Stadiums based on curvilinear geometry: Approximation of the ellipsoid offset surface

Anna Borowska

doi:10.1515/eng-2022-0392

Article Open Access

Stadiums based on curvilinear geometry: Approximation of the ellipsoid offset surface

Anna Borowska

Published/Copyright: December 31, 2022

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From the journal Open Engineering Volume 12 Issue 1

Abstract

This article provides an effective method of determining the coordinates of points and the angles for various surfaces approximating the ellipsoid surface. The stands of the stadium (with a capacity of up to 82,000 seats) with rows shaped by ellipse arcs were designed (in accordance with the European Standard EN 13200-1). An algorithm was proposed to determine the optimal height of the ring steps (with the guarantee of unobstructed vision of the entire playing field from each seat of the stands). The second algorithm ensures a comfortable space in each individual seating place in the stands (deviation of the depth of the row generated by ellipse arcs did not exceed 0.000009 m at any place of the stands). This article encourages readers to program DXF files (read in the AutoCAD system) to streamline design work.

Keywords: offset surface; ellipsoid; stadium stands; DXF file

1 Introduction

Architectural objects are based on sophisticated curvilinear geometry, and although they are not the cheapest, they increasingly appear on city maps (Sydney Opera House, Sky Towers (Abu Dhabi), Beijing National Grand Theatre, Jumeirah Beach Hotel (Dubai)). Such investments will attract tourists over the following years. So, good ideas for reducing the costs of their construction are very important.

To design a structure based on an ellipsoid S ( u , v ) , the coordinates of points of other surfaces approximating the surface S ( u , v ) are necessary (Figures 1 and 2). The approximating surface can be an offset surface or another ellipsoid (with the desired properties) concentric with the base surface. Sections 2.1–2.4 provide mathematical formulas and the effective methodology for acquiring angles and coordinates of points of different surfaces approximating the ellipsoid surface.

Figure 1

(a) Three-tier stadium with covered stands and (b) the ellipsoid-shaped elevation of the stadium with a footbridge.

Figure 2

The stands in the corner (a close-up).

The aim of this study is to program (according to the European Standard EN 13200-1) stadium stands with rows shaped by ellipse arcs. Modern stadium stands are designed in such a way as to place tens of thousands of people in one space and provide them with an uninterrupted view of the entire playing field and a comfortable space in an individual seating place. The problem of providing tens of thousands of people with an unobstructed view of the entire playing field was solved using the line of sight correction (LSC) algorithm (compared with Section 3.3) and the ball vision verification (BVV) algorithm (compared with Section 3.4). The problem of providing them with a comfortable space in an individual seating place was resolved using the row depth correction (RDC) algorithm (compared with Section 3.6). To design and verify the presented algorithms, tests were carried out for different variants of stands.

All figures were programed in C++ (in Visual Studio 2019), saved in DXF format (compared with ref. [1]), and exported (in the AutoCAD system) to DWG file (a standard file format for CAD).

2 The approximation of the ellipsoid offset surface

2.1 Mathematical formulas

Lemma 1

(compared with [2,3], p. 44) If arccos ( x ) = arcsin ( y ) and x + y > 0 , then x 2 + y 2 = 1 .

Let S ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) ( u ∈ [ u 1 , u 2 ] , v ∈ [ v 1 , v 2 ] ) be a parametric representation of a smooth surface in three-dimensional space. The unit normal vector to the surface S at the point R 0 is of the following form (compared with ref. [4], p. 28)

n → unv ( u 0 , v 0 ) = r → u ( u 0 , v 0 ) × r → v ( u 0 , v 0 ) ∣ r → u ( u 0 , v 0 ) × r → v ( u 0 , v 0 ) ∣ , where r → u ( u 0 , v 0 ) = ∂ x ∂ u , ∂ y ∂ u , ∂ z ∂ u u = u 0 , v = v 0 , and r → v ( u 0 , v 0 ) = ∂ x ∂ v , ∂ y ∂ v , ∂ z ∂ v u = u 0 , v = v 0

are the tangent vectors to the surface S at the point R 0 .

For a given smooth surface S , we can define an offset off d S at distance d (compared with ref. [5], p. 341) as follows. On each surface normal, we mark the two points that are at a constant distance d from the surface S . The set of all of these points forms the offset surface off d S . The offset surface off d S ( u , v ) at distance d to S ( u , v ) is obtained as follows:

off d S ( u , v ) = S ( u , v ) ± d ⋅ n → unv ( u , v ) .

Practical applications of the offset surfaces (offset curves) are described in refs [4,6,7].

Let S , S 1 , and S 2 be the concentric ellipsoid surfaces defined as follows (compared with ref. [8], p. 238).

(1) S ( u , v ) : x = a cos ( u ) sin ( v ) , y = b sin ( u ) sin ( v ) , z = c cos ( v ) , u ∈ [ 0 , 2 π ] , v ∈ [ 0 , π ] ,

(2) S 1 ( u , v ) : x = ( a + d ) cos ( u ) sin ( v ) , y = ( b + d ) sin ( u ) sin ( v ) , z = ( c + d ) cos ( v ) ,

(3) S 2 ( u , v ) : x = ( a − d ) cos ( u ) sin ( v ) , y = ( b − d ) sin ( u ) sin ( v ) , z = ( c − d ) cos ( v ) .

Let us assume that P is any point on the ellipsoid S (Figure 3) and l is the normal line to S at the point P . Points P 1 and P 2 lie on the normal line l at distance d from P . Q 1 , Q 2 are the intersection points of the normal l with ellipsoids S 1 and S 2 , respectively. Ellipsoids S 1 and S 2 do not keep a constant distance d relative to the ellipsoid S (distances d P 1 Q 1 = ∣ P 1 Q 1 ¯ ∣ > 0 and d P 2 Q 2 = ∣ P 2 Q 2 ¯ ∣ > 0 ) at points not lying on the axes of the coordinate system (Figure 3).

$Figure 3 The arrangement of points and surfaces S S , S 1 {S}_{1} , S 2 {S}_{2} , off d 1 S {{\rm{off}}}_{d1}S , off d 2 S {{\rm{off}}}_{d2}S .$

Figure 3

The arrangement of points and surfaces S , S 1 , S 2 , off d 1 S , off d 2 S .

2.2 The coordinates of points P 1 and P 2

Let us take the parametric equations (1) of the ellipsoid S ( u , v ) . The coordinates of points P 1 and P 2 lying on the normal l to the ellipsoid S ( u , v ) (at the point P ( x P , y P , z P ) ) and distant from P by the length d can be determined using the following equation of the offset surface off d S ( u , v ) .

(4) off d S ( u , v ) : [ X , Y , Z ] = [ x , y , z ] ± d [ − b c cos ( u ) sin ( v ) , − a c sin ( u ) sin ( v ) , − a b cos ( v ) ] ( b c cos ( u ) sin ( v ) ) 2 + ( a c sin ( u ) sin ( v ) ) 2 + ( a b cos ( v ) ) 2 .

The coordinates of points P 1 and P 2 are as follows:

(5) x P 1 = x P + d b c x P a w , y P 1 = y P + d a c y P b w , z P 1 = z P + d a b z P c w ,

(6) x P 2 = x P − d b c x P a w , y P 2 = y P − d a c y P b w , z P 2 = z P − d a b z P c w ,

where

(7) w = ( b c cos ( u 0 ) sin ( v 0 ) ) 2 + ( a c sin ( u 0 ) sin ( v 0 ) ) 2 + ( a b cos ( v 0 ) ) 2

(8) x P = a cos ( u 0 ) sin ( v 0 ) , y P = b sin ( u 0 ) sin ( v 0 ) , z P = c cos ( v 0 ) .

