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https://en.wikipedia.org/wiki/Lie_sphere_geometry
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Lie sphere geometry
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Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines should be regarded as circles of infinite radius and that points in the plane should be regarded as circles of zero radius. The space of circles in the plane, including points and lines turns out to be a manifold known as the Lie quadric. Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles. To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius. Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous "line-sphere correspondence" between the space of lines and the space of spheres in 3-dimensional space.
| 2022-06-19T14:56:51
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# Lie sphere geometry
**Lie sphere geometry** is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.
The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).
To handle this, curves in the plane and surfaces in space are studied using their **contact lifts**, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.
Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous "line-sphere correspondence" between the space of lines and the space of spheres in 3-dimensional space.
## Basic concepts
The key observation that leads to Lie sphere geometry is that theorems of Euclidean geometry in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangential contact have a more natural formulation in a more general context in which circles, lines and points (resp. spheres, planes and points) are treated on an equal footing. This is achieved in three steps. First an ideal point at infinity is added to Euclidean space so that lines (or planes) can be regarded as circles (or spheres) passing through the point at infinity (i.e., having infinite radius). This extension is known as inversive geometry with automorphisms known as "Mobius transformations". Second, points are regarded as circles (or spheres) of zero radius. Finally, for technical reasons, the circles (or spheres), including the lines (or planes) are given orientations.
These objects, i.e., the points, oriented circles and oriented lines in the plane, or the points, oriented spheres and oriented planes in space, are sometimes called cycles or Lie cycles. It turns out that they form a quadric hypersurface in a projective space of dimension 4 or 5, which is known as the Lie quadric. The natural symmetries of this quadric form a group of transformations known as the Lie transformations. These transformations do not preserve points in general: they are transforms of the Lie quadric, *not* of the plane/sphere plus point at infinity. The point-preserving transformations are precisely the Möbius transformations. The Lie transformations which fix the ideal point at infinity are the Laguerre transformations of Laguerre geometry. These two subgroups generate the group of Lie transformations, and their intersection are the Möbius transforms that fix the ideal point at infinity, namely the affine conformal maps.
These groups also have a direct physical interpretation: As pointed out by Harry Bateman, the Lie sphere transformations are identical with the spherical wave transformations that leave the form of Maxwell's equations invariant. In addition, Élie Cartan, Henri Poincaré and Wilhelm Blaschke pointed out that the Laguerre group is simply isomorphic to the Lorentz group of special relativity (see Laguerre group isomorphic to Lorentz group). Eventually, there is also an isomorphism between the Möbius group and the Lorentz group (see Möbius group#Lorentz transformation).
## Lie sphere geometry in the plane
### The Lie quadric
The Lie quadric of the plane is defined as follows. Let **R**<sup>3,2</sup> denote the space **R**<sup>5</sup> of 5-tuples of real numbers, equipped with the signature (3,2) symmetric bilinear form defined by
$$(x\_{0},x\_{1},x\_{2},x\_{3},x\_{4})\cdot (y\_{0},y\_{1},y\_{2},y\_{3},y\_{4})=-x\_{0}y\_{0}-x\_{1}y\_{1}+x\_{2}y\_{2}+x\_{3}y\_{4}+x\_{4}y\_{3}.$$
The projective space **R**P<sup>4</sup> is the space of lines through the origin in **R**<sup>5</sup> and is the space of nonzero vectors **x** in **R**<sup>5</sup> up to scale, where **x**= (*x*<sub>0</sub>,*x*<sub>1</sub>,*x*<sub>2</sub>,*x*<sub>3</sub>,*x*<sub>4</sub>). The planar Lie quadric *Q* consists of the points \[**x**\] in projective space represented by vectors **x** with **x** · **x** = 0.
To relate this to planar geometry it is necessary to fix an oriented timelike line. The chosen coordinates suggest using the point \[1,0,0,0,0\] ∈ **R**P<sup>4</sup>. Any point in the Lie quadric *Q* can then be represented by a vector **x** = λ(1,0,0,0,0) + **v**, where **v** is orthogonal to (1,0,0,0,0). Since \[**x**\] ∈ *Q*, **v** · **v** = *λ*<sup>2</sup> ≥ 0.
The orthogonal space to (1,0,0,0,0), intersected with the Lie quadric, is the two dimensional celestial sphere *S* in Minkowski space-time. This is the Euclidean plane with an ideal point at infinity, which we take to be \[0,0,0,0,1\]: the finite points (*x*,*y*) in the plane are then represented by the points \[**v**\] = \[0,*x*,*y*, 1, (*x*<sup>2</sup>+*y*<sup>2</sup>)/2\]; note that **v** · **v** = 0, **v** · (1,0,0,0,0) = 0 and **v** · (0,0,0,0,1) = 1.
Hence points **x** = *λ*(1,0,0,0,0) + **v** on the Lie quadric with *λ* = 0 correspond to points in the Euclidean plane with an ideal point at infinity. On the other hand, points **x** with *λ* nonzero correspond to oriented circles (or oriented lines, which are circles through infinity) in the Euclidean plane. This is easier to see in terms of the celestial sphere *S*: the circle corresponding to \[*λ*(1,0,0,0,0) + **v**\] ∈ *Q* (with *λ* ≠ 0) is the set of points **y***S* with **y** · **v** = 0. The circle is oriented because **v**/*λ* has a definite sign; \[*λ*(1,0,0,0,0) + **v**\] represents the same circle with the opposite orientation. Thus the isometric reflection map **x****x** \+ 2 (**x** · (1,0,0,0,0)) (1,0,0,0,0) induces an involution *ρ* of the Lie quadric which reverses the orientation of circles and lines, and fixes the points of the plane (including infinity).
To summarize: there is a one-to-one correspondence between points on the Lie quadric and *cycles* in the plane, where a cycle is either an oriented circle (or straight line) or a point in the plane (or the point at infinity); the points can be thought of as circles of radius zero, but they are not oriented.
### Incidence of cycles
Suppose two cycles are represented by points \[**x**\], \[**y**\] ∈ *Q*. Then **x** · **y** = 0 if and only if the corresponding cycles "kiss", that is they meet each other with oriented first order contact. If \[**x**\] ∈ *S***R**<sup>2</sup> ∪ {∞}, then this just means that \[**x**\] lies on the circle corresponding to \[**y**\]; this case is immediate from the definition of this circle (if \[**y**\] corresponds to a point circle then **x** · **y** = 0 if and only if \[**x**\] = \[**y**\]).
