Therefore, these particulararguments can be applied to the m > 1 case and we have thefollowing theorems (Theorem 3.1 and Theorem 3.2).

Theorem 3.1 (Theorem 3.2 in [6]). Given any triangle abcin 3D-space, there exists a Cartesian coordinate system withorthogonal axes labelled x,y,z, and the z axis vertical, suchthat the triangle is in the positive octant of the coordinatesystem, with one of the vertices of the triangle, say a, at theorigin and the x-axis lying on the plane of the triangle abc.

FIG. 2. G-equilateral triangle abc.

The plane of the �abc defined in the above theorem isdenoted by �. We define a g-equilateral triangle to be a tri-angle abc in 3D-space under the gradient constraint having|ab|g = |bc|g = |ca|g.

Theorem 3.2 (Theorem 3.1 in [6]). A triangle in 3d-space having the minimum value of the Steiner ratio is ag-equilateral triangle.

By this theorem, for such g-equilateral triangles we mayassume that bz < cz, and by symmetry assume cx ≤ 1/2 <

bx. Since scaling does not affect gradients of a triangle orits Steiner ratio, we may assume the g-equilateral �abc hasunit edge length. A G-equilateral triangle is a g-equilateraltriangle abc in the positive octant with a at the origin, thex-axis on the plane � of abc, bz < cz, cx ≤ 1

2 , and |ab|g =|bc|g = |ca|g = 1. For a G-equilateral �abc, LT (�abc) = 2.

For a particular value of m, which determines the max-imum gradient metric, only two independent variables areneeded to describe the vertices a, b, c, and all the inducedfunctions. Let the inclination of the plane abc be β and letcx = k. We define � to be the (k, β) parameter domain deter-mined by 0 ≤ k ≤ 1

2 and 0 ≤ β ≤ π2 (Fig. 2). Clearly, the

G-equilateral �abc, characterized by the parameters k and β,is unique, and the length of the SMT on �abc is a functionof k, β, that is, LS(�abc) = LS(k, β). Summing up the abovediscussions we have

Theorem 3.3. In the gradient metric space

ρ�= inf� {ρ(�)}= inf�∗ {ρ(�∗)|�∗ is a G-equilateral triangle}

= inf�∗

{LS(�∗)LT (�∗)

}

= infk,β

{1

2LS(k, β)

}

= infk,β

{1

2(Las(k, β) + Lbs(k, β)) + Lcs(k, β)

}

= infk,β

{1

2(La(k,β)s(k,β) + Lb(k,β)s(k,β) + Lc(k,β)s(k,β))

}.

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FIG. 3. Angles β, φ, ψ and parameters k, µ, η.

4. FEASIBLE TRIANGLE LABELINGS ANDPARTITIONING OF THE PARAMETER DOMAIN �

From Theorem 3.3 first we need to find the coordinates ofa, b, c and the Steiner point s in terms of k, β. The followingtheorem was proved in Part I. The proof is based on the vari-ational argument and independent of the value of m, hence itholds for the current case, m > 1.

Theorem 4.1 (Theorem 3.5 in [6]). If a triangle abc is aG-equilateral triangle then g(ab) < m and g(ab) ≤ g(bc) ≤g(ac). Hence the only feasible labelings of a G-equilateraltriangle are fff, ffm, ffb, fmm, fmb, and fbb.

Note that the coordinates of a, b, c depend on the labelingsof �abc and the coordinates of s depend on the labelings ofthe SMT. By the definition of G-equilateral triangles a =(0, 0, 0) and c = (k, cy, cz). Recall that in the definition of thegradient metric, µ = m/

√1 + m2 = sin φ [Equation (2)],

where φ is the angle that ac makes with the horizontal plane.Clearly, cz = √

1 − k2 sin β if g(ac) ≤ m, and cz = µ ifg(ac) ≥ m. We introduce a few more parameters in additionto µ. Let

η = µ

sin β, ν = 2 sin ψ − 1 (3)

where ψ = arcsin(η/√

η2 + k2) is the angle as shown inFigure 3. Then, using these parameters the coordinates ofb for different triangle labelings can be determined by thecondition that |ab|g = |bc|g = 1 and b ∈ �. The coordinatesof a, b, c for the three labelings fff, ffb, and fbb are listedin Table 1. Note that the coordinates of a, b, c for the otherthree labelings ffm, fmb, and fmm are not listed in the tablebecause for these labelings the parameters k and β are notindependent.