In practical applications, the (smaller) offset surface off d 2 S ( u , v ) requires correction (Figure 4(a)), because a part of this surface (for u , v ∈ [ 0 , π ∕ 2 ] ) is below the O X Y plane. To get the coordinates x P 2 ≥ 0 , y P 2 ≥ 0 , z P 2 ≥ 0 (for u , v ∈ [ 0 , π ∕ 2 ] ), we have to take into account the following conditions x P 2 ≥ 0 iff w ≥ d b c a , y P 2 ≥ 0 iff w ≥ d a c b , z P 2 ≥ 0 iff w ≥ d a b c (Figure 4(b and c)). The green line segments (perpendicular to the red surface S ( u , v ) ), one end of which is (in Figure 4(a)) below the O X Y plane, were trimmed (in Figure 4(b and c)) at points with the coordinates x P 2 , y P 2 , z P 2 (for w = d a b c ). The fragment of the (blue) surface (mesh) off d 2 S ( u , v ) was reduced to the part above the O X Y plane (Figure 4(b and c)) using the approximation algorithm (using a “divide and conquer” strategy).

$Figure 4 A fragment of the ellipsoid S ( u , v ) S\left(u,v) and the offset surface off d 2 S ( u , v ) {{\rm{off}}}_{d2}S\left(u,v) ( a = 60 a=60 , b = 40 b=40 , c = 20 c=20 , d = 12 d=12 ): (a) a part of the surface off d 2 S ( u , v ) {{\rm{off}}}_{d2}S\left(u,v) is below the O X Y OXY plane and (b and c) a fragment of the surface off d 2 S ( u , v ) {{\rm{off}}}_{d2}S\left(u,v) is above the O X Y OXY plane (after removal of the part of the surface off d 2 S ( u , v ) {{\rm{off}}}_{d2}S\left(u,v) that was below the O X Y OXY plane).$

Figure 4

A fragment of the ellipsoid S ( u , v ) and the offset surface off d 2 S ( u , v ) ( a = 60 , b = 40 , c = 20 , d = 12 ): (a) a part of the surface off d 2 S ( u , v ) is below the O X Y plane and (b and c) a fragment of the surface off d 2 S ( u , v ) is above the O X Y plane (after removal of the part of the surface off d 2 S ( u , v ) that was below the O X Y plane).

2.3 The coordinates of points Q 1 and T 1

Let S ( u , v ) and S 1 ( φ , ξ ) ( u , φ ∈ [ 0 , 2 π ] , v , ξ ∈ [ 0 , π ] ) be the concentric ellipsoids defined in (1) and (2). Let us assume that u 0 , v 0 , φ , ξ ∈ [ 0 , π ∕ 2 ] and set the parameter h , which gives the coordinates of the point Q 1 (an intersection of the normal line l to the surface S ( u , v ) at the point P with the surface S 1 ( φ , ξ ) ):

cos ( u 0 ) sin ( v 0 ) a + h d b c w = ( a + d ) cos ( φ ) sin ( ξ ) sin ( u 0 ) sin ( v 0 ) b + h d a c w = ( b + d ) sin ( φ ) sin ( ξ ) cos ( v 0 ) c + h d a b w = ( c + d ) cos ( ξ ) .

Hence and from Lemma 1, we have A h 2 + B h + C = 0 , where

A = d 2 ( A 1 b 2 c 2 + B 1 a 2 c 2 + C 1 a 2 b 2 ) w 2 , B = 2 a b c d ( A 1 + B 1 + C 1 ) w , C = A 1 a 2 + B 1 b 2 + C 1 c 2 − 1 for A 1 = cos ( u 0 ) sin ( v 0 ) ( a + d ) 2 , B 1 = sin ( u 0 ) sin ( v 0 ) ( b + d ) 2 , C 1 = cos ( v 0 ) ( c + d ) 2 and w ( compared with ( 7 ) ) .

Therefore (for Δ = B 2 − 4 A C > 0 ), we have h 1 = − B + Δ 2 A (the parameter for the point Q 1 ) and h 2 = − B − Δ 2 A . Finally,

x Q 1 = cos ( u 0 ) sin ( v 0 ) a + h 1 d b c w , y Q 1 = sin ( u 0 ) sin ( v 0 ) b + h 1 d a c w , z Q 1 = cos ( v 0 ) c + h 1 d a b w .

Let l be the normal line to the surface S ( u , v ) at the point P ( x P , y P , z P ) (compared with (8)). The line l passes through the point P 1 ( x P 1 , y P 1 , z p 1 ) (compared with (5)). The line t is the orthographic projection of the line l onto the plane (containing the point P and parallel to the O X Y plane). Let us assume that u 0 , v 0 , φ , ξ ∈ [ 0 , π ∕ 2 ] and set the parameter h , which gives the coordinates of the point T 1 ( x T 1 , y T 1 , z T 1 ) (an intersection of the line t with the surface S 1 ( φ , ξ ) (compared with (2))):

x P + h ( x P 1 − x P ) = ( a + d ) cos ( φ ) sin ( ξ ) y P + h ( y P 1 − y P ) = ( b + d ) sin ( φ ) sin ( ξ ) z P = ( c + d ) cos ( ξ ) .

Hence and from Lemma 1, we obtain A h 2 + B h + C = 0 , where A = A 1 ( x P 1 − x P ) 2 + B 1 ( y P 1 − y P ) 2 , B = 2 ( A 1 x P ( x P 1 − x P ) + B 1 y P ( y P 1 − y P ) ) , and C = A 1 x P 2 + B 1 y P 2 − 1 for A 1 = 1 ( a + d ) 2 sin 2 ( ξ ) , B 1 = 1 ( b + d ) 2 sin 2 ( ξ ) , and ξ = arccos z P c + d . Therefore (for Δ = B 2 − 4 A C > 0 ), we have h 1 = − B + Δ 2 A (the parameter for the point T 1 ) and h 2 = − B − Δ 2 A . Finally, for w (compared with (7)):

x T 1 = cos ( u 0 ) sin ( v 0 ) a + h 1 d b c w , y T 1 = sin ( u 0 ) sin ( v 0 ) b + h 1 d a c w , z T 1 = z P = c cos ( v 0 ) .

Figure 5(a) shows a fragment of the (red) ellipsoid S ( u , v ) , the (blue) ellipsoid S 1 ( u , v ) and the (green) offset surface off d 1 S ( u , v ) , yellow line segments P P 1 ¯ and (pink) line segments P Q 1 ¯ . Figure 5(b) presents a fragment of the (orange) ellipsoid S ( u , v ) , the (green) ellipsoid S 1 ( u , v ) , (blue) horizontal line segments P T 1 ¯ , and (pink) line segments P Q 1 ¯ . The orthographic projections of the segments P Q 1 ¯ and P T 1 ¯ onto the O X Y plane lie on one line.

$Figure 5 Fragments of the surfaces S ( u , v ) S\left(u,v) , S 1 ( u , v ) {S}_{1}\left(u,v) , and off d 1 S ( u , v ) {{\rm{off}}}_{d1}S\left(u,v) (a) for a = 80 a=80 , b = 60 b=60 , c = 15 c=15 , d = 15 d=15 and (b) for a = 60 a=60 , b = 40 b=40 , c = 20 c=20 , d = 5 d=5 .$

Figure 5

Fragments of the surfaces S ( u , v ) , S 1 ( u , v ) , and off d 1 S ( u , v ) (a) for a = 80 , b = 60 , c = 15 , d = 15 and (b) for a = 60 , b = 40 , c = 20 , d = 5 .

2.4 The coordinates of points Q 2 and T 2

Let S ( u , v ) and S 2 ( φ , ξ ) ( u , φ ∈ [ 0 , 2 π ] , v , ξ ∈ [ 0 , π ] , a , b , c > d ) be the concentric ellipsoids defined in (1) and (3). Let us assume that u 0 , v 0 , φ , ξ ∈ [ 0 , π ∕ 2 ] and set the parameter h giving the coordinates of the point Q 2 (an intersection of the normal line l to the surface S ( u , v ) at the point P with the surface S 2 ( φ , ξ ) ):

cos ( u 0 ) sin ( v 0 ) a − h d b c w = ( a − d ) cos ( φ ) sin ( ξ ) sin ( u 0 ) sin ( v 0 ) b − h d a c w = ( b − d ) sin ( φ ) sin ( ξ ) cos ( v 0 ) c − h d a b w = ( c − d ) cos ( ξ ) .