It therefore remains to consider the case that neither \[**x**\] nor \[**y**\] are in *S*. Without loss of generality, we can then take **x**= (1,0,0,0,0) + **v** and **y** = (1,0,0,0,0) + **w**, where **v** and **w** are spacelike unit vectors in (1,0,0,0,0). Thus **v** ∩ (1,0,0,0,0) and **w** ∩ (1,0,0,0,0) are signature (2,1) subspaces of (1,0,0,0,0). They therefore either coincide or intersect in a 2-dimensional subspace. In the latter case, the 2-dimensional subspace can either have signature (2,0), (1,0), (1,1), in which case the corresponding two circles in *S* intersect in zero, one or two points respectively. Hence they have first order contact if and only if the 2-dimensional subspace is degenerate (signature (1,0)), which holds if and only if the span of **v** and **w** is degenerate. By Lagrange's identity, this holds if and only if (**v** · **w**)<sup>2</sup> = (**v** · **v**)(**w** · **w**) = 1, i.e., if and only if **v** · **w** = ± 1, i.e., **x** · **y** = 1 ± 1. The contact is oriented if and only if **v** · **w** = – 1, i.e., **x** · **y** = 0.
### The problem of Apollonius
The incidence of cycles in Lie sphere geometry provides a simple solution to the problem of Apollonius. This problem concerns a configuration of three distinct circles (which may be points or lines): the aim is to find every other circle (including points or lines) which is tangent to all three of the original circles. For a generic configuration of circles, there are at most eight such tangent circles.
The solution, using Lie sphere geometry, proceeds as follows. Choose an orientation for each of the three circles (there are eight ways to do this, but there are only four up to reversing the orientation of all three). This defines three points \[**x**\], \[**y**\], \[**z**\] on the Lie quadric *Q*. By the incidence of cycles, a solution to the Apollonian problem compatible with the chosen orientations is given by a point \[**q**\] ∈ *Q* such that **q** is orthogonal to **x**, **y** and **z**. If these three vectors are linearly dependent, then the corresponding points \[**x**\], \[**y**\], \[**z**\] lie on a line in projective space. Since a nontrivial quadratic equation has at most two solutions, this line actually lies in the Lie quadric, and any point \[**q**\] on this line defines a cycle incident with \[**x**\], \[**y**\] and \[**z**\]. Thus there are infinitely many solutions in this case.
If instead **x**, **y** and **z** are linearly independent then the subspace *V* orthogonal to all three is 2-dimensional. It can have signature (2,0), (1,0), or (1,1), in which case there are zero, one or two solutions for \[**q**\] respectively. (The signature cannot be (0,1) or (0,2) because it is orthogonal to a space containing more than one null line.) In the case that the subspace has signature (1,0), the unique solution **q** lies in the span of **x**, **y** and **z**.
The general solution to the Apollonian problem is obtained by reversing orientations of some of the circles, or equivalently, by considering the triples (**x**,*ρ*(**y**),**z**), (**x**,**y**,*ρ*(**z**)) and (**x**,*ρ*(**y**),*ρ*(**z**)).
Note that the triple (*ρ*(**x**),*ρ*(**y**),*ρ*(**z**)) yields the same solutions as (**x**,**y**,**z**), but with an overall reversal of orientation. Thus there are at most 8 solution circles to the Apollonian problem unless all three circles meet tangentially at a single point, when there are infinitely many solutions.
### Lie transformations
Any element of the group O(3,2) of orthogonal transformations of **R**<sup>3,2</sup> maps any one-dimensional subspace of null vectors in **R**<sup>3,2</sup> to another such subspace. Hence the group O(3,2) acts on the Lie quadric. These transformations of cycles are called "Lie transformations". They preserve the incidence relation between cycles. The action is transitive and so all cycles are Lie equivalent. In particular, points are not preserved by general Lie transformations. The subgroup of Lie transformations preserving the point cycles is essentially the subgroup of orthogonal transformations which preserve the chosen timelike direction. This subgroup is isomorphic to the group O(3,1) of Möbius transformations of the sphere. It can also be characterized as the centralizer of the involution *ρ*, which is itself a Lie transformation.
Lie transformations can often be used to simplify a geometrical problem, by transforming circles into lines or points.
### Contact elements and contact lifts
The fact that Lie transformations do not preserve points in general can also be a hindrance to understanding Lie sphere geometry. In particular, the notion of a curve is not Lie invariant. This difficulty can be mitigated by the observation that there is a Lie invariant notion of contact element.
An oriented contact element in the plane is a pair consisting of a point and an oriented (i.e., directed) line through that point. The point and the line are incident cycles. The key observation is that the set of all cycles incident with both the point and the line is a Lie invariant object: in addition to the point and the line, it consists of all the circles which make oriented contact with the line at the given point. It is called a *pencil of Lie cycles*, or simply a *contact element*.
Note that the cycles are all incident with each other as well. In terms of the Lie quadric, this means that a pencil of cycles is a (projective) line lying entirely on the Lie quadric, i.e., it is the projectivization of a totally null two dimensional subspace of **R**<sup>3,2</sup>: the representative vectors for the cycles in the pencil are all orthogonal to each other.
The set of all lines on the Lie quadric is a 3-dimensional manifold called the space of contact elements *Z*<sup>3</sup>. The Lie transformations preserve the contact elements, and act transitively on *Z*<sup>3</sup>. For a given choice of point cycles (the points orthogonal to a chosen timelike vector **v**), every contact element contains a unique point. This defines a map from *Z*<sup>3</sup> to the 2-sphere *S*<sup>2</sup> whose fibres are circles. This map is not Lie invariant, as points are not Lie invariant.
Let *γ*:\[*a*,*b*\] → **R**<sup>2</sup> be an oriented curve. Then *γ* determines a map *λ* from the interval \[*a*,*b*\] to *Z*<sup>3</sup> by sending *t* to the contact element corresponding to the point *γ*(*t*) and the oriented line tangent to the curve at that point (the line in the direction *γ* '(*t*)). This map *λ* is called the *contact lift* of *γ*.
In fact *Z*<sup>3</sup> is a contact manifold, and the contact structure is Lie invariant. It follows that oriented curves can be studied in a Lie invariant way via their contact lifts, which may be characterized, generically as Legendrian curves in *Z*<sup>3</sup>. More precisely, the tangent space to *Z*<sup>3</sup> at the point corresponding to a null 2-dimensional subspace *π* of **R**<sup>3,2</sup> is the subspace of those linear maps (A mod *π*):*π***R**<sup>3,2</sup>/*π* with
*A*(**x**) · **y** \+ **x** · *A*(**y**) = 0
and the contact distribution is the subspace Hom(*π*,*π*/*π*) of this tangent space in the space Hom(*π*,**R**<sup>3,2</sup>/*π*) of linear maps.