We define the collection of points in � that representG-equilateral triangles having the same labeling as triangleregions or simply regions of � as shown in Figure 4. Notethat the label m is a critical label between f and b. That is, anm-edge can be regarded as an f-edge and a b-edge. Hence,an m-edge is a constraint eliminating one degree of freedomwhen defining a region. As a result,

(a) each of the regions represented by fff, ffb, and fbb hastwo degrees of freedom, hence they are 2D-regions;

(b) each of the regions ffm and fmb has one degree of free-dom, and hence each is a 1D-region, that is, each can berepresented by a curve in the partitioning of �. We denotethe two curves by Cffm and Cfmb;

(c) finally, the region fmm has no degrees of freedom, thatis, it is represented by a point, the intersection of Cffm andCfmb.

The expressions for the two curves Cffm and Cfmb can bederived using the m-edge constraints. The expression for thefirst curve Cffm can be derived from the constraint g(ac) =m. By equating cz = √

1 − k2 sin β for the labeling fff andcz = µ for the labeling ffb we obtain the expression forCffm :

√1 − k2 sin β = µ, that is,

(Cffm :) β(k, m) = arcsin

(µ√

1 − k2

)

= arcsin

(m√

1 + m2√

1 − k2

). (4)

Similarly, the expression for the second curve Cfmb can bederived from the constraint g(bc) = m. However, the con-straint g(bc) = m is symmetric to g(ac) = m with respect tothe vertical line k = 1/2. Hence the expression for the curve

TABLE 1. The coordinates of a, b, c for different triangle labelings.

Labeling Coordinates of a, b, c

fff a = (0, 0, 0)

b =(√

3

2

√1 − k2 + k

2,

(1

2

√1 − k2 −

√3

2k

)cos β,

(1

2

√1 − k2 −

√3

2k

)sin β

)

c = (k,√

1 − k2 cos β,√

1 − k2 sin β)

ffb a = (0, 0, 0)

b =(

k

2+ µν

2 sin β,µ cot β

2− kν cos β

2,µ

2− kν sin β

2

)c = (k, µ cot β, µ)

fbb a = (0, 0, 0)

b = (1, 0, 0)

c = (k, µ cot β, µ)

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FIG. 4. Partition patterns of � as m increases.

Cfmb can be directly obtained from Equation (4) by replacingk with (1 − k):

(Cfmb :) β(k, m) = arcsin

(µ√

1 − (1 − k)2

)

= arcsin

(m√

1 + m2√

1 − (1 − k)2

).

(5)

Finally, the two curves meet at k = 1/2 where 1 − k = k.Hence their intersection, denoted by F in Figure 4, is

F(k, β) = F

(1

2,

2µ√3

)= F

(1

2, arcsin

(2m√

3√

1 + m2

)).

Remark 4.1. When m = √3 the region fbb disappears and

F lies at the top-right-hand corner of � (see Fig. 4).

5. SMT LABELINGS IN DIFFERENT REGIONS OF �

In the m ≤ 1 case there are, up to symmetry, five feasiblyoptimal SMT labelings (f/ff), (m/ff), (m/fm), (m/mm), and(b/mm). However, if m > 1, then (m/mm) and (b/mm) areinfeasible but b/ff is feasible as shown in Theorem 5.1.

Theorem 5.1. Up to symmetry there are four feasible SMTlabelings, namely (f/ff),(m/ff), (m/mf), and (b/ff), if m > 1.

Proof. Without loss of generality, assume the singleedge cs is above Hs and the edges bs and as are below Hs.

(i) Suppose cs is a b-edge.First, if one of as and bs, say as, is a b-edge, then a verti-

cally downward perturbation of s keeps Lcs +Las constant byProposition 2.3(ii), but bs is shortened. Second, if both as, bsare m-edges, then s is not optimal because a vertically down-ward perturbation of s would result in a negative directionalderivative as m > 1

L̇cs + L̇as + L̇bs ≤ 1

µ− 2µ = 1 − m2

m√

1 + m2< 0

by Corollary 2.2(ii). Finally, if one of as, bs, say as, is anm-edge but bs is an f-edge, then s lies on the cone Ca. Ifs moves along the horizontal circle Ra,s = Ca

⋂ Hs, thenLcs + Las does not change by Corollary 2.2(i). Hence, tominimise |bs|, bs must be perpendicular to the horizontalcircle Ra,s. Hence bs must lie in the vertical plane through s,which is perpendicular to the tangent to the horizontal circleRa,s = Ca

⋂ Hs; as is also perpendicular to this tangent. Itfollows that as and bs lie in the vertical plane, denoted by

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TABLE 2. Summary of the systems of equations for computing the Steiner point s.

