{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:6b78393e-3876-49d3-b82b-3dd0d4ff3b21", "labels": { "zh-CN": "球极投影", "en": "stereographic projection" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "94c54d1c486d13b1645c8c3c89ae2bb81f00575651f43537fec8ae4bb1f000de", "occurrences": [ { "concept_id": "urn:uuid:6b78393e-3876-49d3-b82b-3dd0d4ff3b21", "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-d59757d7da819e92dd76-zh", "chapter": "7", "section": "7.4", "offset": 6114, "range": [ 14194, 14511 ], "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "球极投影(stereographic projection)", "context": { "range": [ 13761, 14672 ], "quote": "$\\mathbf{C}P^1$ 与其说是“复平面加一个点”，不如看作一个球面，通常称为[黎曼球面(Riemann sphere)](../../../../../concepts/topology/complex-projective-space.md)。球面与复平面及无穷远点之间的关系由[球极投影(stereographic projection)](../../../../../concepts/geometry/stereographic-projection.md)给出：从球面北极向球面上各点作直线，并将这些直线投到平面，可在去掉北极的球面与整个平面之间建立一一对应。北极代表无穷远点，因此 $\\mathbf{C}P^1$ 可与 $S^2$ 等同。示意图如下。", "quote_sha256": "fec56742e2bbcec6432665e30917e3fade57bb40b30c33a2f57ff9bc416c74aa", "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-06da3bb37f386ff60021f900" }, { "concept_id": "urn:uuid:6b78393e-3876-49d3-b82b-3dd0d4ff3b21", "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-d59757d7da819e92dd76-en", "chapter": "7", "section": "7.4", "offset": 7324, "range": [ 14737, 15048 ], "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "stereographic projection", "context": { "range": [ 14197, 15401 ], "quote": "$\\mathbf{C}P^1$ is better viewed as a sphere, often called the [Riemann sphere](../../../../../concepts/topology/complex-projective-space.md), rather than as a plane with one extra point. The sphere is related to the plane and its point at infinity by [stereographic projection](../../../../../concepts/geometry/stereographic-projection.md). Projecting each point of the sphere from the north pole onto the plane gives a one-to-one correspondence between the plane and the sphere with the north pole removed. The north pole represents the point at infinity, so $\\mathbf{C}P^1$ is identified with $S^2$. The picture is shown below.", "quote_sha256": "99e05ced5ce4b4d66193bc0934c57cec48c48c1402db589f3e5e1e2936207640", "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-015511f4a2002744681aafc8" }, { "concept_id": "urn:uuid:6b78393e-3876-49d3-b82b-3dd0d4ff3b21", "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-d59757d7da819e92dd76-zh", "chapter": "7", "section": "7.4", "offset": 7418, "range": [ 15498, 15780 ], "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "球极投影(stereographic projection)", "context": { "range": [ 15406, 15887 ], "quote": "![](images/f3a6282b1c46099e490fe120ede2d79eef98fdf0356c460f13eafdf165456341.jpg) \n图 7.3：通过[球极投影(stereographic projection)](../../../../../concepts/geometry/stereographic-projection.md)得到 $\\mathbf{C}P^1\\setminus\\{\\infty\\}$ 上的复坐标。", "quote_sha256": "cd6cc4d0e01d62adc7d7be70d05c190a083d80b0ab2f0bc324660a8cd77aaac5", "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-8f52bec7d26a16d77eb6a113" }, { "concept_id": "urn:uuid:6b78393e-3876-49d3-b82b-3dd0d4ff3b21", "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-d59757d7da819e92dd76-en", "chapter": "7", "section": "7.4", "offset": 8951, "range": [ 16364, 16640 ], "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "stereographic projection", "context": { "range": [ 16206, 16704 ], "quote": "![](images/f3a6282b1c46099e490fe120ede2d79eef98fdf0356c460f13eafdf165456341.jpg) \nFigure 7.3: Complex coordinates on $\\mathbf{C}P^1\\setminus\\{\\infty\\}$ via [stereographic projection](../../../../../concepts/geometry/stereographic-projection.md).", "quote_sha256": "ef72e66148f513db486eececa250a509794fa723b72bca6b46114623425fb0e4", "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-8732cb704924dfc9e5878011" } ], "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": "stereographic projection", "english": "stereographic projection", "offset": 5207 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "section": "7.4", "signal": "term-mark", "label": "stereographic projection", "english": "stereographic projection", "offset": 6487 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "section": "7.4", "signal": "term-mark", "label": "球极投影", "english": "stereographic projection", "offset": 4032 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "section": "7.4", "signal": "term-mark", "label": "球极投影", "english": "stereographic projection", "offset": 4989 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-06da3bb37f386ff60021f900", "occ-015511f4a2002744681aafc8", "occ-8f52bec7d26a16d77eb6a113", "occ-8732cb704924dfc9e5878011" ], "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": "224d32f5060e4fcb095d7637dc626ae626c4b20b40bdbeef3650db11405044fe", "optional_enrichments": [] }