It follows that an immersed Legendrian curve *λ* in *Z*<sup>3</sup> has a preferred Lie cycle associated to each point on the curve: the derivative of the immersion at *t* is a 1-dimensional subspace of Hom(*π*,*π*/*π*) where *π*=*λ*(*t*); the kernel of any nonzero element of this subspace is a well defined 1-dimensional subspace of *π*, i.e., a point on the Lie quadric.
In more familiar terms, if *λ* is the contact lift of a curve *γ* in the plane, then the preferred cycle at each point is the osculating circle. In other words, after taking contact lifts, much of the basic theory of curves in the plane is Lie invariant.
## Lie sphere geometry in space and higher dimensions
### General theory
Lie sphere geometry in *n*-dimensions is obtained by replacing **R**<sup>3,2</sup> (corresponding to the Lie quadric in *n* = 2 dimensions) by **R**<sup>*n* \+ 1, 2</sup>. This is **R**<sup>n + 3 </sup> equipped with the symmetric bilinear form
$(x\_{0},x\_{1},\ldots x\_{n},x\_{n+1},x\_{n+2})\cdot (y\_{0},y\_{1},\ldots y\_{n},y\_{n+1},y\_{n+2})$
$$=-x\_{0}y\_{0}+x\_{1}y\_{1}+\cdots +x\_{n}y\_{n}+x\_{n+1}y\_{n+2}+x\_{n+2}y\_{n+1}.$$
The Lie quadric *Q*<sub>*n*</sub> is again defined as the set of \[**x**\] ∈ **R**P<sup>*n*+2</sup> = P(**R**<sup>*n*+1,2</sup>) with **x** · **x** = 0. The quadric parameterizes oriented (*n* – 1)-spheres in *n*-dimensional space, including hyperplanes and point spheres as limiting cases. Note that *Q*<sub>*n*</sub> is an (n + 1)-dimensional manifold (spheres are parameterized by their center and radius).
The incidence relation carries over without change: the spheres corresponding to points \[**x**\], \[**y**\] ∈ *Q*<sub>*n*</sub> have oriented first order contact if and only if **x** · **y** = 0. The group of Lie transformations is now O(n + 1, 2) and the Lie transformations preserve incidence of Lie cycles.
The space of contact elements is a (2*n* – 1)-dimensional contact manifold *Z*<sup>2*n* – 1</sup>: in terms of the given choice of point spheres, these contact elements correspond to pairs consisting of a point in *n*-dimensional space (which may be the point at infinity) together with an oriented hyperplane passing through that point. The space *Z*<sup>2*n* – 1</sup> is therefore isomorphic to the projectivized cotangent bundle of the *n*-sphere. This identification is not invariant under Lie transformations: in Lie invariant terms, *Z*<sup>2*n* – 1</sup> is the space of (projective) lines on the Lie quadric.
Any immersed oriented hypersurface in *n*-dimensional space has a contact lift to *Z*<sup>2*n* – 1</sup> determined by its oriented tangent spaces. There is no longer a preferred Lie cycle associated to each point: instead, there are *n* – 1 such cycles, corresponding to the curvature spheres in Euclidean geometry.
The problem of Apollonius has a natural generalization involving *n* \+ 1 hyperspheres in *n* dimensions.
### Three dimensions and the line-sphere correspondence
In the case *n*=3, the quadric *Q*<sub>*3*</sub> in P(**R**<sup>4,2</sup>) describes the (Lie) geometry of spheres in Euclidean 3-space. Lie noticed a remarkable similarity with the Klein correspondence for lines in 3-dimensional space (more precisely in **R**P<sup>3</sup>).
Suppose \[**x**\], \[**y**\] ∈ **R**P<sup>3</sup>, with homogeneous coordinates (*x*<sub>0</sub>,*x*<sub>1</sub>,*x*<sub>2</sub>,*x*<sub>3</sub>) and (*y*<sub>0</sub>,*y*<sub>1</sub>,*y*<sub>2</sub>,*y*<sub>3</sub>). Put *p*<sub>*ij*</sub> = *x*<sub>*i*</sub>*y*<sub>*j*</sub> \- *x*<sub>*j*</sub>*y*<sub>*i*</sub>. These are the homogeneous coordinates of the projective line joining *x* and *y*. There are six independent coordinates and they satisfy a single relation, the Plücker relation
*p*<sub>01</sub> *p*<sub>23</sub> \+ *p*<sub>02</sub> *p*<sub>31</sub> \+ *p*<sub>03</sub> *p*<sub>12</sub> = 0.
It follows that there is a one-to-one correspondence between lines in **R**P<sup>3</sup> and points on the Klein quadric, which is the quadric hypersurface of points \[*p*<sub>01</sub>, *p*<sub>23</sub>, *p*<sub>02</sub>, *p*<sub>31</sub>, *p*<sub>03</sub>, *p*<sub>12</sub>\] in **R**P<sup>5</sup> satisfying the Plücker relation.
The quadratic form defining the Plücker relation comes from a symmetric bilinear form of signature (3,3). In other words, the space of lines in **R**P<sup>3</sup> is the quadric in P(**R**<sup>3,3</sup>). Although this is not the same as the Lie quadric, a "correspondence" can be defined between lines and spheres using the complex numbers: if **x** = (*x*<sub>0</sub>,*x*<sub>1</sub>,*x*<sub>2</sub>,*x*<sub>3</sub>,*x*<sub>4</sub>,*x*<sub>5</sub>) is a point on the (complexified) Lie quadric (i.e., the *x*<sub>*i*</sub> are taken to be complex numbers), then
*p*<sub>01</sub> = *x*<sub>0</sub> \+ *x*<sub>1</sub>, *p*<sub>23</sub> = –*x*<sub>0</sub> \+ *x*<sub>1</sub> *p*<sub>02</sub> = *x*<sub>2</sub> \+ i*x*<sub>3</sub>, *p*<sub>31</sub> = *x*<sub>2</sub> – i*x*<sub>1</sub> *p*<sub>03</sub> = *x*<sub>4</sub> , *p*<sub>12</sub> = *x*<sub>5</sub>
defines a point on the complexified Klein quadric (where i<sup>2</sup> = –1).
