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Real projective line
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Real projective line
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The real projective line can be modeled by the projectively extended real line, which consists of the real line together with a point at infinity; i.e., the one-point compactification of R.

In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity". For solving this problem, points at infinity have been introduced, in such a way that in a real projective plane, two distinct projective lines meet in exactly one point. The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line. It is the set of directions emanating from an observer situated at any point, with opposite directions identified.

An example of a real projective line is the projectively extended real line, which is often called the projective line.

Formally, a real projective line P(R) is defined as the set of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. The automorphisms of a real projective line are called projective transformations, homographies, or linear fractional transformations. They form the projective linear group PGL(2, R). Each element of PGL(2, R) can be defined by a nonsingular 2×2 real matrix, and two matrices define the same element of PGL(2, R) if one is the product of the other and a nonzero real number.

Topologically, real projective lines are homeomorphic to circles. The complex analog of a real projective line is a complex projective line, also called a Riemann sphere.

Definition

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The points of the real projective line are usually defined as equivalence classes of an equivalence relation. The starting point is a real vector space of dimension 2, V. Define on V ∖ 0 the binary relation v ~ w to hold when there exists a nonzero real number t such that v = tw. The definition of a vector space implies almost immediately that this is an equivalence relation. The equivalence classes are the vector lines from which the zero vector has been removed. The real projective line P(V) is the set of all equivalence classes. Each equivalence class is considered as a single point, or, in other words, a point is defined as being an equivalence class.

If one chooses a basis of V, this amounts (by identifying a vector with its coordinate vector) to identify V with the direct product R × R = R2, and the equivalence relation becomes (x, y) ~ (w, z) if there exists a nonzero real number t such that (x, y) = (tw, tz). In this case, the projective line P(R2) is preferably denoted P1(R) or . The equivalence class of the pair (x, y) is traditionally denoted [x: y], the colon in the notation recalling that, if y ≠ 0, the ratio x : y is the same for all elements of the equivalence class. If a point P is the equivalence class [x: y] one says that (x, y) is a pair of projective coordinates of P.[1]

As P(V) is defined through an equivalence relation, the canonical projection from V to P(V) defines a topology (the quotient topology) and a differential structure on the projective line. However, the fact that equivalence classes are not finite induces some difficulties for defining the differential structure. These are solved by considering V as a Euclidean vector space. The circle of the unit vectors is, in the case of R2, the set of the vectors whose coordinates satisfy x2 + y2 = 1. This circle intersects each equivalence classes in exactly two opposite points. Therefore, the projective line may be considered as the quotient space of the circle by the equivalence relation such that v ~ w if and only if either v = w or v = −w.

See also: projectivization

Charts

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The projective line is a manifold. This can be seen by above construction through an equivalence relation, but is easier to understand by providing an atlas consisting of two charts

  • Chart #1:
  • Chart #2:

The equivalence relation provides that all representatives of an equivalence class are sent to the same real number by a chart.

Either of x or y may be zero, but not both, so both charts are needed to cover the projective line. The transition map between these two charts is the multiplicative inverse. As it is a differentiable function, and even an analytic function (outside of zero), the real projective line is both a differentiable manifold and an analytic manifold.

The inverse function of chart #1 is the map

It defines an embedding of the real line into the projective line, whose complement of the image is the point [1: 0]. The pair consisting of this embedding and the projective line is called the projectively extended real line. Identifying the real line with its image by this embedding, one sees that the projective line may be considered as the union of the real line and the single point [1: 0], called the point at infinity of the projectively extended real line, and denoted ∞. This embedding allows us to identify the point [x: y] either with the real number ⁠x/y⁠ if y ≠ 0, or with ∞ in the other case.

The same construction may be done with the other chart. In this case, the point at infinity is [0: 1]. This shows that the notion of point at infinity is not intrinsic to the real projective line, but is relative to the choice of an embedding of the real line into the projective line.

Structure

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Same color means same point.

Points of the real projective line can be associated with pairs of antipodal points on a circle. Generally, a projective n-space is formed from antipodal pairs on a sphere in (n+1)-space; in this case the sphere is a circle in the plane.

