THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE

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1 THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on the fat torus under certain conditions. In particuar, et T be a torus covered by a rectanguar fundamenta domain of dimension w. Let A, B, and C be points in T. Associated with A and B is a minima une region, L(A, B), which wi be described in our paper. If AB 4 + w and C is contained in L(A, B), then the minima path network is the union of the segments connecting A to C and C to B, which are contained in L(A, B). 1. Introduction In this paper, we are going to sove the three-point Steiner probem for a specia case on the fat torus. The Steiner probem asks to find the minima path network connecting n specified points. Given three points in the pane, A, B, and C, if one of the anges of the ABC is greater than 10, then the minima path network in the pane is constructed by connecting the two edges that are adjacent to that ange. If a of the anges of the triange are ess than 10, the minima path network contains an additiona point, caed a Steiner point. There are agorithms that can sove any given n point Steiner probem in the pane. The first such agorithm is Mezak s agorithm. The probem on the torus is significanty harder than the Eucidean probem. In the Eucidean pane there is ony one straight ine that connects two points, but on the torus there are infinitey many straight ines that connect two points. This probem is important because many peope have worked on it. It is aso groundbreaking work for simiar probems, on other surfaces.. Preiminaries We can reate the Steiner probem on the fat torus to the Steiner probem on the Eucidean pane. The fat torus has a rectanguar fundamenta domain in the Eucidean pane that acts as a covering space for the torus. By taking a rectanguar fundamenta domain in the Eucidean pane and sewing it s parae edges together, we can represent the fat torus. Then because on a torus we can create a path network over the seams, to represent the torus in the Eucidean pane, we need to tie the Eucidean pane with fundamenta domains. Let the dimensions of the fundamenta domain be w. Let a fundamenta domain be denoted by T m,n, where m denotes the horizonta position and n denotes the vertica position of the fundamenta domain. Let the origina fundamenta domain the fundamenta domain defined by the positive x and y axis with the origin as a corner be denoted T 0,0. Let the fundamenta domain directy 1

2 KATIE L. MAY AND MELISSA A. MITCHELL above wi be T 0,1, the fundamenta domain directy to the right wi be T 1,0, and the fundamenta domain that is above T 1,0 wi be T 1,1. So, given three points on the torus, each fundamenta domain, T m,n, wi have three points to represent these points. Connecting points in different domains represents the path network crossing over a seam. If a point A is ocated at (x, y), then the transates of A wi be at (x + m, y + nw) where is the ength of the fundamenta domain aong the x-axis, w is the width aong the y-axis, and m and n are indicators as to which fundamenta domain the transate is in. For exampe, if the transate is in T 1,0, then m = 1 and n = 0. Given points A, B, C on the torus, we wi refer to the transates of these points as A, B, and C..1. The Three Point Steiner Probem in Eucidean Space. Given three points A, B, and C in the Eucidean pane, if one of the anges of the ABC is at east 10 then the minima path network is the connection of the two edges forming the obtuse ange, caed the degenerate tree. If the anges are ess than 10, then an extra vertex S is added to form the minima path network, and this type of tree is caed a Steiner tree. Given A and B, a third point C can ie in one of five regions. Two of these regions are Steiner regions, where a the anges in ABC are ess than 10. The ength of the tree can easiy be found in these regions. Two equiatera trianges are constructed with common edge AB denoting the two third points of the equiatera trianges as E-points. The ength of the Steiner tree is the ength of the Simpson ine, which is the ine from C to the E-point on the opposite side of AB. The other three regions are degenerate; two of them connect A and B directy and then connects C to one of the endpoints, and the ast connects A to C and C to B. This ast region is caed the une of A and B, denoted by L(A, B)... Eucidean Space vs. the Fat Torus. The Steiner probem is more difficut than the Steiner probem in Eucidean Space because on the torus there are an infinite number of straight ines that connect to points on the torus, whereas in Eucidean space there is ony one straight ine between two points. To represent the infinite number of ines between two points on the torus, we have tied the Eucidean pane with fundamenta domains that contain transates of a the points on the torus. If A and B are on the torus, then if you choose a point in the Eucidean pane that represents A, then each different segment between A and a representation of B represents a different way to connect A and B on the torus. This unique feature makes the Steiner probem on the torus more difficut than in Eucidean Space. Fortunatey, Keith Penrod has proved that for three points on the fat torus, the minima path network must be contained in one rectanguar fundamenta domain. Whie this resut reduces the number of possibe cases, the probem on the torus is sti more difficut because for three points, there are sti tweve transates of each point to consider in the three-point probem.. Overview We show how to identify the minima path network connecting three fixed points on the fat torus under certain conditions. In particuar, et T be a torus covered by a rectanguar fundamenta domain of dimension w. Let A, B, and C, be points in T. Associated with A and B is a minima une region, L(A, B). If AB 4 + w and C is contained in L(A, B), then the minima path network is the