### Dupin cyclides
Lie sphere geometry provides a natural description of Dupin cyclides. These are characterized as the common envelope of two one parameter families of spheres *S*(*s*) and *T*(*t*), where *S* and *T* are maps from intervals into the Lie quadric. In order for a common envelope to exist, *S*(*s*) and *T*(*t*) must be incident for all *s* and *t*, i.e., their representative vectors must span a null 2-dimensional subspace of **R**<sup>4,2</sup>. Hence they define a map into the space of contact elements **Z**<sup>5</sup>. This map is Legendrian if and only if the derivatives of *S* (or *T*) are orthogonal to *T* (or *S*), i.e., if and only if there is an orthogonal decomposition of **R**<sup>4,2</sup> into a direct sum of 3-dimensional subspaces *σ* and *τ* of signature (2,1), such that *S* takes values in *σ* and *T* takes values in *τ*. Conversely such a decomposition uniquely determines a contact lift of a surface which envelops two one parameter families of spheres; the image of this contact lift is given by the null 2-dimensional subspaces which intersect *σ* and *τ* in a pair of null lines.
Such a decomposition is equivalently given, up to a sign choice, by a symmetric endomorphism of **R**<sup>4,2</sup> whose square is the identity and whose ±1 eigenspaces are *σ* and *τ*. Using the inner product on **R**<sup>4,2</sup>, this is determined by a quadratic form on **R**<sup>4,2</sup>.
To summarize, Dupin cyclides are determined by quadratic forms on **R**<sup>4,2</sup> such that the associated symmetric endomorphism has square equal to the identity and eigenspaces of signature (2,1).
This provides one way to see that Dupin cyclides are cyclides, in the sense that they are zero-sets of quartics of a particular form. For this, note that as in the planar case, 3-dimensional Euclidean space embeds into the Lie quadric *Q*<sub>3</sub> as the set of point spheres apart from the ideal point at infinity. Explicitly, the point (x,y,z) in Euclidean space corresponds to the point
\[0, *x*, *y*, *z*, –1, (*x*<sup>2</sup> \+ *y*<sup>2</sup> \+ *z*<sup>2</sup>)/2\]
in *Q*<sub>3</sub>. A cyclide consists of the points \[0,*x*<sub>1</sub>,*x*<sub>2</sub>,*x*<sub>3</sub>,*x*<sub>4</sub>,*x*<sub>5</sub>\] ∈ *Q*<sub>3</sub> which satisfy an additional quadratic relation
$$\sum \_{i,j=1}^{5}a\_{ij}x\_{i}x\_{j}=0$$
for some symmetric 5 ×; 5 matrix *A* = (*a*<sub>*ij*</sub>). The class of cyclides is a natural family of surfaces in Lie sphere geometry, and the Dupin cyclides form a natural subfamily.
* Walter Benz (2007) *Classical Geometries in Modern Contexts: Geometry of Real Inner Product Spaces*, chapter 3: Sphere geometries of Möbius and Lie, pages 93–174, Birkhäuser, ISBN 978-3-7643-8541-5 .
* Blaschke, Wilhelm (1929), "Differentialgeometrie der Kreise und Kugeln", *Vorlesungen über Differentialgeometrie*, Grundlehren der mathematischen Wissenschaften, vol. 3, Springer.
* Cecil, Thomas E. (1992), *Lie sphere geometry*, Universitext, Springer-Verlag, New York, ISBN 978-0-387-97747-8.
* Helgason, Sigurdur (1994), "Sophus Lie, the Mathematician" (PDF), *Proceedings of the Sophus Lie Memorial Conference, Oslo, August, 1992*, Oslo: Scandinavian University Press, pp. 3–21.
* Knight, Robert D. (2005), "The Apollonius contact problem and Lie contact geometry", *Journal of Geometry*, **83** (1–2), Basel: Birkhäuser: 137–152, doi:10.1007/s00022-005-0009-x, ISSN 0047-2468.
* Milson, R. (2000) "An overview of Lie’s line-sphere correspondence", pp 1–10 of *The Geometric Study of Differential Equations*, J.A. Leslie & T.P. Robart editors, American Mathematical Society ISBN 0-8218-2964-5 .
* Zlobec, Borut Jurčič; Mramor Kosta, Neža (2001), "Configurations of cycles and the Apollonius problem", *Rocky Mountain Journal of Mathematics*, **31** (2): 725–744, doi:10.1216/rmjm/1020171586, ISSN 0035-7596.
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https://en.wikipedia.org/wiki/Babi%C4%99ty_Ma%C5%82e
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Babięty Małe
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Babięty Małe is a village in the administrative district of Gmina Susz, within Iława County, Warmian-Masurian Voivodeship, in northern Poland. It lies approximately 6 kilometres (4 mi) south-east of Susz, 16 km (10 mi) north-west of Iława, and 76 km (47 mi) west of the regional capital Olsztyn.
| 2023-12-08T08:19:48
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# Babięty Małe
**Babięty Małe** \[baˈbjɛntɨ ˈmawɛ\] is a village in the administrative district of Gmina Susz, within Iława County, Warmian-Masurian Voivodeship, in northern Poland. It lies approximately 6 kilometres (4 mi) south-east of Susz, 16 km (10 mi) north-west of Iława, and 76 km (47 mi) west of the regional capital Olsztyn.
## InfoBox
| Babięty Małe | |
| --- | --- |
| Village | |
| Babięty Małe | |
| Coordinates: 53°40′19″N 19°22′22″E / 53.67194°N 19.37278°E / 53.67194; 19.37278 | |
| Country | Poland |
| Voivodeship | Warmian-Masurian |
| County | Iława |
| Gmina | Susz |
| Time zone | UTC+1 (CET) |
| Summer (DST) | UTC+2 (CEST) |
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62,100,397
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https://en.wikipedia.org/wiki/2020_Lion_City_Sailors_FC_season
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2020 Lion City Sailors FC season
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The 2020 season was Lion City Sailors' 25th consecutive season in the Singapore Premier League and the 1st season since privatising from Home United. Along with the Singapore Premier League, the club will also compete in the Singapore Cup.
| 2024-10-01T17:27:31
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# 2020 Lion City Sailors FC season
The 2020 season was Lion City Sailors' 25th consecutive season in the Singapore Premier League and the 1st season since privatising from **Home United**. Along with the Singapore Premier League, the club will also compete in the Singapore Cup.