The real projective line is a complete projective range that is found in the real projective plane and in the complex projective line. Its structure is thus inherited from these superstructures. Primary among these structures is the relation of projective harmonic conjugates among the points of the projective range.

The real projective line has a cyclic order that extends the usual order of the real numbers.

Automorphisms

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The projective linear group and its action

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Matrix-vector multiplication defines a right action of GL2(R) on the space R2 of row vectors: explicitly,

Since each matrix in GL2(R) fixes the zero vector and maps proportional vectors to proportional vectors, there is an induced action of GL2(R) on P1(R): explicitly,[2]

(Here and below, the notation for homogeneous coordinates denotes the equivalence class of the row vector.

The elements of GL2(R) that act trivially on P1(R) are the nonzero scalar multiples of the identity matrix; these form a subgroup denoted R×. The projective linear group is defined to be the quotient group PGL2(R) = GL2(R)/R×. By the above, there is an induced faithful action of PGL2(R) on P1(R). For this reason, the group PGL2(R) may also be called the group of linear automorphisms of P1(R).

Linear fractional transformations

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Using the identification R ∪ ∞ → P1(R) sending x to [x:1] and ∞ to [1:0], one obtains a corresponding action of PGL2(R) on R ∪ ∞ , which is by linear fractional transformations: explicitly, since

the class of in PGL2(R) acts as with the understanding that each fraction with denominator 0 should be interpreted as ∞.[3] and .[4]

Some authors use left action on column vectors which entails switching b and c in the matrix operator.[5][6][7]

Properties

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  • Given two ordered triples of distinct points in P1(R), there exists a unique element of PGL2(R) mapping the first triple to the second; that is, the action is sharply 3-transitive. For example, the linear fractional transformation mapping (0, 1, ∞) to (−1, 0, 1) is the Cayley transform .
  • The stabilizer in PGL2(R) of the point ∞ is the affine group of the real line, consisting of the transformations for all a ∈ R* and b ∈ R.

See also

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Notes

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Real projective line

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The real projective line, denoted RP1\mathbb{RP}^1, is a fundamental object in projective geometry, defined as the set of all one-dimensional subspaces (lines through the origin) of the two-dimensional real vector space R2\mathbb{R}^2. Formally, its points are equivalence classes of nonzero vectors (x,y)∈R2∖{(0,0)}(x, y) \in \mathbb{R}^2 \setminus \{(0,0)\}, where two vectors are identified if one is a nonzero real scalar multiple of the other, often represented in homogeneous coordinates as [x:y][x : y] with not both coordinates zero. Topologically, RP1\mathbb{RP}^1 is homeomorphic to the unit circle S1S^1, obtained by identifying antipodal points on the sphere or by compactifying the real line R\mathbb{R} with a single point at infinity that merges positive and negative infinities. This structure arises naturally in various mathematical contexts, such as the completion of the affine line to handle points at infinity in projective transformations, ensuring that parallel lines intersect at a unique "ideal" point. In algebraic geometry, RP1\mathbb{RP}^1 parameterizes lines in the plane and serves as the simplest non-trivial real projective space, with dimension 1, distinguishing it from higher-dimensional analogs like the real projective plane RP2\mathbb{RP}^2. Its coordinate charts typically consist of two open sets: one covering finite slopes via [1:m][1 : m] for m∈Rm \in \mathbb{R}, and another for the vertical direction via [n:1][n : 1] for n∈Rn \in \mathbb{R}, with transition maps given by inversion m↦1/mm \mapsto 1/m. Notably, RP1\mathbb{RP}^1 is orientable and compact, mirroring the topology of a circle, and it admits a canonical line bundle of rank 1, which plays a role in studying vector bundles over projective spaces. It also provides an entry point for understanding more complex projective varieties, such as the complex projective line CP1≅S2\mathbb{CP}^1 \cong S^2, by analogy over different fields.

