We introduce a method of calculating and rendering shapes in a non-Euclidean 2D space in real-time using hyperbolic and spherical trigonometry. We record the objects’ parameters in a polar coordinate system and use azimuthal equidistant projection to render the space onto the screen. We discuss the complexity of this method, renderings produced, limitations and possible applications of the created software as well as potential future developments

C Frei

C Sutherland

HS Carslaw

J Gray

W Gellert

arXiv

Child, C. H. T.

Osudin, D.

He, Y-H.

City Research Online

              City, University of London Institutional RepositoryCitation: Child, C. H. T. ORCID: 0000-0001-5425-2308, Osudin, D. and He, Y-H. ORCID: 0000-0002-0787-8380 (2019). Rendering Non-Euclidean Space in Real-Time Using Spherical and Hyperbolic Trigonometry. Computational Science – ICCS 2019, 19th International Conference, Proceedings, Part V, 11540, pp. 543-550. doi: 10.1007/978-3-030-22750-0_49 This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link:  http://openaccess.city.ac.uk/id/eprint/22608/Link to published version: http://dx.doi.org/10.1007/978-3-030-22750-0_49Copyright and reuse: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to.City Research Online:            http://openaccess.city.ac.uk/            publications@city.ac.ukCity Research OnlineRendering Non-Euclidean Space in Real-TimeUsing Spherical and Hyperbolic TrigonometryDaniil Osudin, Dr Chris Child, and Prof Yang-Hui HeCity, University of LondonAbstract. We introduce a method of calculating and rendering shapesin a non-Euclidean 2D space in real-time using hyperbolic and spherical trigonometry. We record the objects’ parameters in a polar coordinate system and use azimuthal equidistant projection to render the space onto the screen. We discuss the complexity of this method, renderings produced, limitations and possible applications of the created softwareas well as potential future developments.Keywords: non-Euclidean geometry · spherical trigonometry · hyperbolic trigonom-etry · azimuthal equidistant projection · Polar coordinate system · real-time(a) (b) (c)Fig. 1. Time-lapse images of multiple objects moving through spherical (a), planar (b)and hyperbolic (c) 2D space calculated and rendered by the described software1 Introduction(b) Planar (a) Spherical (c) Hyperbolic Fig. 2. Comparison of parallel lines in the 2D spacesNon-Euclidean geometry is afield that studies any spacethat arises from changing Eu-clid’s fifth postulate [1] orchanging the metric require-ment. In spherical geometry,Fig. 2 (a), all geodesics (short-est paths in a non-planar space) intersect: don’t preserve the distance and appearto ‘bend’ towards each other. In Hyperbolic geometry, Fig. 2 (c), each line hasan infinite number of parallel lines, as they appear to ‘bend’ away from eachother.1      r Θ A O Y x Fig. 3. Point A with polar coor-dinates r and θWe present a method for calculating theobject’s position and its vertices in polarcoordinates using spherical [2] or hyperbolictrigonometry [3] [4]. A polar coordinate systemof the form (r, θ) is used in this model for all cal-culations instead of Cartesian coordinates. Thecentre of the of the screen is taken as a refer-ence point O(0, 0) for the distance coordinate,r, while eastbound is the reference direction forthe bearing coordinate, θ. This allows the same coordinates to be used irrespec-tive of the currect curvature. In order to render the curved space onto a flat 2Dscreen, we are using azimuthal equidistant projection. By definition, distancesand bearing from the centre of the projection are preserved. This works wellwith Polar coordinates, projection is intuitive and can be used with no changesfor both spherical and hyperbolic 2D spaces.2 Method ϴ1 V3 V4 r1 C V2 V1 C’ O O’ Fig. 4. O (0, 0), referencepoint; C (rc, θc), positionand local reference point; Vx(rx, θx), vertices; OO’, refer-ence direction; CC’, local ref-erence directionThe calculations are split into two parts: move-ment of the objects and rendering of the shapes.The screen (rendering space) is limited to a cir-cle of