3 THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE union of the segments connecting A to C and C to B, which are contained in L(A, B). Given two fixed points A and B, and one movabe point, C, the three regions of interest can be found. The une region is the space where ACB is greater than or equa to 10. The degenerate regions are where either CAB or ABC is greater than or equa to 10. The fu Steiner regions are above and beow the une, where the ACB is ess than 10. With three steps we can find when the une path network is minima: Step 1. Maxima Lune Network Finding the maxima possibe ength for a minima path network in L(A, B). Step. Degenerate Case Finding the minima path network in the degenerate regions with the point in this region being a transate of one in the une. Step. Steiner Case Finding the minima path network in the Steiner regions with the point in this region being a transate of one in the une. By using these resuts, we find a condition on A and B such that if it is met, then the minima path network wi be contained in L(A, B). 4. Maximum Path Network Length For a Tree in the Lune We can construct a une by circumscribing two circes, one around each of the two equiatera trianges that have AB as a common edge. The centers of these circes we define as une foci. These two circes intersect at points A and B. The boundary of the une is defined by the two arcs between A and B. Let L(A, B) denote the une of A and B, and et L(A, B) signify the boundary of the une. Because a of the anges of an equiatera triange are 60, the arc between A and B is 10. This impies that a points on L(A, B) form a 10 ange with A and B. Given a point C in the une, the shortest tree to connect A, B, and C is to connect A to C and C to B. Theorem 4.1. Suppose A and B are points in the Eucidean pane. Then amongst a the points C of L(A, B), the sum of the distances AC + CB is maximized at the points equidistant from A and B on L(A, B). Proof. Let M be a point in the L(A, B) that forms a tree with maxima ength. Note that because of symmetry, there woud exist two points with this property. Aso because of symmetry we can find one maxima tree, and the other wi be a refection. First we know that the M that forms the maxima tree cannot be in L(A, B). Otherwise, there woud exist an X on the ine perpendicuar to AB through the point M and between M and L(A, B). Let the point D define the intersection of the perpendicuar ine and AB. Then by the Pythagorean Theorem, we know that then AD + XD = AX AD + DM = AM AX XD = AC MD

4 4 KATIE L. MAY AND MELISSA A. MITCHELL and because we know that XD > MD, AX XD > AM XD AX > AM Therefore AX > AM. Simiary BX > BM. And then AX + BX > AM + BM, contradicting that M defines a maximum ength tree in the une. Therefore M and M, the refection of M about AB, must be on L(A, B). Let O be the une focus of L(A, B) corresponding to the arc containing M, and O to the arc containing M respectivey. Let each of these circes have radius r. Without oss of generaity, consider the point M on L(A, B). We know that the AOB is 10. Let θ be the ange BOM, et α be OBM, and by the aw of sines, OMB is equa to α. Then AOM is 10 - θ, OAM is 10 - α, and AMO is 10 - α. (Picture) If we bisect the anges θ and 10 - θ, then AM = r sin θ BM = r sin 10 θ The maximum ength tree wi occur when the derivative of AM + BM = 0. So Let d dθ (AM + BM) = 0. Then d dθ (AM + BM) = d dθ (r sin θ 10 θ + r sin ) d dθ (AM + BM) = r cos θ 10 θ r cos r cos θ 10 θ r cos = 0 r cos 10 θ = r cos θ cos 10 θ = cos θ 10 θ = θ 10 θ = θ θ = 60 Then because AOB is 10 and θ is 60, OM bisects AOB. Therefore M is equidistant from A and B on the L(A, B), and is a point that maximizes the sum of AC + CB for a C in the une. Aso by refection, we know that the other point that maximizes this sum M is aso equidistant from A and B on the L(A, B). 5. Comparison of Degenerate Regions to Lune Region (Setup the probem.)