Definition and Construction

Homogeneous Coordinates

The real projective line, denoted RP1\mathbb{RP}^1, is formally defined as the quotient space (R2∖{0})/∼(\mathbb{R}^2 \setminus \{\mathbf{0}\}) / \sim, where the equivalence relation ∼\sim identifies points (x,y)(x, y) and (x′,y′)(x', y') if there exists a nonzero scalar λ∈R∖{0}\lambda \in \mathbb{R} \setminus \{0\} such that (x′,y′)=λ(x,y)(x', y') = \lambda (x, y). This construction arises from viewing points in RP1\mathbb{RP}^1 as lines through the origin in R2\mathbb{R}^2. Points in RP1\mathbb{RP}^1 are represented using homogeneous coordinates [x:y][x : y], where [x:y]=[λx:λy][x : y] = [\lambda x : \lambda y] for any λ∈R∖{0}\lambda \in \mathbb{R} \setminus \{0\}. These coordinates provide an algebraic framework for the space, allowing normalization such that, for instance, points with y≠0y \neq 0 can be written as [1:t][1 : t] for t=x/y∈Rt = x/y \in \mathbb{R}. A distinguished point in RP1\mathbb{RP}^1 is the infinite point [1:0][1 : 0], which corresponds to the equivalence class of all vectors (x,0)(x, 0) with x≠0x \neq 0. This construction distinguishes RP1\mathbb{RP}^1 from the affine line R\mathbb{R} by adjoining the point at infinity, effectively compactifying the line and incorporating projective behavior where parallel lines meet.

Geometric Interpretation

The real projective line, denoted RP1\mathbb{RP}^1, can be geometrically interpreted as the set of all one-dimensional subspaces of R2\mathbb{R}^2, or equivalently, the set of all lines passing through the origin in the Euclidean plane. Each such line represents a direction in the plane, without regard to magnitude or specific position along the line, capturing the essence of projective geometry where parallel lines are considered to meet at infinity. This construction emphasizes that points in RP1\mathbb{RP}^1 correspond to equivalence classes of nonzero vectors in R2\mathbb{R}^2 under scalar multiplication by nonzero reals, unifying affine points with ideal points at infinity. A key feature of this interpretation is its duality with points and lines in the affine plane: the real projective line models the space of all lines through a fixed point in the RP2\mathbb{RP}^2, where each point in RP1\mathbb{RP}^1 dualizes to a line in RP2\mathbb{RP}^2 incident to that fixed point. This duality principle, central to , interchanges the roles of points and lines while preserving incidence relations, allowing theorems about points on lines to have dual counterparts about lines through points. serve as an algebraic tool for representing these geometric entities in this dual framework. Visually, RP1\mathbb{RP}^1 can be realized through central projection from the unit circle S1S^1 in R2\mathbb{R}^2, where antipodal points on the circle are identified, effectively quotienting the circle by the action that maps each point to its opposite. This identification arises naturally from the line-through-origin model, as each line intersects the circle at two antipodal points, collapsing them into a single projective point; the resulting space is a closed loop akin to a circle, providing an intuitive embedding of infinite directions. This geometric viewpoint originated in the synthetic approach to developed by Karl Georg Christian von Staudt in his 1847 work Geometrie der Lage, which axiomatized projective structures without relying on metric concepts, establishing the projective line as a foundational primitive for higher-dimensional spaces.