an arbitrary size. When the object’s centremoves past the circumference of the circle, it isrepositioned to the antipodal point on the circlewith the velocity preserved. This is implementedin order to keep the objects in the visible area onthe screen.Shape has a list of position vectors for eachvertex in local coordinates with object’s positionbeing the reference point and reference direction istaken as the reverse of its position vector (Fig. 4).2.1 Rendering the shapeLet K ∈ [−1, 1] ⊂ < s.t. K = 0⇒ Euclidean Geometry; w u v b a c C Fig. 5. Spherical triangleK > 0⇒ Spherical Geometry, r = 1√KTheorem 1. For a sphere of radius r andhence Gaussian curvature K = 1r2 and aspherical triangle on its surface described bypoints u, v and w, connected by great circlesthat form the edges a, b, c (interpreted as sub-tended angles) and an angle C (Fig. 5), thespherical law of cosines [5] states:coscr= cosarcosbr+ sinarsinbrcosC (1) w u v b a c C Fig. 6. Hyperbolic triangleK < 0⇒ Hyperbolic Geometry, k = − 1√KTheorem 2. For a hyperbolic plane withGaussian Curvature K = − 1k2 and a hyper-bolic triangle on its surface described by pointsu, v and w, connected by geodesics that formthe edges a, b and c, as well as an angleC (Fig. 6), the hyperbolic law of cosines [6]states:coshck= coshakcoshbk− sinh aksinhbkcosC (2)Note: To simplify the equations below, all lengths are assumed to have beendivided by r or k depending on the value of K.           Polar coordinate system with a reference point O (0, 0) C(rc, ϴc), object position, reference point of local coordinate system V(rv, ϴv), object’s vertex with local coordinates (rlocal, ϴlocal) OV = rv, OC = rc, CV = rlocal ∠VOO’ =  ϴv, ∠VOC = Δϴ,  ∠COO’ =  ϴc, ∠VCC’ =  ϴlocal ∠OCC’ = α, angle of rotation ∠OCV = β, angle between rlocal  and rc  ϴc O ϴc rv rc rlocal V β ϴlocal C α Δϴ O’ C’  α Δϴ O C β rc ϴv V rv ϴc ϴlocal C’ O’ rlocal (a) (b) Fig. 7. Finding the θ and r coordinates of an object’s vertices through a hyper-bolic/spherical triangle OCV ; Case (a): θlocal + α < pi; case (b): θlocal + α > piCorollary 1. Given: O(0, 0), C(rc, θc), V(rv, θv), OC = rc, CV = rlocal,6 COO’ = θc, 6 OCC’ = α, 6 VCC’ = θlocalFind: rv, θv = ?If K > 0, then: If K < 0, then:cos rv = cos rc cos rlocal+sin rc sin rlocal cosβcosh rv = cosh rc cosh rlocal−sinh rc sinh rlocal cosβ(3)cos∆θv =cos rlocal − cos rc cos rvsin rcsinrvcos∆θv =cosh rc cosh rv − cosh rlocalsinh rc sinh rv(4)In order to find rv, first find β = α + θlocal; if Π < β < 2Π, use theexplementary angle instead to determine to which side of OC the triangle lies.Depending on that ∆θ is then added to or subtracted from θc to find θv.V1 Δϴ2 O C r2 r1 Δϴ1 r1local V2 r2local d rc ϴ1 ϴc ϴ2 (a) (b) V1 d V2 Δϴ2 O’ ϴ2 C r1 ϴ1 ϴc O Δϴ1 rc r1local r2local r2    Polar coordinate system with a reference point O  C(rc, ϴc), object position, reference point of local coordinate system V1(r1, ϴ1), object’s vertex 1 V2(r2, ϴ2), object’s vertex 2 OV1 = r1, OV2 = r2,  OC = rc,  CV1 = r1local, CV2 = r2local,  V1V2 = d, object’s edge ∠V1OO’ = ϴ1, ∠V2OO’ = ϴ2, ∠COO’ = ϴc,  ∠V1OC = Δϴ1, Δϴ1 = |ϴc -  ϴ1|, ∠V2OC = Δϴ2, Δϴ2 = |ϴc -  ϴ2| O’ Fig. 8. Finding the length of edge d and the angle ∆θ. Case (a), ∆θ1 and ∆θ2 diverge,so ∆θ is the sum; case (b), angles converge, so ∆θ is the absolute value of the difference.Corollary 2. Given: O(0, 0), C(rc, θc), V1(r1, θ1), V2(r2, θ2), OC = rc, OV1= r1, OV2 = r2, CV1 = r1local, CV2 = r2local, 6 COO’ = θc, 6 V1OO’ =θ1, 6 V2OO’ = θ2Find: d, ∆θ = ?If angles converge, ∆θ = ‖∆θ1−∆θ2‖; if angles diverge, ∆θ = ‖∆θ1‖+‖∆θ2‖If K > 0, then: If K < 0, then:cos d = cos r1 cos r2+sin r1 sin r2 cos∆θcosh d = cosh r1 cosh r2–sinh r1 sinh r2 cos∆θ(5)r2 V2 Vi d V1 C ri r1 Δϴi Δϴ O ϴ2 ϴi ϴ1 di  Polar coordinate system with a reference point O  V1(r1, ϴ1), V2(r2, ϴ2), vertices Vi(ri, ϴi), intermediate point OV1 = r1, OV2 = r2, OVi = ri,     V1V2 = d, object’s edge V1Vi = di, section of the edge ∠V1OO’ = ϴ1, ∠V2OO’ = ϴ2, ∠ViOO’ = ϴi, ∠OV1Vi = α, ∠V1OV2 = Δϴ, angle between  V1 and V2 ∠V1OVi =  Δϴi,  angle between  V1 and Vi α O’ Fig. 9. Finding intermediate points in or-der to render the edge.Note: