Interactive 4-D Visualization of Stereographic Images From the Double Orthogonal Projection
IInteractive 4-D Visualization of Stereographic Images Fromthe Double Orthogonal Projection.
Michal ZambojCharles University, Faculty of EducationM. D. Rettigov´e 4, 116 39 Prague 1.
Abstract
The double orthogonal projection of the 4-space onto two mutually perpendicular 3-spaces isa method of visualization of four-dimensional objects in a three-dimensional space. We presentan interactive animation of the stereographic projection of a hyperspherical hexahedron ona 3-sphere embedded in the 4-space. Described are synthetic constructions of stereographicimages of a point, hyperspherical tetrahedron, and 2-sphere on a 3-sphere from their doubleorthogonal projections. Consequently, the double-orthogonal projection of a freehand curveon a 3-sphere is created inversely from its stereographic image. Furthermore, we show anapplication to a synthetic construction of a spherical inversion and visualizations of doubleorthogonal projections and stereographic images of Hopf tori on a 3-sphere generated fromClelia curves on a 2-sphere.
Keywords: Four-dimensional visualization, orthogonal projection, stereographic projection, Hopffibration, Clelia curve, spherical inversion, descriptive geometry.
The double orthogonal projection is a four-dimensional generalization of Monge’s projection. InMonge’s projection, an object in the 3-space is orthogonally projected into two mutually perpen-dicular planes (horizontal and vertical, i.e. top and front view). One of these planes is chosen to bethe drawing (or picture) plane and the second is rotated about their intersecting line to the drawingplane. Therefore, each point in the 3-space has two conjugated images in the drawing plane. Inthe double orthogonal projection, an object is in the 4-space, and we project it orthogonally intotwo mutually perpendicular 3-spaces. Let x, y, z , and w be the orthogonal system of coordinateaxes of the 4-space, and Ξ( x, y, z ) and Ω( x, y, w ) be the 3-spaces of projection. If Ω is chosen to bethe modeling 3-space (instead of the drawing plane), then Ξ is rotated about their common plane π ( x, y ) such that z and w have opposite orientations (usually w up and z down). Analogically, eachpoint in the 4-space has two conjugated images in the modeling 3-space. Elementary constructionsand principles of the double orthogonal projection were described in [22], sections, and lighting ofpolytopes in [23], intersections of lines, planes, and 3-spaces with a 3-sphere in [24], and regularquadric sections as intersections of 4-dimensional cones with 3-spaces in [25]. Similarly to Monge’sprojection, in which an observer can reach any point of the drawing plane, in the four-dimensionalcase an observer can reach any point of the modeling 3-space. For this purpose, our constructionsare supplemented by online interactive models [21] created in GeoGebra 5 .A projection between an n -sphere embedded in an ( n + 1)-space from its point N and an n -dimensional hyperplane supplemented with a point at infinity {∞} , not passing through N is calledstereographic projection. Due to its angle-preserving property, it is a convenient tool to visualizespheres. The 3-dimensional case of a stereographic projection of a 2-sphere from the North pole N to a tangent plane at the South pole and its Monge’s projection is depicted in Figure 1. For a1 a r X i v : . [ m a t h . G M ] J u l rief overview see [20], pp. 154–163, and for more general view with the construction of a sphericalinversion used in Section 2.1 see [15], pp. 368–378.Figure 1: A stereographic projection of the southern hemisphere from the North pole N to thetangent plane at the South pole (i.e. the horizontal plane of projection), and its Monge’s projection.Images of a quadrilateral with its sides along parallels and meridians are highlighted.Interactive model available at A stereographic projection of a 3-sphere onto a 3-space was used to create 3-D printed modelsand study properties of four-dimensional polytopes in [19] and later on in [12] to visualize symme-tries of the quaternionic group on a 3-D printed models of monkeys in a hypercube (limbs, head,and tail of the monkey represent faces of cubical cells). Well designed online applets with stereo-graphic projections of a hypercube are at [2] and including other polytopes, sections, and tori in a3-sphere at [8]. A stereographic projection of a 3-sphere is often used to visualize objects usuallystudied in topology. The topology of a 3-sphere is studied in [14]. Several animations and four-dimensional stereographic projection are described in [7], Chapters 6 and 8. The same author in [6]used computer graphics to visualize stereographic images of Pinkall’s tori corresponding