Topological Structure

Atlas and Charts

The real projective line RP1\mathbb{RP}^1 can be endowed with a smooth manifold structure using homogeneous coordinates [x:y][x:y], where (x,y)∈R2∖{(0,0)}(x, y) \in \mathbb{R}^2 \setminus \{(0,0)\} and [x:y]=[λx:λy][x:y] = [\lambda x : \lambda y] for λ≠0\lambda \neq 0. To define this structure, RP1\mathbb{RP}^1 is covered by two open sets that form the domains of coordinate charts. The first open set is U1={[x:y]∈RP1∣y≠0}U_1 = \{ [x:y] \in \mathbb{RP}^1 \mid y \neq 0 \}, with the chart ϕ1:U1→R\phi_1: U_1 \to \mathbb{R} given by ϕ1([x:y])=x/y\phi_1([x:y]) = x/y. The second open set is U2={[x:y]∈RP1∣x≠0}U_2 = \{ [x:y] \in \mathbb{RP}^1 \mid x \neq 0 \}, with the chart ϕ2:U2→R\phi_2: U_2 \to \mathbb{R} given by ϕ2([x:y])=y/x\phi_2([x:y]) = y/x. These sets cover RP1\mathbb{RP}^1 because every point has at least one nonzero homogeneous coordinate. The transition map between these charts is defined on the nonempty intersection U1∩U2=RP1∖{[1:0]}U_1 \cap U_2 = \mathbb{RP}^1 \setminus \{[1:0]\}, which corresponds to R∖{0}\mathbb{R} \setminus \{0\} under ϕ1\phi_1 (or ϕ2\phi_2). Specifically, the map ϕ2∘ϕ1−1:ϕ1(U1∩U2)→ϕ2(U1∩U2)\phi_2 \circ \phi_1^{-1}: \phi_1(U_1 \cap U_2) \to \phi_2(U_1 \cap U_2) is given by ϕ2∘ϕ1−1(t)=1t,t≠0.\phi_2 \circ \phi_1^{-1}(t) = \frac{1}{t}, \quad t \neq 0. This function is a smooth diffeomorphism from R∖{0}\mathbb{R} \setminus \{0\} onto itself, as it is infinitely differentiable with a nonzero derivative −1/t2-1/t^2 everywhere in its domain. The inverse transition ϕ1∘ϕ2−1(s)=1/s\phi_1 \circ \phi_2^{-1}(s) = 1/s is similarly smooth. The atlas A={(U1,ϕ1),(U2,ϕ2)}\mathcal{A} = \{ (U_1, \phi_1), (U_2, \phi_2) \} thus consists of compatible charts, as the transition functions are smooth. Since the charts map to open subsets of R\mathbb{R} and RP1\mathbb{RP}^1 is Hausdorff and second-countable, this equips RP1\mathbb{RP}^1 with the structure of a 1-dimensional smooth manifold without boundary.

Homeomorphism to the Circle

The real projective line RP1\mathbb{RP}^1 is topologically equivalent to the circle S1S^1, as it can be realized as the quotient space S1/∼S^1 / \sim, where ∼\sim denotes the identification of antipodal points z∼−zz \sim -z for z∈S1⊂Cz \in S^1 \subset \mathbb{C}. This construction arises from the geometric interpretation of RP1\mathbb{RP}^1 as the set of lines through the origin in R2\mathbb{R}^2, with the sphere S1S^1 parameterizing directions and the antipodal identification accounting for the unsigned nature of lines. To exhibit an explicit homeomorphism ψ:RP1→S1\psi: \mathbb{RP}^1 \to S^1, consider the continuous surjective map f:S1→S1f: S^1 \to S^1 defined by f(z)=z2f(z) = z^2. The preimage of any point under ff consists precisely of an antipodal pair {z,−z}\{z, -z\}, making ff a two-sheeted covering map whose fibers match the equivalence classes of ∼\sim. Thus, ff descends to a well-defined map ψ:S1/∼→S1\psi: S^1 / \sim \to S^1 by ψ()=z2\psi() = z^2, where $$ denotes the equivalence class. Bijectivity of ψ\psi follows directly: it is surjective because every w∈S1w \in S^1 satisfies w=z2w = z^2 for some z∈S1z \in S^1, and injective since distinct classes [z1][z_1] and [z2][z_2] satisfy z12=z22z_1^2 = z_2^2 only if {z1,−z1}={z2,−z2}\{z_1, -z_1\} = \{z_2, -z_2\}. Continuity of ψ\psi is inherited from the quotient topology, as the quotient map q:S1→S1/∼q: S^1 \to S^1 / \sim is continuous and f=ψ∘qf = \psi \circ q is continuous. The inverse ψ−1:S1→RP1\psi^{-1}: S^1 \to \mathbb{RP}^1 can be constructed by selecting, for each w∈S1w \in S^1, the representative [w][ \sqrt{w} ]
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