## Squad
### SPL squad
*As of 5 January 2020*
| Squad No. | Name | Nationality | Date of Birth (Age) | Previous Club | Contract Since | Contract End |
| --- | --- | --- | --- | --- | --- | --- |
| Goalkeepers | | | | | | | |
| 18 | Hassan Sunny <sup>**\>30**</sup> | | 2 April 1984 | Army United F.C. | 2020 | 2021 |
| 24 | Rudy Khairullah | | 19 July 1994 | Police SA (NFL D1) | 2017 | 2020 |
| 30 | Adib Hakim <sup>**U23**</sup> | | 9 March 1998 | Young Lions FC | 2020 | 2020 |
| Defenders | | | | | | | |
| 2 | Zulqarnaen Suzliman <sup>**U23**</sup> | | 29 March 1998 | Young Lions FC | 2020 | |
| 3 | Tajeli Salamat | | 7 February 1994 | Warriors FC | 2020 | |
| 5 | Kaishu Yamazaki | | 12 July 1997 | Albirex Niigata (S) | 2020 | |
| 6 | Abdil Qaiyyim Mutalib <sup>**\>30**</sup> | | 14 May 1989 | Tampines Rovers | 2015 | 2020 |
| 7 | Aqhari Abdullah | | 9 July 1991 | LionsXII | 2016 | 2020 |
| 12 | Iqram Rifqi | | 25 February 1996 | Youth Team | 2017 | 2020 |
| 15 | Faizal Roslan | | 30 May 1995 | Young Lions FC | 2018 | 2019 |
| 19 | Naqiuddin Eunos <sup>**U23**</sup> | | 1 December 1997 | Young Lions FC | 2020 | 2020 |
| 22 | Ho Wai Loon | | 20 August 1993 | Warriors FC | 2019 | 2019 |
| Midfielders | | | | | | | |
| 8 | Shahdan Sulaiman <sup>**\>30**</sup> | | 9 May 1988 | Tampines Rovers | 2020 | 2021 |
| 10 | Song Ui-young | | 8 November 1993 | Youth Team | 2012 | 2020 |
| 13 | Izzdin Shafiq (Captain) | | 14 December 1990 | Tampines Rovers | 2017 | 2020 |
| 14 | Gabriel Quak | | 22 December 1990 | Warriors FC | 2020 | 2021 |
| 20 | Arshad Shamim <sup>**U23**</sup> | | 9 December 1999 | Youth Team | 2018 | 2019 |
| 27 | Adam Swandi | | 12 January 1996 | Albirex Niigata (S) | 2019 | 2021 |
| 28 | Saifullah Akbar <sup>**U23**</sup> | | 31 January 1999 | Young Lions FC | 2020 | |
| Strikers | | | | | | | |
| 11 | Hafiz Nor <sup>**\>30**</sup> | | 22 August 1988 | Warriors FC | 2018 | 2019 |
| 17 | Shahril Ishak <sup>**\>30**</sup> | | 23 January 1984 | Warriors FC | 2018 | 2019 |
| 23 | Amiruldin Asraf <sup>**U23**</sup> | | 8 January 1997 | Youth Team | 2017 | 2020 |
| 25 | Haiqal Pashia <sup>**U23**</sup> | | 29 November 1998 | Young Lions FC | 2020 | 2020 |
| 29 | Stipe Plazibat | | 31 August 1989 | Hougang United | 2020 | 2021 |
| Players who left during the season | | | | | | | |
| 9 | Andy Pengelly | | 19 July 1997 | Brisbane Strikers FC | 2020 | |
|
### U19 squad
*As of 1 February 2019*
| Squad No. | Name | Nationality | Date of Birth (Age) | Previous Club | Contract Since | Contract End |
| --- | --- | --- | --- | --- | --- | --- |
| Goalkeepers | | | | | | | |
| 40 | Prathip Ekamparam <sup>**U19**</sup> | | 21 August 2001 | Youth Team | 2019 | |
| | Kimura Riki <sup>**U23**</sup> | | 14 November 2000 | Warriors FC | 2020 | |
| Defenders | | | | | | | |
| 44 | Danish Iftiqar <sup>**U19**</sup> | | 16 September 2001 | Youth Team | 2019 | 2020 |
| 55 | Fudhil I'yadh<sup>**U19**</sup> | | 18 August 2001 | NFA U17 | 2019 | 2019 |
| 65 | Aizal Murhamdani<sup>**U19**</sup> | | 26 March 2001 | Youth Team | 2017 | |
| | Naufal Ilham <sup>**U19**</sup> | | 16 August 2002 | FFA U16 | 2020 | 2020 |
| Midfielders | | | | | | | |
| 37 | Anaqi Ismit<sup>**U19**</sup> | | 24 August 2001 | Youth Team | 2019 | 2020 |
| Strikers | | | | | | | |
|
## Coaching staff
Source
| Position | Name | Ref. |
| --- | --- | --- |
| Chairman | Forrest Li | |
| General Manager | Badri Ghent | |
| Team Manager | Herwandy Hamid | |
| Head coach | Aurelio Vidmar | |
| Assistant coach | Noh Rahman | |
| Goalkeeping coach | Chua Lye Heng | |
| Academy director | Luka Lalić | |
| Head of Youth (COE) | Robin Chitrakar | |
| Video analyst | Adi Saleh | |
| Sports Trainer | Fazly Hasan | |
| Sports Performance Coach | Shazaly Ayob | |
| Physiotherapist | Zahir Taufeek | |
| Kitman | | |
## Transfer
### Pre-season transfer
#### In
| Position | Player | Transferred From | Ref |
| --- | --- | --- | --- |
| Coach | Aurelio Vidmar | NA | |
| GK | Hassan Sunny | Army United F.C. (Tier 2) | 2 years contract signed in 2019 |
| GK | Kenji Syed Rusydi | Young Lions FC | Loan Return |
| GK | Adib Hakim | Young Lions FC | |
| GK | Kimura Riki | Warriors FC | |
| DF | Kaishu Yamazaki | Albirex Niigata (S) | Free |
| DF | Zulqarnaen Suzliman | Young Lions FC | |
| MF | Gabriel Quak | Warriors FC | 2 years contract signed in 2019 |
| MF | Shahdan Sulaiman | Tampines Rovers | Undisclosed |
| MF | Saifullah Akbar | Young Lions FC | |
| MF | Naqiuddin Eunos | Young Lions FC | |
| FW | Haiqal Pashia | Young Lions FC | |
| FW | Andy Pengelly | Brisbane Strikers FC (Tier 2) | Free |
|
*Note 1: Kenji Syed Rusydi returned to the team after the loan and move to Tanjong Pagar United.*
#### Out
| Position | Player | Transferred To | Ref |