Labeling System of equations Degree Source

(b/ff) s on Pa,b, ∠(t, as) = ∠(−t, bs) 2 Theorem 6.1(f/ff) ∠asb = ∠bsc = ∠csa = 120◦ 2 Melzak construction [4](m/fm) g(cs) = g(as) = m, bs ⊥ Ea,c 4 Proposition 2.3(ii)(m/ff) g(cs) = m, ∠(t, as) = ∠(−t, bs), cos ∠csa + cos ∠csb = −1 8 Proposition 2.3(i)

Pa,b, through a and b. Moreover, bs ⊥ as otherwise movings along as does not change Lcs+Las but bs could be shortenedagain by the variational argument. However, even if s lies onRa,s and bs ⊥ as, a vertically downward perturbation of swould result in

L̇cs + L̇as + L̇bs ≤ 1

µ− µ − µ

m= 1 − m

m√

1 + m2< 0,

by Proposition 2.1. Hence, s is still not optimal and the onlypossible labeling is (b/ff).

(ii) Suppose cs is an m-edge. Then, an argument similarto (i) shows that neither as nor bs can be a b-edge and thatas and bs cannot both be m-edges. Hence, only (m/ff) and(m/fm) are possible.

(iii) Suppose cs is an f-edge. Note that cs makes an anglewith the positive z-axis greater than an m-edge makes. Hence,if an edge of as or bs, say as, is a b-edge or an m-edge, then avertically downward perturbation of s results in a negative L̇bs

and the total variation is also negative by Corollary 2.2(ii).Hence, the only possible labeling is (f/ff). ■

From Theorem 5.1 there are 10 possible labelings of theSMT of a triangle (not taking symmetry into account), namely(f/ff), (ff/f), (m/ff), (ff/m), (m/fm) (mf/m), (m/mf), (fm/m),(b/ff), and (ff/b). We now proceed to reduce this number tofour for G-equilateral triangles.

Theorem 5.2. For m > 1 the only possible labelings of theSMT of a G-equilateral triangle are (f/ff), (m/ff), (m/fm), and(b/ff).

Proof. We have proved in Part I (Theorem 4.1 in [6]) that(ff/f), (ff/m), (mf/m), (fm/m), and (fm/m) are infeasible form ≤ 1. Because the arguments in the proof are independent ofm, these labelings are also infeasible for m > 1. Argumentssimilar to those in Theorem 4.1 in [6] rule out ff/b. ■

We define the ordered pair (LabLbcLca, Lcs/LbsLas) to bea subregion of �. All points in a subregion of � represent G-equilateral triangles that have the same triangle labeling andin addition have SMTs with the same labeling. Comparedwith the partition of � into regions according to the trian-gle labeling, the partition of a region into Subregions by theSMT labeling is much more complicated. In the latter case,an m-edge in the SMT does not reduce the freedom of pointsin the parameter domain �, that is, there exist 2D-subregionswith an SMT labeling containing an m-edge. The boundary oftwo subregions does not represent a critical SMT labeling but

only indicates a change of labeling of an edge in the SMT:an f-edge becomes an m-edge and/or an m-edge becomesa b-edge. As we indicate below (Table 2), for some SMTlabelings the location of s (the coordinates of s) is the root ofan algebraic system with possible degree up to eight, henceno explicit expression for s exists. It follows that we cannotgive explicit expressions for all the boundary curves of sub-regions but we can plot the curves by numerical computationas shown in Figure 4.

We can consider the partitioning of subregions from thefollowing viewpoint: let β gradually increase for fixed k.When β = 0, trivially the Euclidean metric is the same as thegradient metric and the triangle labeling is fff and the SMTlabeling is f/ff. With β increasing, a configuration is reachedwhere either

(i) the triangle labeling fff becomes ffm and then ffb (thecurve BC in Fig. 4), or

(ii) s lies on the surface of the cone Cc of c and the SMTlabeling f/ff becomes m/ff (the curve BA in Fig. 4).

The former case occurs when k ≤ Bk and the latter caseoccurs when k ≥ Bk , where Bk is the k-coordinate of point B.The system of five equations for determining the coordinatesof s and the parameter pair for B is given below

g(ac) = m, g(cs) = m, ∠(asb) = 120◦, ∠(bsc) = 120◦,

∠(csa) = 120◦. (6)

The solution of the system for B is

B = B(k, β)

=

√2 − √

3

2, arctan

2m√4 − (1 + m2)(2 − √

3)

.

The expression for the curve BC, as a part of the curve Cffm

[Equation (4)], has been given in section 4. The expressionfor the curve BA can be derived from the last four equationsin Equations (6). All the other extreme points as labelled inFigure 4 can be computed and all the other curves in Figure 4can be derived in a similar way.

6. THE STEINER RATIO

As indicated in Theorem 3.3 to obtain ρ� we need onlycompute the SMT on the G-equilateral �abc. To this end,

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we need to compute the Steiner point s with the four feasibleSMT labelings. To locate s a system of three independentequations is required. The equations fall into two classes:

(i) If the gradient of an edge e is m, then a required equationis g(e) = m;

(ii) The equations are provided by the variational argumentthat L̇S = L̇as + L̇bs + L̇cs = 0.