distance d is divided into anumber of equal parts in order to findthe distance di for each of the pointson the edge V1V2. The number of seg-ments depends on the object tessela-tion variable.Corollary 3. Given: O(0, 0), V1(r1, θ1),V2(r2, θ2), Vi(ri, θi), OV1 = r1,OV2 = r2, V1V2 = d, V1Vi =di, 6 V1OO’ = θ1, 6 V2OO’ = θ2,6 V1OV2 = ∆θFind: ri, θi = ?If K > 0, then: If K < 0, then:cosα =cos r2 − cos r1 cos dsin r1 sin dcosα =cosh r1 cosh d− cosh r2sinh r1 sinh d(6)cos ri = cos r1 cos di+sin r1 sin di cosαcosh ri = cosh r1 cosh di–sinh r1 sinh di cosα(7)cos∆θi =cos di − cos r1 cos risin r1 sin ricos∆θi =cosh r1 cosh ri − cosh disinh r1 sinh ri(8)α is calculated to find the angle opposite ri. Then ri and subsequently ∆θican be found using the cosine rule (illustrated on Fig. 9). Then to find actualcoordinates of the point Vi, ri should be multiplied by r or k depending on thevalue of K; ∆θi should be added to or subtracted from angle θ1, depending onthe direction of the edge d, determined previously.2.2 Updating object position Polar coordinate system with a reference point O  Ctx(rtx, ϴtx),  position of the object at time x  OCtx = rtx, object’s r coordinate at time x C’’0C’’1, geodesic of the object’s movement trajectory ∠CtxOO’ = ϴtx, object’s β coordinate at time x ∠OCtxC’tx = βtx, angle of rotation at time x ∠OCtxC’’1 = γtx, direction of movement at time x ∠C’’1CtxC’tx = α, orientation with respect to geodesic βt2 ϴt1 ϴt2 ϴt0 rt0 βt0 C’’0 Ct2 O’ βt1 γ t1 O α C’t0 C’t1 γ t0 γt2 C’t2 C’’1 rt2 rt1 Ct0 Ct1 α α (a) (b) C’t2 Ct1 ϴt1 ϴt2 O ϴt0 rt0 rt1 C’’0 C’t1 βt0 γt2 O’ Ct2 βt2 βt1 γ t1 α C’’1 α α γ t0  C’t0 rt2 Ct0 Fig. 10. Movement of the object along a hyperbolic in Spherical (a) and Hyperbolic (b)space. Orientation with respect to the geodesic is kept the same (angle α is constant)if the object is not rotating.Corollary 4. Given: O(0, 0), Ct0(rt0, θt0), Ct1(rt1, θt1), OCt0 = rt0, Ct0Ct1= rp, 6 Ct0OO’ = θt0, 6 OCt0C” = γt0, 6 OCt0C’t0 = βt0Find: rt1, θt1, γt1, βt1 = ?γt0 should be 0 to pi, if calculated value is γt0 > pi, take the explemntaryangle. This indicates the movement direction with respect to the reference point.Let 6 OCt1Ct0 = γ′t1If K > 0, then: If K < 0, then:cos rt1 = cos rt0 cos rp+sin rt0 sin rp cosαcosh rt1 = cosh rt0 cosh rp+sinh rt0 sinh rp cosα(9)cos∆θ =cos rp − cos rt0 cos rt1sin rt0 sin rt1cos∆θ =cosh rt0 cosh rt1 − cosh rpsinh rt0 sinh rt1(10)cos γ′t1 =cos rt0 − cos rp cos rt1sin rp sin rt1cos γ′t1 =cosh rp cosh rt1 − cosh rt0sinh rp sinh rt1(11)α = βt0 − γt0. α is the difference between rotation direction and the geodesicof movement (C”0C”1), it does not change if the object is not rotating. Hence,βt1 = γt1 + α. Because γ′t1 and γt1 are supplementary angles, γt1 = Π − γ′t1.To find the θ coordinate, either subtract or add ∆θ to the θc depending onwhether the angle α or its explementary angle is used for this calculation.3 Results3.1 ImplementationUsing the method described above and OpenGL, we created a software capableof calculating the objects and rendering the vector graphics in a non-Euclideanspace with constant curvature in the range of −1 ≤ K ≤ 1. Fig. 1 shows thetime-lapses of multiple objects in spherical (a), planar (b) and hyperbolic (c)geometries. They show movement through different geodesics at K = 1, K = 0and K = −1 respectively. Starting positions as well as shape definitions of eachobject are the same across all time-lapses (grid-lines have been created andrendered as separate objects). The software can calculate the object movingin arbitrary direction with arbitrary speed as well as starting from arbitraryposition in the space.Curvature of the world can be modified in real-time using keyboard inputs ina similar manner to controlling the object’s acceleration and orientation. Anotherfeature is the cut-off of the world at a distance of N pixels. This can be seen in thehyperbolic and planar time-lapse images. While these spaces should be infinite,we chose to limit them in order to keep objects within the boundaries of thescreen (non-shaded area). We created a video [7] displaying the implementation.3.2 Complexity AnalysisPositions