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Hyperboloid model

The hyperboloid model is a representation of two-dimensional hyperbolic geometry realized as a surface in three-dimensional Minkowski space, consisting of the upper sheet of the two-sheeted hyperboloid defined by the equation x^2 + y^2 - z^2 = -1 with z > 0.^[1] In this model, points correspond to position vectors on the hyperboloid satisfying the Lorentzian quadratic form q(X) = -1, while geodesics are the nonempty intersections of the hyperboloid with planes through the origin of Minkowski space.^[2] The hyperbolic distance between two points u and v is given by d(u, v) = \arccosh(-\langle u, v \rangle), where \langle \cdot, \cdot \rangle denotes the Minkowski inner product with signature (+, +, -), ensuring the induced Riemannian metric has constant sectional curvature -1.^[3] Developed in the late 19th century by Wilhelm Killing and Henri Poincaré, the hyperboloid model—sometimes called the Minkowski model due to its embedding in Lorentzian space—offers a symmetric and computationally convenient framework for hyperbolic geometry, despite its embedding in three dimensions making visualization somewhat challenging compared to planar models.^[2] Its primary advantages lie in the fact that isometries of the hyperbolic plane correspond directly to linear transformations in the orthogonal group O(2,1) that preserve the quadratic form, facilitating algebraic computations and connections to special relativity, where the hyperboloid represents the worldsheet of constant proper time.^[2] This model is isometric to other standard representations, such as the Poincaré disk (via stereographic projection from the south pole) and the Klein-Beltrami model (via central projection onto the plane z=1), allowing seamless translations between them for different analytical purposes.^[1]

Minkowski Space Foundations

Quadratic form

The Minkowski space \mathbb{R}^{n,1} underlying the hyperboloid model is the real vector space \mathbb{R}^{n+1} equipped with coordinates (x_0, x_1, \dots, x_n), where x_0 plays the role of a time-like coordinate and x_1, \dots, x_n are space-like.^[4] This space is endowed with the indefinite quadratic form

Q(x) = -x_0^2 + \sum_{i=1}^n x_i^2,

which defines the algebraic structure essential for embedding hyperbolic geometry.^[4] The associated symmetric bilinear form, known as the Minkowski inner product, is given by

\langle u, v \rangle = -u_0 v_0 + \sum_{i=1}^n u_i v_i

for vectors u = (u_0, \dots, u_n) and v = (v_0, \dots, v_n).^[4] The indefiniteness of Q arises from its signature (n,1), meaning it takes both positive and negative values, unlike the positive definite form of Euclidean space.^[4] This property classifies nonzero vectors based on the sign of Q(x): timelike if Q(x) < 0, spacelike if Q(x) > 0, and lightlike (or null) if Q(x) = 0.^[4] Such classifications distinguish causal structures in the geometry, with timelike vectors forming the interior of the light cone defined by the level set Q(x) = 0.^[4] The level sets of Q for a nonzero constant c are hyperboloids in \mathbb{R}^{n+1}.^[4] Specifically, for c = -1, the equation Q(x) = -1 describes a two-sheeted hyperboloid, consisting of two connected components separated by the light cone; the upper sheet, where x_0 > 0, serves as the embedding surface for the hyperboloid model of hyperbolic n-space.^[4] For c = 1, the level set Q(x) = 1 yields a one-sheeted hyperboloid, which is connected and lies outside the light cone.^[4] These surfaces illustrate the hyperbolic nature of the geometry induced by the indefinite form.^[4]

