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Moreover, in parallel kinematics robotic machines, unwished rotations about an axis \u2013 termed \u201cparasitic motions\u201d \u2013 need to be excluded. However, due to the non-Abelian nature of spatial rotations, these components cannot be extracted by simple projections as in a vector space. Despite extensive discussion in the literature about the non-uniqueness and distortion of the results due to the nonlinearity of the SO(3) group, they continue to be used due to the absence of alternatives. This paper reviews the existing methods for planar-rotation extraction from 3D rotations, showing their similarities and differences as well as inconsistencies by mathematical analysis as well as two application cases, one of them from biomechanics (flexural knee angle in the sagittal plane). Moreover, a novel, simple, and efficient method based on a pseudo-projection of the Quaternion rotation vector is introduced, which circumvents the ambiguity and distortion problems of existing approaches. In this respect, a novel method for determining the orientation of a box from camera recordings based on a two-plane projection is also proposed, which yields more precise results than the existing Perspective 3-Point Problem from the literature. This paper focuses exclusively on the case of finite rotations, as infinitesimal rotations within a single plane are non-holonomic and, through integration, produce rotation components orthogonal to the plane.<\/jats:p>","DOI":"10.1017\/s0263574724001498","type":"journal-article","created":{"date-parts":[[2024,12,12]],"date-time":"2024-12-12T03:48:21Z","timestamp":1733975301000},"page":"3994-4018","source":"Crossref","is-referenced-by-count":1,"title":["A review of methods for mapping 3D rotations to one-dimensional planar rotations: analysis, comparison, and a novel efficient Quaternion-based approach"],"prefix":"10.1017","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-1736-5929","authenticated-orcid":false,"given":"Mehdi","family":"Ghiassi","sequence":"first","affiliation":[{"name":"University of Duisburg-Essen"}]},{"given":"Andr\u00e9s","family":"Kecskem\u00e9thy","sequence":"additional","affiliation":[{"name":"University of Duisburg-Essen"}]}],"member":"56","published-online":{"date-parts":[[2024,12,12]]},"reference":[{"key":"S0263574724001498_ref6","doi-asserted-by":"publisher","DOI":"10.2514\/1.58558"},{"key":"S0263574724001498_ref21","first-page":"270","article-title":"Assessment of methods to describe the angular position of the pelvis during gait in children with hemiplegic cerebral palsy","volume":"13","author":"Foti","year":"2001","journal-title":"Gait Posture"},{"key":"S0263574724001498_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.gaitpost.2011.09.104"},{"key":"S0263574724001498_ref11","doi-asserted-by":"publisher","DOI":"10.1114\/1.1326030"},{"volume-title":"Computer-Aided Analysis of Mechanical Systems","year":"1988","author":"Nikravesh","key":"S0263574724001498_ref41"},{"key":"S0263574724001498_ref25","doi-asserted-by":"crossref","unstructured":"[25] Haralick, R. 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