in the Hopffibration to simple closed curves on a 2-sphere (see also [17] and [1]). An interactive visualization ofa stereographic projection of a Clifford torus on a 3-sphere is at [5]. Videos and animations of theHopf fibration and its stereographic projection are at [3, 9, 13]. Commented videos with interactiveenvironment discussing quaternions and stereographic projection are at [10]. A stereographic anddouble-stereographic projection of an arbitrary object in the 4-space was described in [4]. As anexample of a recent application, a stereographic projection to a 3-sphere is used in [16] to ana-lyze multiple disconnected anatomical structures mathematically represented as a composition ofcompact finite three-dimensional surfaces.The previous references were based on the analytic representation of points in the fourth dimen-sion. With the use of the double orthogonal projections, we can construct images of four-dimensionalobjects synthetically and use the modeling 3-space to be also the projecting 3-space of a stereogra-phic projection. A construction of a hyperspherical tetrahedron in a special position and its imagesin a stereographic and the double orthogonal projection is in [27]. A synthetic construction andanimation of Hopf fibers in the double orthogonal projection and their stereographic projection arediscussed in [26]. In this paper, we will extend these results in several aspects.Let us describe the visualization of a 3-sphere and stereographic images in the double orthog-onal projection in Figure 2 (left) in an analogy to the three-dimensional case in Figure 1 (right).In Monge’s projection, the conjugated images of a 2-sphere are disks, in the double orthogonalprojection, the conjugated images of a 3-sphere are balls. In 3-D, sections of a 2-sphere with planesparallel to the horizontal plane of projection (parallels) are circles, and their images are segmentsin the vertical plane and circles in the horizontal plane. The stereographic projection from theNorth pole N to the tangent plane at the South pole S projects these parallels to the system ofconcentric circles in the drawing plane. In 4-D, sections of a 3-sphere with 3-spaces parallel to the2-space of projection Ξ are 2-spheres, their Ω-images are disks and Ξ-images are 2-spheres. Thestereographic projection from the point N with the maximal w -coordinate to the tangent 3-spaceΞ at the antipodal point S projects the parallel 2-spheres to the system of concentric 2-spheres inthe modeling 3-space. In both cases, the point N is projected stereographically into the point atinfinity {∞} , and hence images of circles through N become lines. In Figure 2 (right) (see alsoanimated model) is a hyperspherical hexahedron along hyperspherical coordinates. The stereographic image of a point on a 3-sphere is the intersection of the projecting ray with theplane of projection (Figure 3, left). Let us have conjugated images of a 3-sphere with a center Z inthe double orthogonal projection and the Ω-image A of a point A on the 3-sphere. The Ξ-image A lies on the perpendicular to π through A , i.e. ordinal line of the point A . Furthermore, the sectionof the 3-space through A parallel to the 3-space Ω with a 3-sphere is a 2-sphere. Its Ω-image is a2-sphere with the center Z through A , and its Ξ-image is a disk with the same radius in a planeparallel to π . Let N be the abovementioned center of the sterographic projection to the 3-spaceΞ. The stereographic image A s of the point A on the 3-sphere lies on the line N A and also onthe perpendicular to π through the intersection A of π and the line N A . Oppositely, the inverseconstruction of the conjugated images A and A from the stereographic image A s is in Figure 3(right). In this case, we need to find the intersection A of the projecting ray N A s with the 3-sphere.For such construction, we can use a third orthogonal projection into the plane perpendicular to Ξthrough the projecting ray, and rotate it into the modeling 3-space (see [24] for more details). Inthe third view, the image of the 3-sphere is a circle with the center ( Z ), and its intersection withthe rotated line ( N ) A s is the rotated point ( A ). Then, the point A is constructed with the reverseFigure 2: The system of concentric spheres, which are stereographic images of parallel sections ofthe half of the 3-sphere split by the 3-space parallel to Ξ from the point N with the maximal w -coordinate to the tangent 3-space Ξ at the antipodal point S , and its double orthogonal projection.On the right side are conjugated images and stereographic image of a hyperspherical hexahedron,as an analogy to a spherical quadrilateral.Interactive model available at A s of the point A on the 3-sphere givenby its Ω-image A . (right) Construction of the conjugated images A and A of a point A from itsstereographic image A s .Interactive model available at rotation.Figure 4: (left) The double orthogonal projection and stereographic image of a hypersphericaltetrahedron ABCD . (right) The same situation with the circumscribed 2-sphere around