Metric signature conventions

In the hyperboloid model of hyperbolic geometry, the underlying Minkowski space employs an indefinite metric tensor with one of two primary sign conventions: the mostly minus signature (+,-,-,\dots,-), where the time-like component is positive, or the mostly plus signature (-+,+,\dots,+), where it is negative.^[5] These conventions arise from the Lorentzian metric on \mathbb{R}^{n,1}, and the choice determines the form of the quadratic form Q used to define the embedding.^[4] The mostly plus signature (-+,+,\dots,+) leads to a quadratic form Q(\mathbf{x}) = -x_0^2 + \sum_{i=1}^n x_i^2, with the hyperboloid defined by Q(\mathbf{x}) = -1.^[4] Equivalently, this can be expressed as x_0^2 - \sum_{i=1}^n x_i^2 = 1, where the upper sheet corresponds to x_0 > 0, ensuring the position vectors are time-like (negative norm in this signature).^[4] In contrast, the mostly minus signature (+,-,-,\dots,-) yields Q(\mathbf{x}) = x_0^2 - \sum_{i=1}^n x_i^2 = 1 directly, again selecting the upper sheet x_0 > 0 for time-like vectors (positive norm here), which aligns the induced metric to be positive definite on the surface.^[6] The choice affects the interpretation of time-like versus space-like separations but preserves the overall geometry when consistently applied. Historically, the mostly minus convention (+,-,-,\dots,-) is prevalent in general relativity and classical physics texts, as it makes proper time intervals positive, facilitating connections to special relativity.^[5] Conversely, some differential geometry and hyperbolic geometry literature favors the mostly plus signature (-+,+,\dots,+), emphasizing spatial coordinates and aligning with certain algebraic conventions in Lorentz groups.^[4] This preference in geometry texts often stems from treating the model as an abstract Riemannian manifold embedded in pseudo-Euclidean space, independent of physical time.^[6] Regardless of the signature, normalization is achieved by setting the level of Q to \pm 1, ensuring the induced Riemannian metric on the hyperboloid has constant sectional curvature -1, matching the standard hyperbolic space \mathbb{H}^n.^[4] This scaling distinguishes the model from related surfaces like de Sitter space and guarantees isometry with other hyperbolic models, such as the Poincaré disk.^[6]

Model Definition

The hyperboloid sheet

The hyperboloid model of hyperbolic n-space, denoted H^n, is defined as the set

H^n = \{ x = (x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} \mid \langle x, x \rangle_L = -1, \, x_0 > 0 \},

where \langle \cdot, \cdot \rangle_L denotes the indefinite Lorentzian inner product on the Minkowski space \mathbb{R}^{1,n} with mostly plus signature: \langle x, y \rangle_L = -x_0 y_0 + \sum_{i=1}^n x_i y_i. This quadratic form Q(x) = \langle x, x \rangle_L distinguishes the model from Euclidean embeddings by incorporating the pseudo-Euclidean structure of Minkowski space.^[7] Geometrically, H^n forms the upper (or forward) sheet of the two-sheeted hyperboloid Q(x) = -1, with the two sheets comprising the disconnected components of the surface. The hyperboloid is asymptotic to the light cone Q(x) = 0 in the limit as x_0 \to \infty, where the cone separates the space-like and time-like regions and bounds the model's extent. The selection of the upper sheet x_0 > 0 ensures a connected Riemannian manifold suitable for modeling hyperbolic geometry, avoiding the lower sheet x_0 < 0 which is isometric but often excluded for convenience in applications.^[7] The Riemannian structure on H^n arises as an isometric embedding into Minkowski space, with the metric induced by restricting the ambient Lorentzian inner product to the tangent space at each point x \in H^n. Specifically, for a tangent vector v at x satisfying \langle x, v \rangle_L = 0, the metric is ds^2 = \langle v, v \rangle_L, which is positive definite on the tangent space and inherits the pseudo-Riemannian properties selectively. This induced metric equips H^n with the geometry of a simply connected space-form.^[7] The hyperboloid model realizes constant sectional curvature -1, a defining feature of hyperbolic n-space, as verified by direct computation of the Riemann curvature tensor from the embedding or through comparison with other models. This curvature value scales the geometry such that H^n serves as the standard model for \mathbb{H}^n with radius $1$.^[7] In ambient coordinates x = (x_0, \dots, x_n), points on H^n satisfy x_0 = \sqrt{1 + \sum_{i=1}^n x_i^2}, linking the time-like coordinate x_0 to the space-like components and encoding hyperbolic distances via the Lorentzian norm. These coordinates facilitate computations involving projections or transformations while preserving the intrinsic hyperbolic properties, such as the base point at (1, 0, \dots, 0).^[7]