| --- | --- | --- | --- |
| GK | Kenji Syed Rusydi | Tanjong Pagar United | |
| GK | Nazri Sabri | Project Vaults Oxley SC | |
| GK | Haikal Hasnol | | |
| DF | Taufiq Muqminin | | |
| DF | Faritz Abdul Hameed | Tanjong Pagar United F.C. | |
| DF | Juma'at Jantan | Retired | |
| MF | Isaka Cernak | Phrae United F.C. (Tier 2) | |
| MF | Fazli Ayob | | |
| MF | Suhairi Sabri | Tanjong Pagar United F.C. | |
| MF | Muhelmy Suhaimi | | |
| MF | Hami Syahin | Police SA | NS till 2022 |
| FW | Nur Hizami Salim | | |
| FW | Oliver Puflett | Sydney Olympic FC | |
|
#### Extension / Retained
| Position | Player | Ref |
| --- | --- | --- |
| GK | Rudy Khairullah | |
| DF | Abdil Qaiyyim Mutalib | |
| DF | Aqhari Abdullah | |
| DF | Faizal Roslan | |
| DF | Ho Wai Loon | |
| DF | Iqram Rifqi | |
| MF | Izzdin Shafiq | |
| MF | Song Ui-young | 2 years contract signed in Oct 2018 |
| MF | Hami Syahin | NS till 2022 |
| MF | Adam Swandi | 2 years contract signed in Dec 2019 |
| MF | Arshad Shamim | |
| FW | Hafiz Nor | |
| FW | Shahril Ishak | |
| FW | Amiruldin Asraf | |
|
| Position | Player | Ref |
| --- | --- | --- |
|
#### Trial
##### Trial (In)
| Position | Player | Trial From | Ref |
| --- | --- | --- | --- |
|
##### Trial (Out)
| Position | Player | Trial @ | Ref |
| --- | --- | --- | --- |
|
### Mid-season transfer
#### In
| **Position** | **Player** | **Transferred From** | **Ref** |
| --- | --- | --- | --- |
| FW | Stipe Plazibat | Hougang United | Undisclosed <br>2 years contract |
|
#### Out
| **Position** | **Player** | **Transferred To** | **Ref** |
| --- | --- | --- | --- |
| GK | Putra Anugerah | Young Lions FC | Season loan |
| FW | Andy Pengelly | Peninsula Power FC | Free |
| MF | Bill Mamadou | Young Lions FC | Season loan |
#### Loan Out
| **Position** | **Player** | **Transferred To** | **Ref** |
| --- | --- | --- | --- |
| DF | Zulqarnaen Suzliman | Young Lions FC | NS till 2022 |
|
## Friendlies
### Pre-season friendlies
Win Draw Loss
| 15 January 2020 (2020-01-15) 1 | **Johor Darul Ta'zim F.C.** | **4-0** | **Lion City Sailors F.C.** | Johor, Malaysia |
| --- | --- | --- | --- | --- |
| | Gonzalo Cabrera 3' <br>Diogo Luís Santo 12' <br>Syafiq Ahmad 72' <br>Liridon Krasniqi 83' | | | Stadium: Sultan Ibrahim Stadium |
| 15 February 2020 (2020-02-15) 2 | **Singapore Football Club** | **0-4** | **Lion City Sailors F.C.** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | | | Gabriel Quak <br>Shahril Ishak | |
| 18 February 2020 (2020-02-18) 3 | **Yishun Sentek Mariners FC** | **0-10** | **Lion City Sailors F.C.** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | | | Shahril Ishak <br>Song Ui-young <br>Andy Pengelly <br>Shahdan Sulaiman <br>Iqram Rifqi <br>Hafiz Nor <br>Arshad Shamim | |
| 21 February 2020 (2020-02-21) 4 | **Young Lions FC** | **2-5** | **Lion City Sailors F.C.** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Ilhan Fandi <br>Marc Ryan Tan | | Andy Pengelly <br>Iqram Rifqi <br>Haiqal Paisha | |
| 28 February 2020 (2020-02-28) 5 | **Tiong Bahru FC** | **0-8** | **Lion City Sailors F.C.** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | | | Hafiz Nor 7' <br>Song Ui-young 30' 61' <br>Gabriel Quak 31' <br>Haiqal Paisha 33' <br>Andy Pengelly 67' <br>Shahril Ishak 70' 80' | |
| 25 March 2020 (2020-03-25) 6 | **Singapore Football Club** | **Unknown** | **Lion City Sailors F.C.** | Bishan Stadium |
| --- | --- | --- | --- | --- |
#### Tour of Malaysia
| 4 February 2020 (2020-02-04) Friendly | **Selangor FA II** | **6-2** | **Lion City Sailors F.C.** | Selangor |
| --- | --- | --- | --- | --- |
| | Raja Imran | | Song Ui-Young <br>Shahril Ishak | Stadium: Selayang Stadium |
| 8 February 2020 (2020-02-08) Friendly | **Selangor** | **3-1** | **Lion City Sailors F.C.** | Selangor |
| --- | --- | --- | --- | --- |
| | Ifedayo Olusegun | | | Stadium: Selayang Stadium |
## Team statistics
### Appearances and goals
*As of 15 March 2019*
| No. | Pos. | Player | Sleague | | Singapore Cup | | Total | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| | | | Apps. | Goals | Apps. | Goals | Apps. | Goals |
| 3 | DF | Tajeli Salamat | 14 | 2 | 0 | 0 | 14 | 2 |
| 5 | DF | Kaishu Yamazaki | 14 | 2 | 0 | 0 | 14 | 2 |
| 6 | DF | Abdil Qaiyyim Mutalib | 9 | 0 | 0 | 0 | 9 | 0 |
| 7 | DF | Aqhari Abdullah | 10 | 0 | 0 | 0 | 10 | 0 |
| 8 | MF | Shahdan Sulaiman | 13 | 1 | 0 | 0 | 13 | 1 |
| 10 | MF | Song Ui-young | 12 | 9 | 0 | 0 | 12 | 9 |
| 11 | FW | Hafiz Nor | 14 | 1 | 0 | 0 | 14 | 1 |
| 12 | MF | Iqram Rifqi | 1 | 0 | 0 | 0 | 1 | 0 |
| 13 | MF | Izzdin Shafiq | 12 | 0 | 0 | 0 | 12 | 0 |
| 14 | MF | Gabriel Quak | 14 | 5 | 0 | 0 | 14 | 5 |
| 15 | DF | Faizal Roslan | 8 | 0 | 0 | 0 | 8 | 0 |
| 17 | FW | Shahril Ishak | 9 | 3 | 0 | 0 | 9 | 3 |
| 18 | GK | Hassan Sunny | 11 | 0 | 0 | 0 | 11 | 0 |