Hence and from Lemma 1, we obtain the equation A h 2 + B h + C = 0 , where

A = d 2 ( A 1 b 2 c 2 + B 1 a 2 c 2 + C 1 a 2 b 2 ) w 2 , B = − 2 d a b c ( A 1 + B 1 + C 1 ) w , C = A 1 a 2 + B 1 b 2 + C 1 c 2 − 1 for A 1 = cos ( u 0 ) sin ( v 0 ) ( a − d ) 2 , B 1 = sin ( u 0 ) sin ( v 0 ) ( b − d ) 2 , C 1 = cos ( v 0 ) ( c − d ) 2 and w ( compared with ( 7 ) ) .

Therefore (for Δ = B 2 − 4 A C > 0 ), we have h 1 = − B + Δ 2 A , and h 2 = − B − Δ 2 A (the parameter for the point Q 2 ). Finally:

(9) x Q 2 = cos ( u 0 ) sin ( v 0 ) a − h 2 d b c w , y Q 2 = sin ( u 0 ) sin ( v 0 ) b − h 2 d a c w , z Q 2 = cos ( v 0 ) c − h 2 d a b w .

Remark 1

By using the equalities (9) to determine the coordinates x Q 2 , y Q 2 , z Q 2 (for u , v ∈ [ 0 , π ∕ 2 ] ), we obtain x Q 2 ≥ 0 iff h 2 ≤ a w d b c and y Q 2 ≥ 0 iff h 2 ≤ b w d a c and z Q 2 ≥ 0 iff h 2 ≤ c w d a b .

Let l be the normal line to the surface S ( u , v ) at the point P ( x P , y P , z P ) (compared with (8)). The line l passes through the point P 2 ( x P 2 , y P 2 , z p 2 ) (compared with (6)). The line t is the orthographic projection of the line l onto the plane (containing the point P and parallel to the O X Y plane). Let us assume that u 0 , v 0 , φ , ξ ∈ [ 0 , π ∕ 2 ] . The coordinates of the point T 2 ( x T 2 , y T 2 , z T 2 ) (an intersection of the line t with the surface S 2 ( φ , ξ ) (compared with (3))) are as follows:

x T 2 = cos ( u 0 ) sin ( v 0 ) a − h 2 d b c w , y T 2 = sin ( u 0 ) sin ( v 0 ) b − h 2 d a c w , z T 2 = z P = c cos ( v 0 ) ,

where h 2 = − B − Δ 2 A (for Δ = B 2 − 4 A C > 0 ),

A = A 1 ( x P 2 − x P ) 2 + B 1 ( y P 2 − y P ) 2 , B = 2 ( A 1 x P ( x P 2 − x P ) + B 1 y P ( y P 2 − y P ) ) , C = A 1 x P 2 + B 1 y P 2 − 1 , A 1 = 1 ( a − d ) 2 sin 2 ( ξ ) , B 1 = 1 ( b − d ) 2 sin 2 ( ξ ) , ξ = arccos z P c − d , w ( compared with ( 7 ) ) .

Figure 6(a) shows a fragment of the (red) ellipsoid S ( u , v ) , the (green) ellipsoid S 2 ( u , v ) , the (blue) offset surface off d 2 S ( u , v ) , (black) line segments P P 2 ¯ , and (pink) line segments P Q 2 ¯ . Figure 6(b) presents a fragment of the (green) ellipsoid S ( u , v ) , the (orange) ellipsoid S 2 ( u , v ) , (blue) horizontal line segments P T 2 ¯ , and (pink) line segments P Q 2 ¯ .

$Figure 6 Fragments of the surfaces S ( u , v ) S\left(u,v) , S 2 ( u , v ) {S}_{2}\left(u,v) , and off d 2 S ( u , v ) {{\rm{off}}}_{d2}S\left(u,v) : (a) for a = 80 a=80 , b = 60 b=60 , c = 20 c=20 , d = 8 d=8 , (b) for a = 60 a=60 , b = 40 b=40 , c = 20 c=20 , d = 5 d=5 .$

Figure 6

Fragments of the surfaces S ( u , v ) , S 2 ( u , v ) , and off d 2 S ( u , v ) : (a) for a = 80 , b = 60 , c = 20 , d = 8 , (b) for a = 60 , b = 40 , c = 20 , d = 5 .

3 Stadiums based on curvilinear geometry

The form of the stands of a contemporary stadium is shaped in such a way as to place tens of thousands of people in one space and provide them with: an uninterrupted view of the entire playing field, a comfortable space in an individual seating place, and safe conditions for communication and evacuation (compared with ref. [9], p. 97). The paper proposes the construction of the stadium stands (in accordance with the guidelines and standards [10,11]) based on curvilinear geometry (i.e., with rows along the ellipse arcs) and with a constant height of the ring steps (different for each ring).

3.1 Symbols and dimensions of the stadium

The following dimensions of the stadium comply with the standards [10,11].

2 d x = 105 m ( 2 d y = 68 m ) – the length (and width) of the playing field; a x = 10 m ( a y = 8.5 m ) – the nominal width of the auxiliary area on the ends (on the sides); 2 d a x × 2 d a y = 130 m × 90 m – the arena dimensions ( d a x = d x + a x + 2.5 m , d a y = d y + a y + 2.5 m “The normative dimensions of the arena of a modern football stadium cannot be smaller than 130 m × 90 m ” (compared with ref. [9], p. 11)), S = 0.8 m – the depth of a row; G = 0.25 m – the distance of the eye point from the back edge of a row; Δ O k = S − G = 0.55 m , H O s = 1.2 m – the (normative) eye-height for a sitting position (compared with ref. [9], p. 97); C x = 0.12 m – the distance from the eye point to the top of the head (compared with ref. [9], p. 133); Δ y ∈ { 0.5 , 1.0 , … , 20 } ( Δ x ∈ { 0.5 , 1.0 , … , 20 } ) – increment of the semi-minor (semi-major) axis of an ellipse e B i ( e A i ) for the side stands (for the stands behind the goal).

The stadium stands consist of three rings (I, II, and III) and are divided into parts as follows: the side stands (BA and BB), the stands behind the goal (AA and AB), and the stand in the corner (C) (Figure 7(a and b)). Three types of arcs were used for the design: the arcs of concentric ellipses e B i (for stands BA and BB), the arcs of concentric ellipses e A i (for stands AA and AB), and the arcs of concentric circles e C i (for the stand C). The center of the ellipses is in the center of the pitch, and the center of the circles is in the corners of the auxiliary arena.

$Figure 7 The stadium stands (for Δ x = 3 m \Delta x=3\hspace{0.33em}{\rm{m}} , Δ y = 7 m \Delta y=7\hspace{0.33em}{\rm{m}} , n , m , k = 20 n,m,k=20 ): (a) depth of the rows (top view), (b) depth of the rows (SW Isometric view), (c) color palette, (d) the arrangement of seats in the stands, and (e) a view of a sector in the stand behind the goal.$

Figure 7

The stadium stands (for Δ x = 3 m , Δ y = 7 m , n , m , k = 20 ): (a) depth of the rows (top view), (b) depth of the rows (SW Isometric view), (c) color palette, (d) the arrangement of seats in the stands, and (e) a view of a sector in the stand behind the goal.