For example, for labeling (m/ff) the first equation in thesystem is g(cs) = m. Let t be the tangent vector of the circleRc,s at s; then the second equation is L̇S = 0 when s movesalong Rc,s. This implies as and bs meet t, the tangent to Rc,s atthe same angle. Similarly, the third equation is L̇S = 0 when smoves along the generating line cs. The systems of equationsfor the labelings f/ff, m/ff and m/fm with the constraint m ≤ 1have been derived in [8]. Because the derivation of thesesystems is independent of m, they also hold for m > 1. Hence,the remaining case to discuss is the system for b/ff.

Theorem 6.1. The Steiner point s with labeling (b/ff) form > 1 is determined by the following conditions:

(i) s lies on the vertical plane Pa,b through a and b;(ii) ∠(t, as) = ∠(−t, bs)(≤ π/2) where t is the horizontal

vector at s on Pa,b;(iii) L̇S = 0 for s moving vertically downwards.

Proof. (i) By an argument similar to the proof(i) inTheorem 5.1.

(ii) By Corollary 2.2(i).(iii) Let s move vertically downwards. Then L̇s =√

1 + m2/m − 2 cos(π/2 − γ ) = 0 by the variationalargument and Corollary 2.2(ii). ■

Corollary 6.2. If the SMT labeling is b/ff, then the coordi-nates of s are

xs = xa(zs − za)

g∗√x2a + y2

a

, ys = ya(zs − za)

g∗√x2a + y2

a

,

zs = 1

2(za + zb + g∗

√(xa − xb)2 + (ya − yb)2), (7)

where g∗ = 1 + m2/√

3m2 − 1 is the gradient of as and bs.

All the systems and their degrees are listed in Table 2.The coordinates of s for labeling (b/ff) has been given in

Corollary 6.2. The coordinates of s for labeling (f/ff) can bedetermined by the well-known Melzak construction [4]. Wedo not have explicit expressions of the coordinates of s for thelabelings (m/fm) and (m/ff), and have to use approximations[8]. As a result, we cannot give an explicit expression for thelength of the SMT on �abc. However we have the followingresult.

Theorem 6.3. Suppose �abc is G-equilateral and supposem > 1. Then

(i) ρ(�abc) = √3/2 in subregion (fff,f/ff).

(ii) ρ(�abc) = 1/2+√3m2 − 1/4m in subregion (fbb,b/ff).

(iii) ∂ρ(�abc)/∂β > 0 in subregions (fff,m/ff) and (fff,b/ff).(iv) ∂ρ(�abc)/∂β < 0 in subregions (fbb,m/ff) and

(fbb,m/fm).(v) ∂ρ(�abc)/∂β < 0 in all subregions of region ffb.

(vi) ∂ρ(�abc)/∂k > 0 in line segment PJ or PF in Figure 4.

Proof. Most of these statements, except for the SMTlabeling in (ii) and (iii), have been proved in Part I for m ≤ 1and they still hold for m > 1 because the arguments are inde-pendent of m. Now we prove case (ii) (fbb,b/ff) and case (iii)(fff,b/ff).

(ii) Using g(as) = √1 + m2/

√3m2 − 1, sx = 1

2 and sy = 0

we obtain sz = √1 + m2/2

√3m2 − 1, and this together

with cz = m/√

1 + m2 gives the required result.(iii) Since for any k in the subregion (fff,f/ff) the edge ab

subtends a constant angle ∠(asb), the point s lies on thearc asb of a circle of constant radius. Therefore for aparticular k, increasing β results in a reduction of lengthof as ∪ bs and an increase of sz. Since ac is a b-edge,cz = m/

√1 + m2 for the entire subregion, and cz − sz

decreases with increasing β. ■

Corollary 6.4. ρ� is achieved at point P = (0, π/2), that

is, ρ� = infk,β ρ(k, β) = ρ(0, π/2).

Proof. The result follows from Theorems 6.3 and 3.3.■

Proof of the main theorem (Theorem 1.3). If m ≤ √3,

then the labeling pair at the point (0, π/2) in � is (ffb,m/fm).Clearly, the triangle labeling ffb means a = (0, 0, 0), c =(0, 0, µ), b = (

√1 − µ/2, 0, µ/2) by Table 2, and the SMT

labeling m/fm means g(as) = g(cs) = m, g(bs) = 0. Hence,it immediately follows that

ρ� = ρ1 = 1

2+

√4 + 3m2 − 1

4√

1 + m2.

Similarly, if m ≥ √3, then the labeling pair at the point

(0, π/2) in � is (ffb,f/ff). Hence, the SMT labeling f/ff implies∠asc = 120◦ and g(as) = g(cs) = √

3, and it immediatelyfollows that

ρ� = ρ2 = 3m + √9m2 + 12

4√

3m2 + 3.

■

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