Induced hyperbolic metric

The induced hyperbolic metric on the hyperboloid sheet H^n = \{ u \in \mathbb{R}^{n+1} \mid \langle u, u \rangle = -1, \, u_0 > 0 \} arises from restricting the Minkowski metric of the ambient space \mathbb{R}^{1,n} with Lorentzian inner product \langle u, v \rangle = -u_0 v_0 + \sum_{i=1}^n u_i v_i to the tangent spaces of H^n. This restriction yields a positive definite Riemannian metric of constant sectional curvature -1, endowing H^n with the standard hyperbolic structure.^[4]^[7] In geodesic polar coordinates centered at a base point, with radial coordinate r \geq 0 and angular coordinates on the unit sphere S^{n-1}, the line element takes the form

ds^2 = dr^2 + \sinh^2 r \, d\theta^2,

where d\theta^2 denotes the standard metric on S^{n-1}. This infinitesimal metric measures lengths of tangent vectors on H^n and follows directly from parametrizing the hyperboloid via hyperbolic functions, such as u_0 = \cosh r and the spatial components involving \sinh r times unit vectors.^[4]^[8] The geodesic distance d(u, v) between distinct points u, v \in H^n is given by

d(u, v) = \arccosh(-\langle u, v \rangle),

derived by parametrizing the unique geodesic connecting u and v as the intersection of H^n with the plane spanned by u and v in \mathbb{R}^{n+1}. Along this geodesic, the parameter t satisfies \langle \gamma(t), \gamma(t) \rangle = -1 and traces an arc where the Minkowski inner product yields the hyperbolic angle, leading to \cosh t = -\langle u, v \rangle at the endpoints. This formula originates from applying the hyperbolic law of cosines to the right triangle formed by u, v, and the origin in the embedding space, where the "angle" at the origin corresponds to the spatial separation.^[9]^[10] The distance d(u, v) is real and positive for u \neq v, since -\langle u, v \rangle \geq 1 with equality only when u = v, as ensured by the Cauchy-Schwarz inequality in the Lorentzian metric for future-directed timelike vectors. The triangle inequality d(u, w) \leq d(u, v) + d(v, w) holds due to hyperbolic addition formulas, such as \cosh(a + b) = \cosh a \cosh b + \sinh a \sinh b \geq \cosh(|a - b|), which imply the subadditivity of the arccosh function on [1, \infty).^[9]^[10] This embedding preserves the hyperbolic structure up to isometry, as the hyperboloid model realizes H^n as a Riemannian submanifold of \mathbb{R}^{1,n} whose intrinsic geometry matches the unique (up to scaling) simply connected space of constant curvature -1.^[10]^[4]

Basic Geometry

Geodesics

In the hyperboloid model of hyperbolic n-space, denoted H^n, geodesics are defined as the intersections of the hyperboloid \{ x \in \mathbb{R}^{n,1} : \langle x, x \rangle = -1, x_0 > 0 \} with two-dimensional linear subspaces of the ambient Minkowski space \mathbb{R}^{n,1} that pass through the origin.^[3]^[11] These subspaces are precisely the planes spanned by a point p \in H^n on the geodesic and a nonzero tangent vector v \in T_p H^n at that point.^[12] Such intersections yield the shortest paths on the hyperboloid, analogous to straight lines in Euclidean geometry.^[11] A unit-speed parametrization of a geodesic \gamma passing through an initial point p \in H^n in the direction of a unit space-like vector q \in T_p H^n (satisfying \langle p, q \rangle = 0 and \langle q, q \rangle = 1) is given by