In the side stands: the arc of the ellipse e B 0 shapes the front edge of the first row, the arcs of ellipses e B i (for i ∈ [ 1 , R ] , where n + m + k = R – the number of rows in the stands) define the back edges of the consecutive rows, and the arcs of ellipses e Δ B i (for i ∈ [ 1 , R ] ) model the eye lines in the consecutive rows.

e B 0 : x B 0 = a B 0 cos ( u ) , y B 0 = b B 0 sin ( u ) , where b B 0 = d a y + Δ y and a B 0 = d a x cos ( t 1 ) for t 1 = arcsin d a y b B 0 , u ∈ [ t 1 , 9 0 ∘ ] (the ellipse e B 0 reaches the point ( d a x , d a y ) for the angle t 1 ),

e B i : x B i = a B i cos ( u ) , y B i = b B i sin ( u ) , where a B i = a B 0 + i S , b B i = b B 0 + i S , u ∈ [ t 1 , 9 0 ∘ ] , e Δ B i : x Δ B i = a Δ B i cos ( u ) , y Δ B i = b Δ B i sin ( u ) , where b Δ B i = d a y + Δ y + Δ O k + ( i − 1 ) S , a Δ B i = d a x cos ( t 1 ) + Δ O k + ( i − 1 ) S for u ∈ [ t 1 , 9 0 ∘ ] .

The arcs of ellipses e A i ( i ∈ [ 0 , R ] ), e Δ A i ( i ∈ [ 1 , R ] ) (for u ∈ [ 0 ∘ , v 1 ] , v 1 = arccos d a x a A 0 , and a A 0 = d a x + Δ x , (the ellipse e A 0 reaches the point ( d a x , d a y ) for the angle v 1 )) shaping the stands behind the goal were defined in a similar way.

The arcs of circles e C i ( i ∈ [ 1 , R ] ) model the back edges of the rows of the stand in the corner:

e C i : x C i = a C i cos ( u ) + d a x , y C i = a C i sin ( u ) + d a y , where a C i = i S , u ∈ [ v 1 , t 1 ] .

The arcs of circles e Δ C i ( i ∈ [ 1 , R ] ) model the eye lines in the rows of the stand in the corner:

e Δ C i : x Δ C i = a Δ C i cos ( u ) + d a x , y Δ C i = a Δ C i sin ( u ) + d a y , where a Δ C i = Δ O k + ( i − 1 ) S , u ∈ [ v 1 , t 1 ] .

3.2 Depth of a row

Remark 2

While determining the length of the ellipse arc, we obtain an elliptic integral that cannot be expressed by elementary functions (compared with ref. [12], p. 315–318). When designing the stadium stands, the length of the ellipse arc, measured along a broken line (composed of equal segments) inscribed in the ellipse arc, is more useful.

Symbols: S r ( ϕ ) – the depth of a row r (for the angle ϕ ) measured along the perpendicular segment to the back edge of the row; Δ S r ( ϕ ) = S − S r ( ϕ ) – row depth deficit (for the angle ϕ ).

The values of S r ( ϕ ) were determined for the side stands and the stands behind the goal (for three rings). It was assumed that each ring consists of 25 rows. To plot the back edges of the rows, the arcs of ellipses e B i (for the side stands) and e A i (for the stands behind the goal) (for i ∈ [ 1 , R ] (compared with Section 3.1) and Δ x , Δ y ∈ { 0.5 , 1 , 1.5 , 2 , … , 20 } ) were used. The values of S r ( ϕ ) were calculated as the lengths of the line segments ∣ P i Q 2 ¯ ∣ for i ∈ [ 1 , R ] (the point P i lies on the ellipse e B i ( e A i ) for the angle ϕ ). Points P i were measured every Δ ϕ i (corresponding to a length of 0.25 m) along a broken line inscribed in the arc of the ellipse e B i ( e A i ) defining the back edge of the row. That is, on the ellipse arc, segments with a length 0.25 m were measured (using an approximation method), and the angles Δ ϕ i were calculated. Coordinates of the point Q 2 (for the determined P i ) were calculated from the formula (9) for d = 0.8 m , c = 1 , and v 0 = 9 0 ∘ . The results are presented in Tables 1 and 2 and in Figures 7 and 8.

Table 1

Test results for the side stands (for Δ y ∈ { 0.5 , 1 , 1.5 , 2 , 3 , 5 , 7 , 10 , 15 , 20 } , n , m , k = 25 )

Δ y (m)	0.5	1	1.5	2	3	5	7	10	15	20
b B 1 (m)	46.3	46.8	47.3	47.8	48.8	50.8	52.8	55.8	60.8	65.8
a B 1 (m)	440.45849	314.23717	258.79431	226.01759	187.58951	149.92023	130.51249	113.85143	99.07076	90.87730
t 1 [ ∘ ]	8 1 ∘ 29 ′ 5 3 ″	7 8 ∘ 1 ′ 5 2 ″	7 5 ∘ 24 ′ 2 7 ″	7 3 ∘ 13 ′ 3 1 ″	6 9 ∘ 38 ′ 9 ″	6 4 ∘ 9 ′ 2 9 ″	5 9 ∘ 55 ′ 3 6 ″	5 4 ∘ 54 ′ 1 1 ″	4 8 ∘ 35 ′ 2 5 ″	4 3 ∘ 48 ′ 4 7 ″
S 1 ( t 1 ) (m)	0.792945	0.787394	0.782800	0.778960	0.773080	0.766664	0.765147	0.768314	0.779317	0.789723
Δ S 1 ( t 1 ) (m)	0.007055	0.012606	0.017200	0.021040	0.026920	0.033336	0.034853	0.031686	0.020683	0.010277
dist Δ S 1 > 0.01 (m)	+	7	15.25	20	25.25	29.75	31.25	30.75	25.5	3.25
dist Δ S 1 > 0.02 (m)	+	+	+	1.5	8.75	14.75	16.5	15.25	1.75	+
dist Δ S 1 > 0.03 (m)	+	+	+	+	+	3.25	5.0	2.0	+	+
PTI	6,048	5,994	6,076	6,081	6,169	6,290	6,393	6,540	6,799	6,924
PTII	6,207	6,403	6,555	6,674	6,870	7,170	7,286	7,611	8,098	8,398
PTIII	6,516	6,816	7,062	7,254	7,417	7,927	8,328	8,683	9,382	9,861

Table 2

Test results for the stands behind the goal (for Δ x ∈ { 0.5 , 1 , 1.5 , 2 , 3 , 5 , 7 , 10 , 15 , 20 } , n , m , k = 25 )

Δ x (m)	0.5	1	1.5	2	3	5	7	10	15	20
a A 1 (m)	66.3	66.8	67.3	67.8	68.8	70.8	72.8	75.8	80.8	85.8
b A 1 (m)	365.69134	260.29010	213.87164	186.36042	153.99161	122.04356	105.42506	91.00067	77.99210	70.63463
v 1 [ ∘ ]	7 ∘ 5 ′ 2 ″	9 ∘ 59 ′ 1 1 ″	1 2 ∘ 11 ′ 3 3 ″	1 4 ∘ 2 ′ 5 ″	1 7 ∘ 4 ′ 5 7 ″	2 1 ∘ 47 ′ 1 2 ″	2 5 ∘ 28 ′ 2 8 ″	2 9 ∘ 55 ′ 3 5 ″	3 5 ∘ 39 ′ 3 3 ″	4 0 ∘ 7 ′ 9 ″
S 1 ( v 1 ) (m)	0.795898	0.793302	0.791561	0.790433	0.789478	0.790640	0.793516	0.797715	0.799885	0.796214
Δ S 1 ( v 1 ) (m)	0.004102	0.006698	0.008439	0.009567	0.010522	0.009360	0.006484	0.002285	0.000115	0.003786
dist Δ S 1 > 0.01 (m)	+	+	+	+	1.25	+	+	+	+	+
PTI	4,185	4,248	4,288	4,327	4,397	4,482	4,471	4,647	4,930	5,248
PTII	4,428	4,593	4,627	4,660	4,845	5,126	5,361	5,677	6,037	6,499
PTIII	4,680	4,810	5,002	5,163	5,437	5,853	6,103	6,581	7,128	7,758

$Figure 8 The depth of the first row in the stands: (a) BAI, BBI (for Δ y ∈ { 0.5 , 1 , 1.5 , 2 , … , 20 } \Delta y\in \left\{0.5,1,1.5,2,\ldots ,20\right\} ), (b) AAI, ABI (for Δ x ∈ { 0.5 , 1 , 1.5 , 2 , … , 20 } \Delta x\in \left\{0.5,1,1.5,2,\ldots ,20\right\} ), and (c) color palette.$

Figure 8

The depth of the first row in the stands: (a) BAI, BBI (for Δ y ∈ { 0.5 , 1 , 1.5 , 2 , … , 20 } ), (b) AAI, ABI (for Δ x ∈ { 0.5 , 1 , 1.5 , 2 , … , 20 } ), and (c) color palette.