| 19 | MF | Naqiuddin Eunos | 14 | 0 | 0 | 0 | 14 | 0 |
| 20 | MF | Arshad Shamim | 10 | 2 | 0 | 0 | 10 | 2 |
| 22 | DF | Ho Wai Loon | 1 | 0 | 0 | 0 | 1 | 0 |
| 24 | GK | Rudy Khairullah | 3 | 0 | 0 | 0 | 3 | 0 |
| 25 | FW | Haiqal Pashia | 9 | 0 | 0 | 0 | 9 | 0 |
| 27 | MF | Adam Swandi | 11 | 4 | 0 | 0 | 11 | 4 |
| 28 | MF | Saifullah Akbar | 11 | 2 | 0 | 0 | 11 | 2 |
| 29 | FW | Stipe Plazibat | 8 | 9 | 0 | 0 | 8 | 9 |
| **Players who have played this season but had left the club or on loan to other club** | | | | | | | | | | | | | | |
| 2 | DF | Zulqarnaen Suzliman | 2 | 0 | 0 | 0 | 2 | 0 |
| 9 | FW | Andy Pengelly | 2 | 1 | 0 | 0 | 2 | 1 |
|
## Competitions
### Overview
| Competition | Record | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| | P | W | D | L | GF | GA | GD | Win % |
| Singapore Premier League | 14 | 8 | 3 | 3 | 44 | 18 | +26 | 057.14 |
| Total | 14 | 8 | 3 | 3 | 44 | 18 | +26 | 057.14 |
### Singapore Premier League
Win Draw Loss
| 6 March 2020 (2020-03-06) 1 | **Tanjong Pagar United** | **1-1** | **Lion City Sailors F.C.** | Jurong East Stadium |
| --- | --- | --- | --- | --- |
| | Yann Motta 26' <br>Shodai Nishikawa 20' <br>Syahadat Masnawi 47' <br>Takahiro Tanaka 56' <br>Faritz Abdul Hameed 86' | Report | Andy Pengelly 45' <br>Hafiz Nor 59' <br>Kaishu Yamazaki 77' | Attendance: 2,723<br>Referee: Lim Liang Yi<br>Man of the Match: Kenji Syed Rusydi |
| 18 March 2020 (2020-03-18) 2 | **Tampines Rovers** | **4-0** | **Lion City Sailors F.C.** | Our Tampines Hub |
| --- | --- | --- | --- | --- |
| | Boris Kopitović 60' <br>Zehrudin Mehmedović 69' <br>Jordan Webb 78' <br>Shah Shahiran 82' <br>Daniel Bennett 16' | Report | Kaishu Yamazaki 10' <br>Hafiz Nor 77' | Attendance: 0 <br>Referee: G. Letchman<br>Man of the Match: Jordan Webb |
| 18 October 2020 (2020-10-18) 3 (Rescheduled) | **Lion City Sailors F.C.** | **4-0** | **Geylang International** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Stipe Plazibat 7' 85' <br>Song Ui-young 45+5' (pen) <br>Shahril Ishak 90' <br>Naqiuddin Eunos 66' <br>Tajeli Salamat 78' | Report | Zainol Gulam 45' | Attendance: 0 <br>Referee: Farhan Mohd<br>Man of the Match: Stipe Plazibat |
| 24 October 2020 (2020-10-24) 4 (Rescheduled) | **Albirex Niigata (S)** | **3-2** | **Lion City Sailors F.C.** | Jurong East Stadium |
| --- | --- | --- | --- | --- |
| | Tomoyuki Doi 19' <br>Ryosuke Nagasawa 35' <br>Reo Nishiguchi 94' | Report | Song Ui-young 59' <br>Gabriel Quak 88' <br>Stipe Plazibat 90+3' <br>Saifullah Akbar 90+4' | Attendance: 0 <br>Referee: Muhammad Taqi<br>Man of the Match: Kotaro Takeda |
| 1 November 2020 (2020-11-01) 5 (Rescheduled) | **Lion City Sailors F.C.** | **5-0** | **Young Lions FC** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Adam Swandi 71' <br>Stipe Plazibat 76' 82' (pen) <br>Gabriel Quak 90+1' 27' <br>Saifullah Akbar 37' <br>Haiqal Pashia 41' | Report | Jacob Mahler 46' (o.g.) <br>Shahib Masnawi 17' <br>Nur Adam Abdullah 32' | Attendance: 0 <br>Referee: Jansen Foo<br>Man of the Match: Gabriel Quak |
| 4 November 2020 (2020-11-04) 6 (Rescheduled) | **Lion City Sailors F.C.** | **1-1** | **Hougang United** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Shahdan Sulaiman 60' 30' <br>Saifullah Akbar 67' <br>Tajeli Salamat 82' | Report | Shawal Anuar 33' <br>Farhan Zulkifli 72' <br>Justin Hui 83' <br>Lionel Tan 90+2' | Attendance: 0 <br>Referee: Ahmad A'Qashah<br>Man of the Match: Shawal Anuar |
| 7 November 2020 (2020-11-07) 7 (Rescheduled) | **Balestier Khalsa** | **1-7** | **Lion City Sailors** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Kristijan Krajcek 63' <br>Faizal Raffi 78' | Report | Stipe Plazibat 10' 27' 31' <br>Song Ui-young 18' <br>Saifullah Akbar 33' <br>Tajeli Salamat 56' <br>Adam Swandi 65' | Attendance: 0 <br>Referee: Juherman Zaiton<br>Man of the Match: Stipe Plazibat |
| 13 November 2020 (2020-11-13) 8 (Rescheduled) | **Lion City Sailors F.C.** | **6-1** | **Tanjong Pagar United** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Kaishu Yamazaki 14' <br>Stipe Plazibat 19' (pen) 45+2' <br>Song Ui-young 40' <br>Adam Swandi 84' 90' <br>Tajeli Salamat 34' <br>Abdil Qaiyyim Mutalib 77' | Report | Suhairi Sabri 51' <br>Suria Prakash 85' | Attendance: 0 <br>Referee: Syarqawi Buhari<br>Man of the Match: Shahdan Sulaiman |
| 17 November 2020 (2020-11-17) 9 (Rescheduled) | **Hougang United** | **1-3** | **Lion City Sailors** | Hougang Stadium |
| --- | --- | --- | --- | --- |
| | Charlie Machell 90+2' (pen) <br>Daniel Martens 45' <br>Maksat Dzhakybaliev 77' <br>Farhan Zulkifli 80' | Report | Gabriel Quak 12' <br>Hafiz Nor 57' <br>Shahdan Sulaiman 87' | Attendance: 0 <br>Referee: Lim Liang Yi<br>Man of the Match: Gabriel Quak |