\gamma(t) = \cosh t \, p + \sinh t \, q,

for t \in \mathbb{R}.^[12] This curve lies entirely on the hyperboloid, as \langle \gamma(t), \gamma(t) \rangle = -1 holds for all t, and it traces the intersection with the plane spanned by p and q.^[12] The arc length of \gamma from t = 0 to t = s is exactly s, obtained by integrating the induced Riemannian metric element ds along the path, which yields the hyperbolic distance between \gamma(0) and \gamma(s).^[3]^[12] Geodesics in this model are complete, extending infinitely in both directions without boundary, and minimizing, providing the unique shortest path between any two points on H^n.^[11] For any two distinct points p, q \in H^n, there exists a unique such geodesic connecting them, determined by the plane they span with the origin.^[3] This uniqueness follows from the positive definiteness of the induced metric and the simply connected nature of hyperbolic space.^[11] In the two-dimensional case of H^2, geodesics can be visualized as hyperbolic arcs within specific coordinate planes, such as the (x_1, x_3)-plane where the intersection with the hyperboloid x_1^2 + x_2^2 - x_3^2 = -1, x_3 > 0, yields branches of the hyperbola x_1^2 - x_3^2 = -1.^[3] These curves illustrate the exponential divergence characteristic of hyperbolic geometry, with distances growing rapidly away from the starting point.^[11]

Horospheres

In the hyperboloid model of hyperbolic space \mathbb{H}^n, a horosphere is defined as the intersection of the upper hyperboloid sheet \{ x \in \mathbb{R}^{n,1} : \langle x, x \rangle = -1, x_0 > 0 \} with an affine hyperplane tangent to the light cone \{ y \in \mathbb{R}^{n,1} : \langle y, y \rangle = 0 \} at an ideal point on the boundary at infinity.^[4]^[13] Such ideal points correspond to light-like directions, and the tangency ensures the hyperplane is parallel to the asymptotic direction of geodesics approaching that point.^[14] Horospheres exhibit several key properties in this model. They are flat hypersurfaces with zero Gaussian curvature, making them intrinsically isometric to Euclidean space \mathbb{E}^{n-1}.^[14]^[4] Additionally, each horosphere is equidistant from its defining ideal point, serving as a boundary-parallel surface that foliates \mathbb{H}^n into a family of parallel horospheres approaching the same point at infinity.^[15] This foliation covers the entire space without overlap, except at the ideal boundary.^[13] The metric induced on a horosphere from the hyperbolic metric is Euclidean, reflecting its flat geometry.^[14] However, when viewed extrinsically in the ambient hyperbolic space, regions on the horosphere exhibit area growth that is exponential in the hyperbolic distance from a reference point on the horosphere; specifically, for constant curvature -1, the area scales as e^{(n-1)d} where d is the hyperbolic distance along the foliation direction.^[14]^[4] This exponential expansion underscores the diverging nature of hyperbolic geometry near the boundary. Horospheres are intimately related to Busemann functions in the hyperboloid model. The Busemann function b_\xi : \mathbb{H}^n \to \mathbb{R} associated with an ideal point [\xi](/page/Xi) at infinity is defined as b_\xi(x) = \lim_{t \to \infty} \left( d(x, \gamma(t)) - t \right), where \gamma is a geodesic ray from a base point to \xi, and d is the hyperbolic distance.^[15] Horospheres centered at \xi are precisely the level sets \{ x \in \mathbb{H}^n : b_\xi(x) = c \} for constants c \in \mathbb{R}, providing a signed distance function orthogonal to the horospheres.^[15]^[4] In the case of \mathbb{H}^2, a horosphere in the hyperboloid model appears as the intersection of the hyperboloid sheet with a plane in \mathbb{R}^{2,1} tangent to the light cone at an ideal point, resulting in a hyperbolic arc that asymptotically approaches the boundary light rays while maintaining the Euclidean metric intrinsically.^[13] This view emphasizes the horosphere's role as a flat frontier parallel to the ideal boundary, distinct from its curved projections in other models.^[14]