Figure 7 shows the stadium stands plotted for unfavorable values Δ x = 3 m and Δ y = 7 m (for Δ x = 3 m , we obtain S 1 ( v 1 ) = 0.789478 m and Δ S 1 > 0.01 m for 1.25 m (for 2.5 seats), and for Δ y = 7 m , we have S 1 ( t 1 ) = 0.765147 m and Δ S 1 > 0.03 m for 5 m (for 10 seats) (compared with Tables 1 and 2)). The floor in the consecutive rows was painted along the line segments perpendicular to the back edge of the row in a color determined by the length of the line segment (according to the color palette) (see Figure 7(c)). Figure 7(a and b) shows that the greatest deficit Δ S 1 ( ϕ ) = 0.034853 m occurred in the stand BBI in the first row (for ϕ = 5 9 ∘ 55 ′ 3 6 ″ ). The values of Δ S r ( ϕ ) can be reduced by moving the center of the ellipses e B i ( e A i ). Figure 7(d and e) illustrates the arrangement of seats in the stands.

Table 1 presents test results for the side stands (for the values Δ y ∈ { 0.5 , 1 , 1.5 , 2 , 3 , 5 , 7 , 10 , 15 , 20 } , n , m , k = 25 ). Since the greatest deficit Δ S r ( ϕ ) for the considered stadium stands always occurred for ϕ = t 1 in the first row, the values of b B 1 ( a B 1 , S 1 ( t 1 ) , Δ S 1 ( t 1 ) , dist Δ S 1 > 0.01 (it is the number of meters in the first row, where Δ S 1 ( ϕ ) > 0.01 m ), dist Δ S 1 > 0.02 , dist Δ S 1 > 0.03 ) were given for the first row. The values of PTI (it is the number of seats in the stands 2 ⋅ (BAI+BBI)), PTII, and PTIII are indicative and were determined as follows. (a) First, the stands BA and BB were divided into sectors (Figure 7(b and d)). That is, for each ring (I, II, and III), the angles μ i (defining the centers of the stairs) were determined as follows. In the last row of each ring (starting at ϕ = 9 0 ∘ , along a broken line inscribed in the ellipse arc generating the back edge of the row), 14 times the 0.5 m long segment (0.5 m for 14 people in a row (compared with ref. [11], p. 14)) was measured. Next, the operations listed below were repeated until the angle t 1 was reached: once the 1.2 m long segment (for stairs (compared with ref. [9], p. 203)) and 28 times the 0.5 m long segment (0.5 m for 28 people in a row (compared with ref. [11], p. 14)) were measured and (in the middle of the stairs) the consecutive angle μ i was determined and remembered. In lower rows, the established angles μ i were used and care was taken to ensure that the stairs were 1.2 m wide (at the expense of seats in sectors) (Figure 7(d and e)). (b) Next, in each row of each sector, the number of segments with a length of 0.5 m was counted (one seat = 0.5 m). Entrances, the television, press and VIP areas were omitted. The number of seats PT (for the entire stadium for n , m , k = 25 ) was determined for Δ y = 0.5 and Δ x = 10 .

PT = 2 ⋅ [ 2 ( BA + BB ) ] + 2 ⋅ [ 2 ( AA + AB ) ] + 4 ⋅ [ C ] = = 2 ⋅ 18,771 + 2 ⋅ 16,905 + 4 ⋅ 3,395 = 84,932

Table 2 presents test results for the stands behind the goal. Figure 8 illustrates the values of Δ S 1 ( ϕ ) for the first rows of stands (a) BAI, BBI, (b) AAI, ABI (for Δ x , Δ y ∈ { 0.5 , 1 , 1.5 , 2 , … , 20 } ). The values of Δ x and Δ y increase every 0.5 m . The row depth S 1 ( ϕ ) was measured for the points P i determined every Δ ϕ i = const = 0.0 5 ∘ ( P i lie on the arc of the ellipse e B i ( e A i ) defining the back edge of the first row). The reader can compare the floor colors in the first row in Figure 7(a and b), the belt colors for Δ y = 7 m (in Figure 8(a)), and the belt colors for Δ x = 3 m in Figure 8(b).

3.3 Lines of sight of the focus point

The focus is a point on the pitch of which unobstructed vision provides a full view of the entire playing field. In the case of a football arena, this is usually the closest point on the sideline. The observation of this point is possible only when the line of sight connecting this point with the point of the observer’s eye (let us denote by P F i ) runs tangent to the top of the viewer’s head (from the row below) or above it (compared with ref. [9], p. 97, Figure 9(b)). We propose a simple algorithm (determining the optimal height of the ring steps), which ensures an unobstructed view of each point of the pitch from any point of the stands shaped by ellipse arcs.

$Figure 9 The profile of the stand BBI (for n = 4 n=4 ) in the plane (containing two points P n {P}_{n} , FO, and perpendicular to the OXY plane): (a) for the approximate value of H n {H}_{n} and (b) for H n {H}_{n} determined by the LSC algorithm.$

Figure 9

The profile of the stand BBI (for n = 4 ) in the plane (containing two points P n , FO, and perpendicular to the OXY plane): (a) for the approximate value of H n and (b) for H n determined by the LSC algorithm.

The line of sight correction (LSC) algorithm determines the optimal height H _n of the steps of the ring, i.e., such a minimum value (identical for the entire ring) that it is possible to observe the entire playing field from any eye point of the ring. The LSC algorithm is based on the method of the farthest eye point (a graphical method of modeling a rectilinear profile of the spectator stand (compared with ref. [9], p. 134)) and is adapted to the stands shaped by ellipse arcs. The LSC algorithm optimizes the initial lines of sight of the focus point F (ILSF is the line of sight lying on the plane (containing two points P F i and F and perpendicular to the O X Y plane) passing through two points: the observer’s eye point P F i and the top of the viewer’s head P f i from the previous row). In Figure 9(a and b), the ILSF lines are colored in red, P n – the observer’s eye point from the row n . Two types of focus points were analyzed: (a) FO (the closest point on the nearest sideline), and (b) FL, FR (the two closer corners of the playing field).

The LSC algorithm (the variant for the stand BBI)

Focus – the point FO ( d x , d y , 0 ) at the corner of the playing field (Figure 10).

$Figure 10 ILSF lines for the n n th row in the stand BBI, Focus – the point F O ( d x , d y , 0 ) FO\left({d}x,dy,0) .$

Figure 10

ILSF lines for the n th row in the stand BBI, Focus – the point F O ( d x , d y , 0 ) .

