{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:6VRML5HQJP73Z4P4P7LH5KWGPD","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2b2413b9fc8ec68a0d3afeb32f79b01d55e8689acf4c85b0c174dfa8cbfb040d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-06-09T19:15:54Z","title_canon_sha256":"2503828c6f229ac9876b6ca613cdeca1444f1de5c515679d0f43d065b02dc9bc"},"schema_version":"1.0","source":{"id":"1706.03095","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.03095","created_at":"2026-05-18T00:42:38Z"},{"alias_kind":"arxiv_version","alias_value":"1706.03095v1","created_at":"2026-05-18T00:42:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.03095","created_at":"2026-05-18T00:42:38Z"},{"alias_kind":"pith_short_12","alias_value":"6VRML5HQJP73","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"6VRML5HQJP73Z4P4","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"6VRML5HQ","created_at":"2026-05-18T12:31:03Z"}],"graph_snapshots":[{"event_id":"sha256:23fb4a4817f69fdb1a61e2a38c0a789f3901d7ee56d06fe21b4c72c9f2cb4414","target":"graph","created_at":"2026-05-18T00:42:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we study the shape space of curves with values in a homogeneous space $M = G/K$, where $G$ is a Lie group and $K$ is a compact Lie subgroup. We generalize the square root velocity framework to obtain a reparametrization invariant metric on the space of curves in $M$. By identifying curves in $M$ with their horizontal lifts in $G$, geodesics then can be computed. We can also mod out by reparametrizations and by rigid motions of $M$. In each of these quotient spaces, we can compute Karcher means, geodesics, and perform principal component analysis. We present numerical examples inc","authors_text":"Eric Klassen, Martin Bauer, Zhe Su","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-06-09T19:15:54Z","title":"The Square Root Velocity Framework for Curves in a Homogeneous Space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03095","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:648f41839596211cbb263c495ec17eee5089c0c59d22094d04b95a74e2874da9","target":"record","created_at":"2026-05-18T00:42:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2b2413b9fc8ec68a0d3afeb32f79b01d55e8689acf4c85b0c174dfa8cbfb040d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-06-09T19:15:54Z","title_canon_sha256":"2503828c6f229ac9876b6ca613cdeca1444f1de5c515679d0f43d065b02dc9bc"},"schema_version":"1.0","source":{"id":"1706.03095","kind":"arxiv","version":1}},"canonical_sha256":"f562c5f4f04bffbcf1fc7fd67eaac678d9a78cbea706c671dac7beb0e1846012","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f562c5f4f04bffbcf1fc7fd67eaac678d9a78cbea706c671dac7beb0e1846012","first_computed_at":"2026-05-18T00:42:38.514311Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:38.514311Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lh1dbdB9gTX+6q6jLx8RIaACu/dwcfyZnUHckzowjIuCfz4FFfG9Raw2dSMp9LMI60sgYGNS9jS9llyMwmfODA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:38.514886Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.03095","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:648f41839596211cbb263c495ec17eee5089c0c59d22094d04b95a74e2874da9","sha256:23fb4a4817f69fdb1a61e2a38c0a789f3901d7ee56d06fe21b4c72c9f2cb4414"],"state_sha256":"e5aa1744f73ae8237456b55900bc3a73961eff08418b41ae56baf4c5aa7009af"}