Isometries

Lorentz group

The isometries of the hyperboloid model are the linear transformations of \mathbb{R}^{n+1} that preserve the quadratic form Q(x) = -x_0^2 + \sum_{i=1}^n x_i^2, forming the indefinite orthogonal group O(n,1) under the mostly plus signature convention.^[4] These transformations restrict to diffeomorphisms of the hyperboloid H^n = \{ x \in \mathbb{R}^{n+1} : Q(x) = -1, x_0 > 0 \} that preserve the induced Riemannian metric, thereby acting as isometries of the hyperbolic space.^[16] The group O(n,1) consists of four connected components, distinguished by the sign of the determinant (orientation-preserving or reversing) and the preservation of the forward light cone (time-orientation). The proper orthochronous component, denoted SO^+(n,1), comprises the orientation-preserving isometries that maintain the time direction and acts transitively on H^n, meaning any point in H^n can be mapped to any other via an element of this subgroup.^[16] Furthermore, SO^+(n,1) acts transitively on the unit tangent bundle of H^n, allowing any tangent vector at a point to be mapped to any other tangent vector of the same length at another point.^[4] The Lie algebra of O(n,1) is \mathfrak{so}(n,1), whose infinitesimal generators correspond to spatial rotations in the Euclidean subspaces (compact generators) and Lorentz boosts, which are hyperbolic rotations in the time-space planes (noncompact generators).^[17] The exponential map from \mathfrak{so}(n,1) to SO^+(n,1) is surjective, reflecting the connected nature of the proper orthochronous subgroup.^[16]

Classification of isometries

The isometries of the hyperboloid model can be classified based on their action and fixed points, corresponding to elements of the orthogonal group O(n,1) that preserve the upper sheet of the hyperboloid. Orientation-reversing isometries include reflections, which are orthogonal reflections across hyperplanes orthogonal to space-like vectors in the ambient Minkowski space; these fix the entire hyperplane pointwise and reverse orientation.^[18] Orientation-preserving isometries fall into three main categories: elliptic, hyperbolic (or boosts), and parabolic. Elliptic isometries, often called rotations, act as rotations in space-like planes within the Minkowski space, generating compact subgroups isomorphic to SO(n); they fix a point in the hyperbolic space and rotate the orthogonal subspace around it.^[11] For example, in the hyperboloid model of H^n, such a rotation preserves a geodesic through the fixed point and acts as a Euclidean rotation on the tangent space at that point.^[4] Hyperbolic isometries, known as boosts or translations, operate as hyperbolic rotations in time-like planes, displacing points exponentially along a unique invariant geodesic (the axis); they have no fixed points in the space but fix two ideal points at infinity.^[19] In the hyperboloid model of H^2 embedded in Minkowski space with metric signature (2,1), a boost along the x-axis takes the form

\begin{pmatrix} \cosh \theta & \sinh \theta & 0 \\ \sinh \theta & \cosh \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

in the basis (x_0, x_1, x_2), where θ parameterizes the translation distance, preserving the Lorentz inner product and mapping the hyperboloid to itself.^[19] Parabolic isometries correspond to symmetries of horospheres, fixing exactly one ideal point at infinity and inducing Euclidean motions (translations and rotations) on the horosphere; they have no fixed points in the hyperbolic space and translate along horocycles.^[11] In the hyperboloid model, these arise from linear transformations involving null directions in Minkowski space, preserving the light cone structure.^[4] Discrete subgroups of these isometries, such as Fuchsian groups in dimension 2 or Poincaré groups in higher dimensions, generate fundamental domains and tessellations in the hyperboloid model, often used to study hyperbolic manifolds.^[19]