Initialization:

( 1) double Cx = 0.12, HnB = Cx+0.00001;

( 2) double HOkn = HOk1+(n-1)*HnB, h = HOkn;

( 3) double b = day+dlt_y, t1 = asin(day/b);

( 4) double a = dax/cos(t1)+dlt_Ok, b += dlt_Ok;

( 5) double an = a+(n-1)*S, bn = b+(n-1)*S;

( 6) double ai = an-S, bi = bn-S, u = acos(dx/an);

( 7) ellipse* ellB = new ellipse(an,bn);

Iterations:

( 8) while (u >= t1) {

( 9) xn = an*cos(u); yn = bn*sin(u); zn = h;

(10) A1 = xn-dx; B1 = yn-dy;

(11) A = pow(bi*A1,2)+pow(ai*B1,2);

(12) B = pow(bi,2)*2*dx*A1+pow(ai,2)*2*dy*B1;

(13) C = pow(bi*dx,2)+pow(ai*dy,2)-pow(ai*bi,2);

(14) D = B*B-4*A*C;

(15) if(D >= 0) {

(16) h1 = (-B+sqrt(D))/(2*A);

(17) fi = acos((dx+h1*(xn-dx))/ai);

(18) xfi = ai*cos(fi); yfi = bi*sin(fi);

zfi = h-HnB+Cx;

(19) Sn = sqrt(pow(xn-xfi,2)+pow(yn-yfi,2));

(20) LOkn = sqrt(pow(xn-dx,2)+pow(yn-dy,2));

(21) distn = ((HOk1+(n-1)*HnB)*Sn)/(HnB-Cx);

(22) if (LOkn-distn < 0) {

(23) distn = LOkn;

(24) HnB = (distn*Cx+HOk1*Sn)/(distn-Sn*(n-1)); } }

(25) xy = ellB->Node_ellipse_prev((u*180)/PI,0.25);

(26) u = acos(xy->x/an); delete xy; } delete ellB;

(27) Hn = HnB;

Description of the LSC algorithm: In the LSC algorithm, all distances are measured on a plane containing two points P n and F O and perpendicular to the O X Y plane. The points P n ( x n , y n , z n ) , P f i ( x f i , y f i , z f i ) (respectively in the n th and ( n − 1 ) th row) lie on one initial line of sight of the focus point F O (in Figure 10, the segments P n P f i ¯ are red). The point P n lies on the eye line in the n th row (the ellipse arc e Δ B n ) at the height H O k n , and the point P f i lies above the eye line in the ( n − 1 ) th row (the ellipse arc e Δ B n − 1 ) at the height H O k n − 1 + C x . P n X Y is the orthographic projection of the point P n onto the O X Y plane.

Symbols in the algorithm (Figure 9): LOkn ( L O k n = ∣ P n X Y F O ¯ ∣ ) – the horizontal distance of the eye point P n from the focus point F O ; distn ( dist n = ∣ P n X Y P 0 ¯ ∣ ) – the horizontal distance of the eye point P n from the point P 0 ( x 0 , y 0 , z 0 ) (an intersection of the ILSF line with the plane of the pitch); Sn ( S n = ∣ P n X Y P f i X Y ¯ ∣ ) – the horizontal distance of the eye point P n from the point P f i ; HnB – the height of the ring steps (the auxiliary variable); Hn ( H n ) – the optimal height of the ring steps (the return value); HOk1 ( H O k 1 ) – the elevation of the eye point of the first row; HOkn ( H O k n ) – the elevation of the eye point of the n th row; an ( a Δ B n ) – the semi-major axis of the ellipse e Δ B n ; ai ( a Δ B n − 1 ) – the semi-major axis of the ellipse e Δ B n − 1 .

Initialization: As the initial value of HnB, any value from the range ( C x , H n ] can be taken.

Iterations: The LSC algorithm makes calculations for the consecutive eye points P n (on the ellipse arc e Δ B n for u ∈ [ arccos d x a Δ B n , t 1 ] – the variant BBI) measured every 0.25 m , along a broken line inscribed in the ellipse arc (in Figure 10, points P n were measured every 2 m ). For each eye point P n , the LSC determines the eye point P f i (in the ( n − 1 ) th row (lines 10–18)) and distances S n , L O k n , dist n as follows.

The ellipse e Δ B n : x n = a Δ B n cos ( u ) , y n = b Δ B n sin ( u ) .

Let us write down the equations of the line l X Y , which is the orthographic projection of the line l (passing through the points P n ( x n , y n , z n ) and F O ( d x , d y , 0 ) ) onto the O X Y plane and determine the coordinates x f i , y f i of the point P f i X Y (an intersection of the line l X Y with the ellipse e Δ B n − 1 ).

x f i = d x + h 1 ( x n − d x ) = a Δ B n − 1 cos ( φ ) , y f i = d y + h 1 ( y n − d y ) = b Δ B n − 1 sin ( φ ) .

We determine h 1 (as in the program (lines 10–16)) and obtain the coordinates of the point P f i X Y . Next, we calculate the lengths of S n = ∣ P n X Y P f i X Y ¯ ∣ (line 19) and L O k n = ∣ P n X Y F O ¯ ∣ (line 20). The value of dist n (line 21) results from the following equation:

(10) H O k n H O k n − H n + C x = dist n dist n − S n ,

where H O k n = H O k 1 + ( n − 1 ) H n (Figure 9(a)) for H n = HnB (current value). If ( L O k n − dist n < 0 ) (i.e., the ILSF line is over the focus point F O (line 22)), the HnB value should be increased. The LSC algorithm optimizes the value of dist n to L O k n ( = ∣ P n X Y F O ¯ ∣ – the optimal value) and for the new dist n determines HnB from the equation (10) (Figure 9(b)). Then

HnB = H n = dist n ⋅ C x + H O k 1 ⋅ S n dist n − ( n − 1 ) S n .

Remark 3

(a) The critical point P n k is the eye point (in the n th row of the stand BBI), for which the optimal value of H n was determined.

(b) The optimal value of H n (for the critical point P n k ) guarantees that the focus F O will be visible to all eye points from the stand BBI, provided that the following condition is met:

(11) S n − 1 S n ≤ ( H O k n − 1 + C x ) H O k n − 1 ,

where S n and S n − 1 (the depths of the n th and ( n − 1 ) th rows, respectively) are determined for the critical point P n k .

It was checked that for the stands BBI (and ABI) (for Δ x , Δ y ∈ { 0.5 , 1 , 1.5 , 2 , … , 20 } , n = 20 , H O k 1 = 2 m , C x = 0.12 m ), the condition S i − 1 S i ≤ ( H O k i − 1 + C x ) H O k i − 1 is fulfilled for all rows i = 3 , … , n . For the aforementioned stands, the height H n determined for the critical point P n k (from the n th row) is sufficient to make the point F O visible from any eye point in these stands.

In order to determine the optimal value of H n , we have to execute the LSC algorithm (after appropriate modifications in determining the coordinates of the point P f i X Y ) from each point P n (from the n th row) of the stands BAI, BBI, AAI, ABI, and CI, and select the largest value of H n .

In Figure 11(a), the ILSF lines for the focus F O (in the stands BAI, BBI, AAI, ABI, and CI, for all rows) are plotted. We can see that the height H n (determined by the LSC algorithm) guarantees seeing the closest point on the nearest sideline from each eye point in the aforementioned stands. Figure 11(b) presents the ILSF lines for the focus point F O (in the stands BA, BB, AA, AB, and C, for all rows ( n , m , k = 20 )). For each ring, the optimal height of steps ( H n , H m , and H k ) was determined. In each row, the ILSF lines are painted in a different color (according to the color palette).

Figure 11

Lines ILSF for the focus point F O : (a) in the stands BAI, BBI, AAI, ABI, and CI (for n rows), (b) in the stands BA, BB, AA, AB, and C (for R = n + m + k rows), and (c) color palette.

Tests showed that in the case of the proposed stands (shaped by arcs of ellipses), seeing the focus F O (the closest point on the nearest sideline) from any eye point in the stands does not guarantee seeing the focuses F L and F R (the closer corners of the playing field). Figure 12 is plotted for all rows in all stands (for H O k 1 = 2 m , Δ x , Δ y = 0.5 ). In Figure 12(a), the value of H n ( H m , H k ) was calculated using the LSC algorithm for the focus F O ; hence, the ILSF lines intersect the sideline ( H n is not sufficient). In Figure 12(b), the value of H n ( H m , H k ) was calculated using the LSC algorithm for the focuses F L and F R (the ILSF lines do not cross the sideline). The ILSF lines where the eye point P F i is in the stand C and the point P f i is in the stand BB (or AB) are plotted in black.

