{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:75e9bd12-9b58-4080-8f6c-d42cdf594c77", "labels": { "zh-CN": "实射影空间", "en": "real projective space" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "ff3e2a6e72e510c66f55669b1b879c7232b3da2a6da6046d7421add6c6e5069f", "occurrences": [ { "concept_id": "urn:uuid:75e9bd12-9b58-4080-8f6c-d42cdf594c77", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-10/04-induced-topologies", "anchor": "first-8126cdda28079489caa9-zh", "chapter": "10", "section": "10.4", "offset": 12725, "range": [ 17122, 17369 ], "source_revision": "c963ef9644060098249c7d9aa964e77eb8940ecd1a6784da795e98becb6a8f3f", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "实射影空间(real projective space)", "context": { "range": [ 16088, 17866 ], "quote": null, "quote_sha256": "0294e1092af6899278e54ed33ebf607a37343858baa4539800652d6664fbb601", "environment": null, "excerpt_omitted": true }, "parallel_readings": [ { "route": "en/chapter-10/04-induced-topologies", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-5491dac676fdab198d049be3" }, { "concept_id": "urn:uuid:75e9bd12-9b58-4080-8f6c-d42cdf594c77", "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-6cbaf7163cfb07bdaad4-zh", "chapter": "7", "section": "7.4", "offset": 4271, "range": [ 12351, 12663 ], "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "实射影空间(real projective space)", "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-1aae1b5f54498e821eb9c847" }, { "concept_id": "urn:uuid:75e9bd12-9b58-4080-8f6c-d42cdf594c77", "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-6cbaf7163cfb07bdaad4-en", "chapter": "7", "section": "7.4", "offset": 4850, "range": [ 12263, 12568 ], "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 space", "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-5a232ff472d2c358a746b872" } ], "registered_positions": [ { "route": "zh/chapter-10/04-induced-topologies", "lang": "zh-CN", "section": "10.4", "signal": "term-mark", "label": "实射影空间", "english": "real projective space", "offset": 11508 }, { "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 space", "english": "real projective space", "offset": 3306 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "section": "7.4", "signal": "term-mark", "label": "实射影空间", "english": "real projective space", "offset": 2762 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#example-10-15\"", "href": "#example-10-15", "range": [ 16129, 16150 ], "relation": "references", "target_route": "zh/chapter-10/04-induced-topologies", "target_anchor": "example-10-15", "resolved": true, "reason": null, "source_revision": "c963ef9644060098249c7d9aa964e77eb8940ecd1a6784da795e98becb6a8f3f", "source_route": "zh/chapter-10/04-induced-topologies", "occurrence_id": "occ-5491dac676fdab198d049be3" }, { "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-1aae1b5f54498e821eb9c847" }, { "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-5a232ff472d2c358a746b872" } ], "reading_order": [ "occ-5491dac676fdab198d049be3", "occ-1aae1b5f54498e821eb9c847", "occ-5a232ff472d2c358a746b872" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 3, "located_occurrences": 3, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "25b01313923f04cfd7f8c50ddc2366f59ad11711dd3e705c58cc1506c23b9aa1", "optional_enrichments": [] }