{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:ce618650-9f53-49e2-ab6f-718ffe06a89e", "labels": { "zh-CN": "实射影直线", "en": "real projective line" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "18f146b99a0b619e4cbc4afc3d0ef22962a6a27468097fce81d651c4412fb64b", "occurrences": [ { "concept_id": "urn:uuid:ce618650-9f53-49e2-ab6f-718ffe06a89e", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "anchor": "first-cdbf26767d2e19aff4dc-zh", "chapter": "7", "section": "7.4", "offset": 3364, "range": [ 11444, 11754 ], "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "实射影直线(real projective line)", "context": { "range": [ 11437, 11829 ], "quote": "图 7.1：[实射影直线(real projective line)](../../../../../concepts/topology/real-projective-line.md) $\\mathbf{R}P^1$", "quote_sha256": "a24915bcc5c2f7b926479196f178147aea12f386cac24a6253b1fc0c66d3837e", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-1e5804abed2fd87b3e2fcabe" }, { "concept_id": "urn:uuid:ce618650-9f53-49e2-ab6f-718ffe06a89e", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "anchor": "first-cdbf26767d2e19aff4dc-en", "chapter": "7", "section": "7.4", "offset": 3779, "range": [ 11192, 11495 ], "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "real projective line", "context": { "range": [ 11175, 11570 ], "quote": "Figure 7.1: The [real projective line](../../../../../concepts/topology/real-projective-line.md) $\\mathbf{R}P^1$", "quote_sha256": "fa2f8ab213f3c069f1ca7680d6f052f7ec562affd7e3265c651eff60cafc4b6d", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-12ae4d62cc8d5c0a15bdbb80" }, { "concept_id": "urn:uuid:ce618650-9f53-49e2-ab6f-718ffe06a89e", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "anchor": "source-7-4-real-projective-space", "chapter": "7", "section": "7.4", "offset": 3884, "range": [ 11964, 12239 ], "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "实射影直线(real projective line)", "context": { "range": [ 11905, 12899 ], "quote": "每条这样的直线都与单位圆交于一对对跖点。因此，所有这类直线可以用一个半圆参数化，并把半圆两端视为同一点。这个空间称为[实射影直线(real projective line)](../../../../../concepts/topology/real-projective-line.md) $\\mathbf{R}P^1$。在更高维中，$\\mathbf{R}^n$ 内过原点的直线所成的空间称为[实射影空间(real projective space)](../../../../../concepts/topology/real-projective-space.md) $\\mathbf{R}P^{n-1}$；它可以表示为 $\\mathbf{R}^n$ 中将对跖点视为同一点的单位球面。回顾[第 6.2.3 节](../chapter-06/02-spin-groups-in-three-and-four-dimensions#section-6-2-3)，$SO(3)$ 可与 $\\mathbf{R}P^3$ 等同。", "quote_sha256": "526ca7ab23035d12dadb94cea83e3520eb823c394b2688c166b4dcfbef13f4f1", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-71a0dd4eb271134012f40fe8" }, { "concept_id": "urn:uuid:ce618650-9f53-49e2-ab6f-718ffe06a89e", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "anchor": "source-7-4-real-projective-space", "chapter": "7", "section": "7.4", "offset": 4419, "range": [ 11832, 12100 ], "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "real projective line", "context": { "range": [ 11646, 12907 ], "quote": "Each such line intersects the unit circle at two antipodal points. Thus the set of lines can be parametrized by a semicircle with its two endpoints identified. This space is called the [real projective line](../../../../../concepts/topology/real-projective-line.md) $\\mathbf{R}P^1$. In higher dimensions, the space of lines through the origin in $\\mathbf{R}^n$ is the [real projective space](../../../../../concepts/topology/real-projective-space.md) $\\mathbf{R}P^{n-1}$; it can be represented by the unit sphere in $\\mathbf{R}^n$ with antipodal points identified. In particular, recall from [Section 6.2.3](../chapter-06/02-spin-groups-in-three-and-four-dimensions#section-6-2-3) that $SO(3)$ is identified with $\\mathbf{R}P^3$.", "quote_sha256": "729f0f81662375d9482c2161fef3095023b5fa4f75eebf6792794a3216634e39", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-5517d03e721d4d24f1b23c93" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "section": "7.4", "signal": "term-mark", "label": "real projective line", "english": "real projective line", "offset": 2582 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "section": "7.4", "signal": "term-mark", "label": "real projective line", "english": "real projective line", "offset": 3031 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "section": "7.4", "signal": "term-mark", "label": "实射影直线", "english": "real projective line", "offset": 2202 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "section": "7.4", "signal": "term-mark", "label": "实射影直线", "english": "real projective line", "offset": 2531 } ], "statements": [], "resolved_references": [ { "raw": "[第 6.2.3 节](../chapter-06/02-spin-groups-in-three-and-four-dimensions#section-6-2-3)", "href": "../chapter-06/02-spin-groups-in-three-and-four-dimensions#section-6-2-3", "range": [ 12784, 12868 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "target_anchor": "section-6-2-3", "resolved": true, "reason": null, "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "occurrence_id": "occ-71a0dd4eb271134012f40fe8" }, { "raw": "[Section 6.2.3](../chapter-06/02-spin-groups-in-three-and-four-dimensions#section-6-2-3)", "href": "../chapter-06/02-spin-groups-in-three-and-four-dimensions#section-6-2-3", "range": [ 12770, 12858 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "target_anchor": "section-6-2-3", "resolved": true, "reason": null, "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "occurrence_id": "occ-5517d03e721d4d24f1b23c93" } ], "reading_order": [ "occ-1e5804abed2fd87b3e2fcabe", "occ-12ae4d62cc8d5c0a15bdbb80", "occ-71a0dd4eb271134012f40fe8", "occ-5517d03e721d4d24f1b23c93" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 4, "located_occurrences": 4, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "70b722986d53fb59e22ed60ccadc0918a888402a587134ffb0214553d2725944", "optional_enrichments": [] }