ABCD .Interactive model available at
In Figure 4 (left), the same method is used to construct the stereographic images A s , B s , C s , D s of the vertices of a hyperspherical tetrahedron ABCD (generalization of a spherical triangle). Theedges of the tetrahedron
ABCD are circular arcs and faces are spherical triangles. These properties The faces are not depicted due to insufficient possibilities of the surface parametrization in GeoGebra, but thereader can turn on the visibility of the corresponding spheres in the stereographic projection in the online model. A s B s lie on the circle A s B s A (cid:48) s , where A (cid:48) s is the stereographic image of thepoint A (cid:48) antipodal to A , and so its conjugated images A (cid:48) and A (cid:48) are the mirror images of A and A about Z and Z , respectively. The conjugated images of the edges are constructed point-by-point from their stereographic images and create elliptical arcs in a general position. Note that wecould construct them, however, more laboriously, as the intersections of the 3-sphere and planes.Further on, a 2-sphere circumscribed around a tetrahedron ABCD is visualized in Figure 4(right). While in the stereographic projection it is simply a 2-sphere in the true shape, the con-jugated images in the double orthogonal projection are ellipsoids. They can be constructed asthe intersections of a 3-sphere with the 3-space defined by the noncospatial points
A, B, C , and D (see [24] for details), or, as highlighted in the Ω-image, as the intersections of the conical hypersur-face with the vertex N with a 3-sphere.Figure 5: My drawing was not a picture of a hat. It was a picture of a boa constrictor digesting anelephant on a 3-sphere in the double orthogonal projection (slightly modified from [18]).Interactive model available at The interactive environment, in combination with the construction of a stereographic projectionfrom a 3-space onto a 3-sphere (Figure 3, right), brings a possibility of freehand drawing on a 3-sphere. With the use of the trace tool in GeoGebra (or similar point-by-point construction in othersoftware), we move the point in the stereographic projection, and its dependent conjugated imagesdraw their traces. This way, we can draw a stereographic image (its Ξ-image) of a curve, or anyother picture, in the modeling 3-space, and simultaneously its double orthogonal projection. Afour-dimensional curve on a 3-sphere drawn by hand on a 2-sphere in a stereographic projection isin Figure 5.
A spherical inversion may be obtained as a composition of two stereographic projections (Figure 6).For this purpose, we choose the projecting 3-space Σ of the stereographic projection through thecenter Z of the 3-sphere and parallel to Ξ. A point A in the 3-space Σ ∪ {∞} is stereographicallyprojected in the projection p N from the center N to the point A ◦ on the 3-sphere. The secondstereographic projection p S from the center S projects the point A ◦ to the point A (cid:48) in the 3-space Σ ∪ {∞} . Since Σ is parallel to Ξ, all the Ξ-images of objects in the 3-space Σ are in the true5igure 6: Spherical inversion of a tetrahedron ABCD and its circumscribed 2-sphere as a com-position of stereographic projections from antipodal poles between a 3-sphere and its equatorial3-space.Interactive model available at shape. The section γ of a 3-sphere and the 3-space Σ is a 2-sphere overlapping with the Ξ-imageof the 3-sphere with the center Z in the modeling 3-space. The composition of p N and p S isthe spherical inversion, and p S ( p N ( A )) = A (cid:48) . Especially, all the points on γ are fixed. Moreover, p S ( p N ( ∞ )) = p S ( N ) = Z , and p S ( p N ( Z )) = p S ( S ) = ∞ . Note, that for each point A (cid:54) = Z, {∞} inΣ, we could choose a projecting plane of the line AZ perpendicular to Ξ which cuts the 2-sphere γ in a circle c , and the final composition would be a circle inversion. The situation is depicted inMonge’s projection in Figure 7, in which the circle inversion is in the orthogonal projection intothe horizontal plane. Triangles A Z N and S Z A (cid:48) are in their true shape in the front view, andthey are apparently similar. Therefore, | A Z || Z N | = | S Z || Z A (cid:48) | , and so | A Z || A (cid:48) Z | = | Z N || Z S | = r , A in the orthogonal projection to the projecting plane ofthe line N A .where r is the radius of c, γ , and also the 3-sphere. From the given construction, it also holds that | A Z | = | A Z | = | AZ | and | A (cid:48) Z | = | A (cid:48) Z | , and we have | AZ || A (cid:48) Z | = r . The last formula leads to the standard definition of the spherical inversion about the 2-sphere γ with the center Z and radius r .Figure 6 also shows a tetrahedron ABCD with a circumscribed sphere and their inversion. Theconformity of stereographic projections is inherited in their composition to the spherical inversion.The lines not passing through the center of the inversion become circles. Consequently, the edgesof the image tetrahedron A (cid:48) B (cid:48) C (cid:48) D (cid:48) are circular arcs with preserved mutual angles. The image ofthe sphere circumscribed to ABCD is a sphere circumscribed to A (cid:48) B (cid:48) C (cid:48) D (cid:48) . The Hopf fibration is a mapping between a 3-sphere and a 2-sphere. In particular, it sends a point P ( x P , y P , z P , w P ) ∈ R on a 3-sphere to the point P (cid:48) (2( x P z P + y P w P ) , − x P w P + y P z P ) , x P − y P − z P − w P ) ∈ R on a 2-sphere. Oppositely, a point on a 2-sphere in spherical coordinates P (cid:48) (sin ψ P cos ϕ P , sin ψ P sin ϕ P , cos ψ P )for ψ P ∈ (cid:104) , π (cid:105) , ϕ P ∈ (cid:104) , π ) corresponds to a set of points c P (cos ψ P ϕ P + β ) , cos ψ P ϕ P + β ) , sin ψ P β, sin ψ P β )for β ∈ (cid:104) , π ), which forms a great circle (fiber) on a 3-sphere. If P (cid:48) lies on a closed curve on the2-sphere, its corresponding Hopf fibers form a Hopf torus on the 3-sphere. For the visualization of7igure 8: (Up) Conjugated images of the Hopf tori generated by the Clelia curves for s = 0 . s = 1 Viviani curve (Center), s = 2 (Right). Highlighted is a point (blue) on the Cleliacurve (black), conjugated images of its fiber (red) and conjugated images of the Hopf torus (green).(Down) Stereographic image of a fiber (purple) and Hopf torus (gray).Interactive model available at Hopf tori, it is convenient to use the stereographic projection that preserves circles. This way wecan construct and study tori on the 3-sphere in the 4-space given by a curve on the 2-sphere in the3-space, and vice-versa.For the sake of visualization in the double orthogonal projection, we perform several adjust-ments. First, we swap the reference axes y and z , and hence the 3-spaces of projection will beΞ( x, y, z ) (upper) and Ω( x, z, w ) (lower) with the common plane π ( x, z ). To avoid overlappingof the conjugated images, we translate the center of the abovementioned unit 3-sphere from theorigin to the point (0 , , , x, z, w ), so its Ξ-image is in the true shape. Consequently, points ofthe 3-sphere are stereographically projected from the center (0 , , ,
1) to the 3-space Ω( x, z, w ). Asynthetic construction of the Hopf fiber of any point on the 2-sphere and a Hopf torus of a circleon a 2-sphere and their stereographic images in this setting are described in [26]. This choice reflects the possibility of a definition of the Hopf fibration in the complex number plane, and so thecommon plane π ( x, z ) corresponds to the real parts of coordinates ( x + ı y, z + ı w ) of points.
8e extend these results and highlight interactive possibilities for the so-called Clelia curvesgiven parametrically on the (translated) 2-sphere in the form k ( ψ ) = (sin( sψ ) cos ψ, sin( sψ ) sin ψ + 1 , cos( sψ )) , where s ∈ R defines the specific curve, for ψ ∈ I ⊂ R (see [11] for details and further generalizations).Conjugated images and stereographic projections of the Hopf tori corresponding to the Clelia curvesfor s = , , and 2 are visualized in Figure 8. In the interactive model, the user can choose theparameter s of the curve and change the corresponding Hopf torus, and also move with a point P on k ( ψ ) and its corresponding fiber on the 3-sphere by manipulating the parameter ψ . An example ofa simple straightforward observation is, that a point, in which the curve on the 2-sphere intersectsitself, becomes a circle (or line) in which the torus on the 3-sphere intersects itself. The combination of the double orthogonal projection and stereographic projection is an accessibletool for the investigation of a 3-sphere in the 3-space. The simplicity of the synthetic constructionof the stereographic image of a point and the use of interactive 3-D modeling software imply thatwe can actually draw sketches and shapes on the 3-sphere embedded in the 4-space. We havealso shown a construction of the well-known relationship between the stereographic projection andthe spherical inversion. Our results were applied to the construction of Hopf tori on a 3-spheregenerated by Clelia curves on a 2-sphere. The visualizations were presented on interactive 3-Dmodels, which are easily extendible for further theoretical and practical applications in varioussoftware.
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