History

Precursors and early formulations

The foundations of hyperbolic geometry, which underpins the hyperboloid model, trace back to efforts in the 18th and 19th centuries to challenge Euclid's parallel postulate. In 1766, Johann Heinrich Lambert explored a geometry where the sum of angles in a triangle is less than π, conceptualizing it on an "imaginary sphere" with radius involving √(-1), where distances become hyperbolic functions; this provided an early analytic precursor to non-Euclidean spaces, though Lambert aimed to disprove such geometries via contradiction.^[20]^[2] The explicit discovery of hyperbolic geometry occurred independently in the early 19th century, prompting a search for concrete models to demonstrate its consistency within Euclidean framework. Nikolai Lobachevsky published the first systematic treatment in 1829, developing trigonometry and axioms for a geometry allowing multiple parallels, while János Bolyai outlined a similar "absolute geometry" in 1832 as an appendix to his father's work.^[21]^[22] These advances, building on earlier hints like those from Johann Heinrich Lambert and Ferdinand Karl von Prillwitz (as Taurinus in 1826), spurred mathematicians to seek embeddings in higher-dimensional Euclidean or projective spaces.^[2] Wilhelm Killing advanced this quest in the late 1870s and 1880s by embedding the hyperbolic plane in three-dimensional space using quadratic forms. Drawing from Karl Weierstrass's 1872 lectures on coordinates, Killing introduced Weierstrass coordinates in 1878–1880 to represent the Lobachevskian plane on a hyperboloid sheet via the quadratic form x² + y² - z² = -1, enabling computations in non-Euclidean spaces; he formalized this in his 1885 monograph Die Nicht-Euklidischen Raumformen, treating the hyperboloid as a surface of constant negative curvature.^[23]^[2] Henri Poincaré independently formulated the hyperboloid model around 1880–1881 in unpublished notes, later publishing in 1881 on the invariance of hyperbolic structures under quadratic forms and linking it to Fuchsian groups in his 1882 paper on automorphic functions.^[23] These works connected the model to projective geometry and complex analysis, projecting the hyperboloid onto disk representations for visualizing group actions.^[2] Contemporary contributions further refined coordinate systems and applications. In 1882, Homersham Cox developed homogeneous coordinates for "imaginary geometry," applying them to force systems and implicitly supporting hyperboloid embeddings. Alfred Clebsch and Ferdinand Lindemann, in their 1891 edition of Clebsch's lectures, discussed quadratic relations like x₁² + x₂² - 4k² x₃² = -4k², exposing the model's projective properties and metric implications.^[24] Alexander Macfarlane, in his 1894 Papers on Space Analysis, treated the hyperboloid explicitly as a metric space, deriving the hyperbolic law of cosines (cosh c = cosh a cosh b - sinh a sinh b cos C) from versor algebra and area ratios, emphasizing its utility for angle definitions in non-Euclidean contexts.^[25]

Modern interpretations

In the early 20th century, the hyperboloid model gained prominence through its integration into the framework of special relativity, particularly via Hermann Minkowski's formulation of spacetime in 1907. Minkowski linked the hyperboloid to the future light cone in Minkowski space, representing the space of four-velocities where points on the hyperboloid correspond to observers with constant proper time, providing a geometric interpretation of relativistic kinematics.^[26]^[27] Building on this, H. Jansen explicitly focused on the hyperboloid as a model for hyperbolic geometry in his 1909 paper, offering the first detailed exposition of its properties for the two-dimensional case by deriving geodesics, distances, and angles directly from the Lorentzian metric on the hyperboloid sheet.^[28] This work emphasized the model's advantages in computations involving the Lorentz group, bridging pure geometry with relativistic applications. Vladimir Varićak further advanced these ideas in 1912 by applying the hyperboloid model to relativistic kinematics, reinterpreting velocity addition and transformations using hyperbolic functions on the model to avoid complex numbers and align with non-Euclidean interpretations of Lorentz boosts.^[29] Post-relativity developments included historical analyses such as W.F. Reynolds' 1993 recount, which traced the model's evolution from 19th-century precursors to its relativistic embeddings while highlighting its computational tractability.^[2] The model also integrates with other hyperbolic representations, such as the Poincaré disk, through projective mappings that preserve the metric up to conformal factors, enabling transitions between models for visualization and analysis.^[30] In contemporary applications, the hyperboloid model facilitates the rendering of hyperbolic tilings in computer graphics, where its embedding in Minkowski space allows efficient generation of infinite tessellations projected onto Euclidean displays for games and visualizations.^[31] Additionally, in quantum field theory, the forward hyperboloid serves as a quantization surface in point-form approaches, enabling Lorentz-invariant formulations of scalar fields on spacetimes with compactly generated Cauchy horizons.^[32] These uses illustrate the model's evolution from a geometric curiosity to a versatile tool in physics and computation.^[2]

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