$Figure 12 ILSF lines (plotted for the focuses F L FL , F R FR ) for all eye points in all stands for the values of H n {H}_{n} , H m {H}_{m} , H k {H}_{k} determined by the LSC algorithm for focuses (a) F O FO , (b) F L FL , F R FR , (c) color palette.$

Figure 12

ILSF lines (plotted for the focuses F L , F R ) for all eye points in all stands for the values of H n , H m , H k determined by the LSC algorithm for focuses (a) F O , (b) F L , F R , (c) color palette.

To find the optimal value of H n , we have to execute the LSC algorithm (for the focuses F L and F R ) from each point P n (from the n th row) of the stands BAI, BBI, AAI, ABI, and CI, and select the largest value of H n . In addition, we need to check (a) the ILSF line (passing through the eye point P n (in the stand CI) and the point P f i (in the stand BBI)) for the focus F L and (b) the ILSF line (passing through the eye point P n (in the stand CI) and the point P f i (in the stand ABI)) for the focus F R .

Table 3 presents the optimal heights of the ring steps for the stadiums with the following parameters ( Δ x = 10 , Δ y ∈ { 0.5 , 1 , 2 , … , 10 } , H O k 1 = 1.2 m , n , m , k = 20 ). The values of H n ( F O ) were determined for the focus F O , the values of H n ( F R F L ) were obtained for the focuses F L and F R and H O k n is the elevation of the eye point of the n th row (determined for the optimal H n ). Analogous calculations were made for rings II and III.

Table 3

Heights for rings I, II, and III (for Δ x = 10 , Δ y ∈ { 0.5 , 1 , 2 , … , 10 } , H O k 1 = 1.2 , n , m , k = 20 )

Δ y	0.5	1	2	3	4	5	6	7	8	9	10
H n ( F O )	0.356464	0.352438	0.345274	0.338902	0.333097	0.327733	0.32273	0.318034	0.313665	0.309664	0.306011
H n ( F L F R )	0.357095	0.353325	0.34622	0.339649	0.333574	0.327964	0.322792	0.318034	0.313665	0.309664	0.306011
H O k n	7.98481	7.91317	7.77818	7.65334	7.5379	7.43131	7.33305	7.24264	7.15963	7.08362	7.01421
H m ( F O )	0.428275	0.424208	0.416733	0.409817	0.403307	0.397131	0.391257	0.385756	0.38065	0.375925	0.37157
H m ( F L F R )	0.428828	0.424915	0.417365	0.410215	0.40348	0.397162	0.391257	0.385756	0.38065	0.375925	0.37157
H O k m	16.5614	16.4115	16.1255	15.8576	15.6075	15.3745	15.1582	14.9578	14.7726	14.6021	14.4456
H k ( F O )	0.473431	0.469526	0.46215	0.455112	0.448332	0.4418	0.435615	0.429812	0.424388	0.41934	0.414659
H k ( F L F R )	0.473916	0.470084	0.462548	0.455287	0.448362	0.4418	0.435615	0.429812	0.424388	0.41934	0.414659
H O k k	26.0397	25.8132	25.3765	24.9634	24.5747	24.2105	23.8705	23.554	23.2604	22.9889	22.7388

3.4 The borderline position of the eye

European Standard EN 13200-1 specifies the maximum range of sight ( RS = 190 m ) for football stadiums. For this distance, the minimum angle of view of the ball (with a diameter of 0.22 m ) is ψ = 0 ∘ 4 ′ (compared with ref. [13], p. 84, 85). The height of the eye point in the last row can reach up to 45 m (above the arena level). Hence, the line of sight of the farthest point of the playing field can be sloped at a large angle. Therefore, the actual length of the segment (from the point of the farthest eye (raised to the height of H O k k ) to the farthest corner of the playing field) is important (compared with ref. [9], p. 89).

The ball vision verification algorithm (in short BVV). The variant for the stand C.

(a) Plot a fragment of the sphere e ψ (compared with (12)) with a radius of RS = 190 m and the center at the farthest corner of the playing field.

(12) e ψ : x ψ = a ψ cos ( u ) sin ( v ) − d x , y ψ = a ψ sin ( u ) sin ( v ) − d y , z ψ = a ψ cos ( v ) ,

(13) P F i 1 : x F i 1 = a Δ C i cos ( φ 1 ) + d a x , y F i 1 = a Δ C i sin ( φ 1 ) + d a y , z F i 1 = H O k i ,

(14) P F i 2 : x F i 2 = a Δ C i cos ( φ 2 ) + d a x , y F i 2 = a Δ C i sin ( φ 2 ) + d a y , z F i 2 = H O k i .

(b) Determine on the eye line in the i th row (the ellipse arc e Δ C i (compared with Section 3.1)) two eye points P F i 1 and P F i 2 (compared with (13) and (14)) adjacent to each other (at a distance of 0.25 m ). H O k i – the elevation of the eye point in the i th row. (c) Determine the point P F i 1 ψ ( x F i 1 , y F i 1 , z P F i 1 ψ ) – an intersection of the line h φ 1 (passing through the points P F i 1 and P F i 1 X Y ) with the sphere e ψ . The point P F i 1 X Y is the orthographic projection of the point P F i 1 onto the O X Y plane. In the same way, determine the point P F i 2 ψ ( x F i 2 , y F i 2 , z P F i 2 ψ ) (an intersection of the line h φ 2 with the sphere e ψ ). (d) If the condition

(15) ( z P F i 1 ψ ≥ H O k i and z P F i 2 ψ ≥ H O k i )

is met, then the segment P F i 1 P F i 2 ¯ (in the i th row) is inside the sphere e ψ (it is within the range of view of RS). Then the color of the segment follows the color of the eye line in the stand. Otherwise, the segment P F i 1 P F i 2 ¯ is outside the range of view RS (the segment is red).

The aforementioned algorithm should be repeated for consecutive pairs of points P F i 1 and P F i 2 adjacent to each other in the i th row (for i = n + m + k , … , q , where q is the first row (from the top) in which for all pairs of points P F i 1 and P F i 2 the condition (15) is met). Figure 13(a) shows the eye lines in the stands ( Δ x = 10 , Δ y = 0.5 , H O k 1 = 1.2 , n , m , k = 20 ). The places where the range of sight RS = 175 m was exceeded are marked in red.

$Figure 13 (a) Borderline positions of the eye lines (for RS = 175 m {\rm{RS}}=175\hspace{0.33em}{\rm{m}} ) and (b) stadium stands with 81,928 seats.$

Figure 13

(a) Borderline positions of the eye lines (for RS = 175 m ) and (b) stadium stands with 81,928 seats.

The largest football stadiums (except Salt Lake Stadium (Kolkata)) have a capacity of up to 100,000 seats. It has been checked that verified stadium (stadium template) can accommodate up to 82,000 seats (orange and blue seats in Figure 13(b)). It was assumed that each spectator (each point of the eye) must see the entire playing field from a distance of up to 190 m. The stadium stands (in Figure 13(b)) were plotted for the following parameters: Δ y = 10 m , Δ x = 0.5 m , n = 25 , m = 27 , k = 31 , H n = 0.365873 m , H m = 0.445959 m , H k = 0.501235 m , H O k 1 = 1.2 m , H O k n = 9.98094 m , H O k m = 22.0218 m , H O k k = 37.5601 m , and the number of seats: 81,928.