of each vertex need to be calculated, requiring O(v) time, where v isthe number of vertices. Subsequently, intermediate points have to be computed,requiring O(i) time to find all of the points on a single edge, where i is thelevel of tessellation. Complexity to render the world with s number of shapes istherefore O(s ∗ v ∗ i). The best case would be equal to O(n) complexity, if two ofthe terms are negligibly small. The worst case can be approximated to O(n3) ifall terms were comparably large. Spatial complexity for shape rendering is onlyO(v ∗ i) as previous shape’s data is rewritten to store the next shape’s data. Soeither O(n) in the best case or O(n2) in the worst case.Only one movement calculation per object is required and the previous po-sition record is overwritten, both spatial and time complexity is O(n), where nis the number of objects in the world.Trigonometric and hyperbolic functions in the calculations are slower to com-pute than simpler operations, hence additional cost (implementation dependent).For example the AGM iteration [8] method is faster than the previously commonTaylor series method.4 DiscussionImplementation does not affect performance up to a certain number of objectsor tessellation amount. The focus was on implementing the method correctlyand having it work continuously under any curvature in the range −1 ≤ K ≤ 1.The next step in the project’s development is improving the execution timeusing parallelised calculations. Subsequent calculation of the points creates abottleneck, which can be solved by performing some calculations directly on theGPU. Other approaches are considered as well, including lookup tables to speedup trigonometric calculations, for example, Frank Rochet’s implementation [9];or finding intermediate points from a geodesic equation.Potential applications for the software include education about non-Euclideangeometry (more intuitive than standard projections: Poincare disk and UpperHalf-Plane models); cartography [10] (the engine could be modified to efficientlyconvert data into different projections); ecology [11] and climatology [12] (mod-elling dynamic systems); Astrophysics (modelling systems of cosmological ob-jects and gravitational fields) and video games (game engine for a real-timecontinuous non Euclidean space, unlike HyperRogue [13], which uses step bystep implementation).References1. T. L. Heath, Euclid’s Elements. Dover, 1956. (translated).2. I. Todhunter, Spherical Trigonometry For the use of colleges and schools. ProjectGutenberg License, 1886. (republished November 12, 2006).3. H. S. Carslaw, The Elements of Non-Euclidean Plane Geometry and Trigonometry.Longmans, Green and co., 1916.4. T. Traver, “Trigonometry in the hyperbolic plane,” 2014 (accessed December 2017).Manuscript.5. W. Gellert, S. Gottwald, M. Hellwich, H. Ka¨stner, and H. Ku¨stner, The VNRConcise Encyclopedia of Mathematics, 2nd ed. Van Nostrand Reinhold: New York,1989. ch. 12.6. J. Gray, Non-euclidean geometry—A re-interpretation. Historia Mathematica,1979. 236–258.7. D. Osudin, C. Child, and Y. Hui-He, “Rendering non-euclideanspace in real-time using spherical and hyperbolic trigonometry,” 2019.https://youtu.be/A1ZCFh5qfNg.8. R. P. Brent, “Multiple-precision zero-finding methods and the complex-ity of elementary function evaluation,” 2010 (accessed August 26, 2018).http://arxiv.org/abs/1004.3412v2.9. F. Rochet, “Fast trigonometry functions using lookup tables,” 2004(accessed August 30, 2018). http://www.flipcode.com/archives/Fast Trigonometry Functions Using Lookup Tables.shtml.10. G. Gartner and H. Huang, “Recent research developments in modern cartographyin europe,” Issue 1: EuroCarto 2015, 2015.11. C. Sutherland, “Modelling non-euclidean movement and landscape connectivity inhighly structured ecological networks,” British Ecological Society, 2014.12. C. Frei, “Interpolation of temperature in a mountainous region using nonlinearprofiles and non-euclidean distances,” Royal Meteorological Society, 2013.13. Zeno Rogue Games, “Hyperrogue,” 2019. http://roguetemple.com/z/hyper/.

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