5 THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE Horizonta and Vertica Case. Given AB parae to the x-axis, for any C in the Lune, et L (A, B) be a transate of the une that contains C. The maximum vaue of AC+CB wi occur when C = M, where M is a point that is on the (A, B) and equidistant from A and B, as discussed above. If b is haf of the ength of AB, then AM + MB = 4b. If C is in L(A,B), the cosest C that is in the degenerate regions of AB is the same point as a transate of A (picture). If this is the case, then AB + BC =, the ength of the fundamenta domain. So if 4b <, then AC + CB wi aways be minima for C in the une of a given A and B. 4b A simiar case is when AB is parae to the y-axis. If < w the width of the fundamenta domain, then AC + CB wi be minima for a given A and B with C in L(A, B). 5.. Non-Horizonta Case. If AB is not parae to the??? x or y-axis,??? then there is an ange, α, between AB and the positive x-axis. The midpoint of AB, D, wi be ocated at (b cos α, b sin α), and transates of D wi be at (b cos α+, b sin α), (b cos α, b sin α + w), and (b cos α +, b sin α + w). These are the ony transates we have to consider because the minima path network must be contained in one fundamenta domain. The point B wi be at (b cos α, b sin α). Define the circe of rotation, P, to be the circe centered at D with radius b. The circe of rotation wi circumscribe of L(A, B) about the point D. (Picture) Theorem 5.1. Suppose A and B are points on a fundamenta domain of dimension w, in the xy-pane and et α be the ange between AB and the positive x-axis. Suppose C is a transate of C in L(A, B) in T 0,1. Then AB + BC wi be ess than or equa to b + b b cos α +. Proof. By the distance formua, we know that the distance from B to D is BD = (b cos α ) + (b sin α). Then the distance from B to the edge of P (define.. the circe of rotation) is the distance from B to the center of P minus the radius of the circe, or (b cos α ) + (b sin α) b. Then BC must be ess than or equa to the distance from B to P because C must be inside P. Then if we add AB to both sides and simpify, AB + BC b + (b cos α ) + (b sin α) b AB + BC b + (b cos α ) + (b sin α) AB + BC b + b cos α b cos α + + b sin α AB + BC b + b b cos α + By a simiar argument, for a transate C in T 1,0, AB+BC b+ b bw sin α + w. Theorem 5.. If the ratio AB > 1 cos α( cos α ) then AC + CB < AB + BC for every C in L(A, B) and C, a transate of C in T 0,1.

6 6 KATIE L. MAY AND MELISSA A. MITCHELL Proof. We know that AB + BC b + b b cos α +, and that AB + BC has maxima vaue 4b. So when AC + CB < AB + BC 4b < b + b b cos α + 4b b < b b cos α + ( 4b b) < b b cos α + ( 4 1) b < b b cos α + ( 4 ) b b < b cos α ( )b < b cos α ( 16 8 )b > cos α b + ( 16 8 )b > cos α b b (16 8 ) b > cos α b (8 4 ) b > cos α 6 If x = b, then to sove for x Then by the quadratic formua x ( 8 4 ) 1 6 x cos α = 0 x ( 8 4 ) x cos α = 0 6 b = cos α ± b = cos α ± cos α(1 ± = b cos α 4(1)( ) cos α ( 16 8 ) cos α )

7 THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE7 And since ony the positive vaues have meaning in this case (a better way to say this?) cos α(1 + 1 ( 16 8 cos = α )) AB Simiary, for C in T 1,0, if the ratio w AB > 1 sin α( sin α ) then AC + CB < AB + BC for every C in L(A, B). Theorem 5.. Given the points A, B, on the fat torus and C in L(A, B), a transate of C in T 1,1 is not the cosest transate of C. (A better way to say this...?) Proof. Consider D the center of the circe of rotation in T 0,0, D in T 0,1, and D in T 1,1. Assume D and D ie on the x-axis. If b is the radius of the circe of rotation, b cannot be arger than 1 + w, because otherwise AB does not ie in a singe fundamenta domain. Consider when b = 1 + w. Let H be the circe of radius 1 + w centered and D and et Q a circe of the same radius centered at D. Let d be a point on the circe of rotation centered at D, such that 0 drr < π 4. The point d is contained in the circe H because if you consider the ine x = 1, then the boundary of the circe is on the eft but the point d is on the right. Aso the point d is not contained in the circe Q because d is beow the tangent ine y = w x + w, so d cannot be inside Q. This impies that D is coser to d than D. Simiary, if d is on the arc of the circe so that π 4 < drr π, then D in the vertica domain wi be coser than D. And if α = π 4, then d is the midpoint of the diagona of the fundamenta domain, which impies that d is equidistant from the three transates of D. Now if b be ess than 1 + w and π 4 α π. Under these conditions, the maximum vaue for D and d is at α = π 4. What now? 6. Comparison of Steiner Regions to Lune Region In this section we wi be comparing the ength of the maxima une region tree, the union of AM and MB, with trees in the Steiner region constructed with a transate of C, a point in the une region. For the ength of the Steiner tree we wi use the Simpson ine, the ine from E to C. Again we wi denote b as haf of the ength of AB. Given this, the maximum ength tree in the une is 4b. Denote une focus above AB to be O and corresponding transate to be O. Theorem 6.1. Suppose, A and B are points in a fundamenta domain of a torus of dimension w, AB makes an ange α with the positive x-axis. A new set of axes