3.5 Plotting a profile of the stadium

Figure 14 shows a vertical plane containing the points T R ( ϕ ) and T R − 1 ( T R ( ϕ ) – the point on the ellipse arc e B R for the angle ϕ = 8 5 ∘ (at the height equal to H O k k − 1.2 m ); T R − 1 – the point on the ellipse arc e B R − 1 (at the height equal to H O k k − H k − 1.2 m ), the segment T R X Y T R − 1 X Y ¯ is perpendicular to the ellipse e B R (i.e., perpendicular to the back edge of the last row), R = n + m + k ). The row depths S i and the horizontal distances between P F i (the eye point in i th row) and P F i − 1 (the eye point in ( i − 1 ) th row) were determined using the formulas (compared with (9), Section 2.4). The heights of the steps of the rings are constant (different for each ring): H n = 0.357095 m , H m = 0.428828 m , H k = 0.473916 m , and H O k k = 26.039685 m . Figure 14 shows the ILSF lines for the focus F (the point on the nearest sideline).

$Figure 14 The profile of the stand BA ( Δ x = 10 m \Delta x=10\hspace{0.33em}{\rm{m}} , Δ y = 0.5 m \Delta y=0.5\hspace{0.33em}{\rm{m}} , n , m , k = 20 n,m,k=20 , H O k 1 = 1.2 m HO{k}_{1}=1.2\hspace{0.33em}{\rm{m}} , C x = 0.12 m Cx=0.12\hspace{0.33em}{\rm{m}} , and ϕ = 8 5 ∘ \phi =8{5}^{\circ } ).$

Figure 14

The profile of the stand BA ( Δ x = 10 m , Δ y = 0.5 m , n , m , k = 20 , H O k 1 = 1.2 m , C x = 0.12 m , and ϕ = 8 5 ∘ ).

3.6 Row depth correction

In Section 3.2, the depth S r ( ϕ ) and the deficit Δ S r ( ϕ ) were measured (for the row r and the angle ϕ ) along a line perpendicular to the back edge of the row. Figure 7 shows the stadium stands for unfavorable values Δ x = 3 m and Δ y = 7 m (where Δ S 1 ( ϕ ) = 0.034853 m for ϕ = 5 9 ∘ 55 ′ 3 6 ″ ). To correct the depth S r ( ϕ ) , we have to move the center of ellipses. Figure 15 shows the top view of the stands BA and BB. Virtual arcs of ellipses e B i ( i = 0 , … , 60 ) (representing the front and back edges of 60 rows of the stands BA, BB) were connected by colored segments (perpendicular to the back edges) measured every 0.05 m . The segments were colored depending on the length S r ( ϕ ) (according to the color palette (Figure 15(c))).

Figure 15

The stands BA, BB: (a) top view, (b) top view (corrected), and (c) color palette.

The row depth correction (RDC) algorithm corrects the values of S r ( ϕ ) in all rows of stands BA, BB with the desired accuracy Δ E . Let us note (Figure 15) that it is enough to correct the value of S 1 ( t 1 ) , i.e., the length of the segment P F i P f i ¯ (where P F i is a point on the ellipse arc e B 1 (for the angle ϕ = t 1 ), P f i is a point on the ellipse arc e B 0 , and the segment P F i P f i ¯ is perpendicular to the arc e B 1 ).

The RDC algorithm

( 1) double dlt_E = 0.00001, dlt_b = 0;

( 2) while (true) {

( 3) b = day+dlt_y+dlt_b; t1 = asin((day+dlt_b)/b);

( 4) a = dax/cos(t1);

( 5) ai = a+S; bi = b+S; ell1 = new ellipse(ai,bi);

( 6) xFi = ai*cos(t1); yFi = bi*sin(t1)-dlt_b;

( 7) Pfi = ell1->Node_ellipse_xQ2yQ2((t1*180)/PI,S);

( 8) S1 = sqrt(pow(xFi-Pfi->x,2)+pow((yFi-(Pfi->y-dlt_b)),2));

( 9) delete Pfi; delete ell1;

(10) if((S-S1) <= dlt_E) break; dlt_b += 1; }

The RDC algorithm determines the value of Δ b B (an increment of the semi-minor axis of the ellipse e B i ) by which the center of the ellipses e B i should be shifted. At initialization, we set the values Δ E = 0.00001 m and Δ b B = 0 . In consecutive iterations, the algorithm determines: the length of the semi-minor axis b B 0 (for the current value of the increment Δ b B ), the coordinates of points P F i and P f i (lines 6 and 7), the value of S 1 ( t 1 ) (line 8) and checks the condition ( S − S 1 ( t 1 ) ≤ Δ E ). If the condition is met, the algorithm exits the loop. Otherwise, it increases the value of Δ b B by 1 m and repeats the calculations for the semi-minor axis b B 0 longer by 1 m . The coordinates of the point P f i are determined from the formula (9) (compared with Section 2.4).

For stands BA and BB (for n , m , k = 20 , Δ x = 3 m , Δ y = 7 m ), with a given accuracy of Δ E = 0.00001 m , the RDC algorithm determined the value of Δ b B = 240 m . For Δ b B = 0 (before the correction), the angle t 1 = 5 9 ∘ 55 ′ 3 6 ″ and the maximum value of Δ S 1 ( t 1 ) = 0.034853 m (Figure 15(a)). After the correction (for Δ b B = 240 m ) the angle t 1 = 7 7 ∘ 25 ′ 4 5 ″ and the maximum value of Δ S 1 ( t 1 ) = 0.000009 m (Figure 15(b)). Figure 15(b) shows how the lengths of S 1 ( ϕ ) (in the first row) were changed for ϕ ∈ [ t 1 , 9 0 ∘ ] . The first belt (from the bottom) corresponds to the value Δ b B = 0 and each next one was determined for the semi-minor axis b B 0 increased by 10 m .

4 Applications of the method

Nowadays, designers are looking for more and more sophisticated forms for representative architectural objects (Sydney Opera House, Cayan Tower (Dubai), 30 St Mary Axe (London), La Seine Musicale (Paris)). The designing of such objects requires appropriate design technologies, including ones for creating virtual prototypes, simulations, and numerical and visual data analysis. Also, when designing less impressive buildings, it is important to be able to obtain mathematical models for individual elements, fully dimension them, change the parameter values at each stage, and immediately obtain their visualization and a numerical report.

The presented geometric method can be used to model fragments of architectural objects with an ellipsoidal shape (elevations, roofs, and domes). Light lattice structures (compared with ref. [14]) or reinforced double-layer lattice structures (compared with ref. [7,15]) are especially appreciated. The method provides mathematical formulas for the coordinates of points and angles for the base surface and various surfaces approximating it. The proposed technique (programing DXF files) provides visualization (in the AutoCAD system) and full dimensioning (in TXT file). The combination of these methods can help designers effectively. We can find a lot of the Computer-aided design systems (CATIA, Digital Project, Pro/ENGINEER) that include parametric design tools and offer designers the possibility to create parametric scripts. But it is very expensive software. Only large design companies can afford to buy it. The proposed technology is much cheaper and allows the programer to design complex curvilinear forms in a short time (e.g., an ellipsoidal stadium with stands shaped by ellipse arcs).

5 Summary

The problem of placing up to 82,000 of people in the stands (with rows along the arcs of ellipses) of the stadium has been resolved. The problem is very precisely defined due to the established norms (compared with ref. [10,11]). First, the analysis of the depth of the rows was carried out. Next, a comfortable space was provided for each seat (in all stands) with an accuracy of 0.000009 m (compared with the RDC algorithm). An unobstructed view of the entire playing field from any place in the stands (at a distance not exceeding 190 m) was ensured, using the LSC algorithm and the BVV algorithm. The mentioned algorithms are effective because the formula (9) was used. In addition, the elevation of the stadium (in the shape of an ellipsoid) made of a light, two-layer lattice (Figure 1) was proposed. The design is a template (can be used for different parameter values). The geometric method and programing of DXF files with visualization in the AutoCAD system were used.

Conflict of interest: The author states no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article (and its supplementary information files).

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Received: 2022-10-03

Revised: 2022-11-23

Accepted: 2022-11-30

Published Online: 2022-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/eng-2022-0392

Keywords for this article

offset surface; ellipsoid; stadium stands; DXF file

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