| 22 November 2020 (2020-11-22) 10 (Rescheduled) | **Lion City Sailors F.C.** | **2-3** | **Albirex Niigata (S)** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Song Ui-young 22' (pen) <br>Tajeli Salamat 45' <br>Aqhari Abdullah 58'<br>Saifullah Akbar 78'<br>Shahdan Sulaiman 90'<br>Hafiz Nor 90+1' | Report | Tomoyuki Doi 12' <br>Fairoz Hasan 34' <br>Yasuhiro Hanada 77' <br>Kotaro Takeda 25'<br>Kenta Kurishima 73' | Attendance: 0 <br>Referee: Farhan Mohd<br>Man of the Match: Tomoyuki Doi |
| 25 November 2020 (2020-11-25) 11 (Rescheduled) | **Geylang International** | **0-3** | **Lion City Sailors F.C.** | Our Tampines Hub |
| --- | --- | --- | --- | --- |
| | Harith Kanadi 52' | Report | Song Ui-young 71' <br>Gabriel Quak 86' <br>Kaishu Yamazaki 90+1' <br>Arshad Shamim 49' | Attendance: 0 <br>Referee: Syarqawi Buhari<br>Man of the Match: Kaishu Yamazaki |
| 29 November 2020 (2020-11-29) 12 (Rescheduled) | **Young Lions FC** | **0-4** | **Lion City Sailors F.C.** | Jurong West Stadium |
| --- | --- | --- | --- | --- |
| | | Report | Arshad Shamim 33' 57' <br>Song Ui-young 40' (pen) <br>Gabriel Quak 72' <br>Abdil Qaiyyim Mutalib 4' <br>Kaishu Yamazaki 12' | Attendance: 0 <br>Referee: Lim Liang Yi<br>Man of the Match: Gabriel Quak |
| 2 December 2020 (2020-12-02) 13 (Rescheduled) | **Lion City Sailors F.C.** | **1-1** | **Tampines Rovers** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Shahril Ishak 45' <br>Arshad Shamim 19' <br>Song Ui-young 72' | Report | Zehrudin Mehmedović 63' <br>Amirul Adli 90' | Attendance: 0 <br>Referee: Nathan Chan<br>Man of the Match: Zehrudin Mehmedović |
| 5 December 2020 (2020-12-05) 14 (Rescheduled) | **Lion City Sailors F.C.** | **5-2** | **Balestier Khalsa** | Bishan Stadium |
| --- | --- | --- | --- | --- |
| | Shahril Ishak 20' <br>Saifullah Akbar 34' <br>Danish Uwais 39' (o.g) <br>Song Ui-young 80' 90+2' <br>Rudy Khairullah 66' | Report | Sime Zuzul 41' 88' <br>Ahmad Syahir 47' | Attendance: 0 <br>Referee: Jansen Foo<br>Man of the Match: Hafiz Nor |
| Pos | Team | Pld | W | D | L | GF | GA | GD | Pts | Qualification or relegation |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 1 | Albirex Niigata (S) (C) | 14 | 10 | 2 | 2 | 32 | 14 | +18 | 32 | |
| 2 | Tampines Rovers | 14 | 8 | 5 | 1 | 27 | 11 | +16 | 29 | Qualification for AFC Champions League group stage |
| 3 | Lion City Sailors | 14 | 8 | 3 | 3 | 44 | 18 | +26 | 27 | Qualification for AFC Cup group stage |
| 4 | Geylang International | 14 | 6 | 2 | 6 | 18 | 22 | 4 | | 20 |
| 5 | Balestier Khalsa | 14 | 5 | 4 | 5 | 22 | 28 | 6 | 19 | |
| 6 | Hougang United | 14 | 4 | 3 | 7 | 19 | 24 | 5 | | 15 |
| 7 | Young Lions | 14 | 3 | 0 | 11 | 12 | 38 | 26 | | 9 |
| 8 | Tanjong Pagar United | 14 | 0 | 5 | 9 | 14 | 33 | 19 | | 5 |
Source: Singapore Premier League
Rules for classification: 1) points; 2) goal difference; 3) number of goals scored; 4) number of wins
(C) Champions
Notes:
### Singapore Cup
Win Draw Loss
1. Due to Covid 19, this match is played behind closed door.
2. The match is postponed following the coronavirus pandemic.
1. "FIRST TEAM STAFF".
2. "Vidmar become dragon's tamer".
3. "Hassan Sunny, 35, back Home with Protectors after leaving Thai side Army United".
4. "Besides Hassan, they have recruited Japanese defender Kaishu Yamazaki".
5. "YAMAZAKI Kaishu will be transferred to Home United".
6. "Gabriel Quak Instagram post comments section". Archived from the original on 2021-12-24.
7. "Besides Hassan, they have recruited forward Gabriel Quak".
8. "Shahdan Sulaiman cross over to Protectors".
9. "Saifullah Akbar Instagram profile change to "Home United FC🐉"".
10. "Vidmar become dragon's tamer".
11. "welcome former Home United FC goal stopper, Nazri Sabri back to PVOSC".
12. "Isaka Cernak signs for newly promoted Thai League 2 side Phrae United".
13. "Izzdin Shafiq retained".
14. "Song Ui-young rejects Persija for Home stay, eyes Singapore colours". Archived from the original on 2018-10-31. Retrieved 2019-10-19.
15. "Sailors offered him a two-year extension".
16. "Hassan Sunny, playing with Shahril Ishak for 2020".
17. "Stipe Plazibat moving back to Bishan for 2 years contract".
18. "The club would like to announce that we have parted ways with Andy Pengelly, after coming to a mutual agreement to terminate his contract".
19. "2019 #NPLQLD Men's Golden Boot and record breaker Andy Pengelly is back!".
20. "Bill Mamadou joins Young Lions".
21. "LSC not be able to call upon the services of Zulqarnaen Suzliman due to his National Service commitments".
## InfoBox
Lion City Sailors
| 2020 season | |
| --- | --- |
| Chairman | Forrest Li |
| Head coach | Aurelio Vidmar |
| Stadium | Bishan Stadium |
| Singapore Premier League | 3rd |
| | |
| | |
| | |
| | |
|
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