8 8 KATIE L. MAY AND MELISSA A. MITCHELL are setup with the center of the une centered at the origin, then DD, the segment from the center of the une to the center of the transated une makes an ange β with the positive x -axis. (Note that β is aso the ange between a horizonta ine through O and OO.) Suppose E is the E-point for A and B beow AB, then for any C in a transate of L(A, B), EC + 8b sin β + 16b b. Proof. By the distance formua, we know that the distance from E to M is ( cos β) + ( sin β + b + b). Then the distance from C to M is the radius of the circe, or b. Therefore EC must be ess than or equa to EM C M, when we simpify we get EC + 8b sin β + 16b b By a simiar argument, for a transate in the vertica domain EC w + 8bw sin(90 β) + 16b b. Lemma 6.. r 5 r sin β /0 > 0 if and ony if EC > AC +CB for positive r. Proof. We know that the maximum ength of AC + CB inside the L(A, B) is 4b. We want that ength to be ess than EC. 4b < + 8b sin β + 16b b ( 6b ) < + 8b sin β + 16b b 6b < + 8b sin β + 16b 0b 8b sin β > 0 If we divide this by 0 and note that r = b/, then we get r 5 r sin β /0 > 0. Lemma 6.. If C is in the Steiner region, then cos(β + 0 ) r/

9 THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE9 Proof. We want to find what ange β is when M is on the boundary between the Steiner region and the degenerate region. Note that β is aso the ange between the ine parae to the x-axis through point Mand MM. Denote K to be the intersection of the ine parae to the x-axis through point M and BM. We draw BMM. The ength of MM is, and the ength of MB is b. Denote BMM as δ. We know that cos δ = b/ = b. It is easy to find that KMB is 0. In order for point M to be in the Steiner region β arccos r 0. From this we see that cos(β + 0 ) r. Theorem 6.4. Let r = b, if r then AC + CB < EC for every C in L(A, B) and C, a transate of C in the horizonta domain. Proof. From before, we know that r 5 r sin β 0 > 0. We can us the quadratic formua to sove for r. r > 1 5 sin β ± 5 (sin β) We do not want r to be negative, so r > 5 sin β + 5 (sin β) + 0. Now we want to sove cos(β + 0 ) r formua we get, for sin β. By using the cosine sum

10 10 KATIE L. MAY AND MELISSA A. MITCHELL cos β cos 0 sin β sin 0 r cos β 1 r sin β 1 (sin β) 1 r sin β ( 1 (sin β) ) ( 1 sin β + r ) 4 (1 (sin β) ) 1 4 (sin β) + r sin β + 4r (sin β) + r sin β + 16r We can use the quadratic formua to sove for sin β. r 4r ± sin β 4 16r 9 1 We do not want sin β to be negative, so sin β r + 4r. 4 When we put this in for sin β in our r equation. We find that r Therefore if r 14, then EC > AC + CB. A simiar resut can be found for the points in the T 0,1 fundamenta domain, exchanging w for and etting β be the ange between the negative x-axis and the ine from O to O. 7. Summary We have found that given a torus covered by a fundamenta domain of, and given two points A and B separated by a distance of. If a point ies in the L(A, B) then the minima path network is the union of segments connecting A to C and C to B. 8. Concusion These resuts are not tight resuts, but they are a start. We have experimented with this probem in Geosketchpad and in Mape, and it appears that these resuts are true for greater vaues of AB, amost for any A and B contained in one fundamenta domain. In the future we hope to find tighter resuts, use this to competey

11 THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE 11 sove the three point probem on the torus, and to extend the three point probem to n points on the fat torus.

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