A projection is a mapping between linear spaces, which preserves collinearity, and whose restriction to any line is injective or constant. Motivated by applications in photogrammetry, we prove that any projection from a subset M of a (little) Desarguesian projective space to another such space is a product of a central projection and an isomorphism, provided M is not too small and the image of M is not contained in one line. 1. EINLEITUNG Photographische Bilder werden meist als Resultate von Zentralprojektionen angesehen. Dies ist gerechtfertigt fiir Bilder einer Lochkamera; bei einer Kamera mit Objektiv jedoch weicht die Abbildung gew6hnlich etwas vonder Zentralprojektion ab. Im Idealfall l~iBt sich diese Verzeichnung genannte Abweichung bestimmen, so dab man sie bei einer Bildauswertung rechnerisch ausgleichen kann. Einige Schwierigkeiten bei der Bestimmung der Verzeichnung werden in [13] beschrieben. Nun sind aber viele Rekonstruktionsaufgaben schon unter der Voraussetzung 16sbar, dab die photographische Abbildung durch eine lineare Abbildung f: ~4 ~ ~3 gegeben ist, wenn man sowohl im abgebildeten Raum als auch in der Bildebene homogene Koordinaten einfiihrt. Man kann n~imlich f bestimmen, wenn man sechs PaBpunkte im Raum, keine vier davon in einer Ebene, samt ihren Bildpunkten kennt und die Kamera sich nicht auf der kubischen Normkurve durch die PaBpunkte befindet (siehe [2] und [7] sowie §4 der vorliegenden Arbeit). Aus diesem Zusammenhang ergibt sich die Bedeutung des folgenden Satzes. SATZ 3. Seien P und P' reelle projektive Rdume, M ~ P often und ~o: M ~ P' eine Abbildung derart, dab die Bilder kollinearer Punkte stets kollinear sind, die Einschrfinkung yon q) auf Geraden injektiv oder konstant ist und Bild q~ drei nicht kollineare Punkte enthfilt. Dann wird q) in homogenen Koordinaten durch eine lineare Abbildung induziert. Dieser Satz wurde f/Jr den Fall dimP < m von H. Lenz [16, Hilfssatz 3] bewiesen. Seine Behauptung folgt auch aus [17, Theorem], wenn man zus~itzlich dim P --dim P' < mund ~o stetig und injektiv voraussetzt; sie folgt aus [4, Satz 1] oder [20, Satz 2.3], falls es einen Teilraum Z c P mit M = P\Z gibt und die Bilder yon Geraden und punktierten Geraden stets Geraden oder Punkte sind. Da mir die Arbeit yon H. Lenz unbekannt war, suchte und Geometriae Dedicata 44: 53-66, 1992.
Lumbar spine biomechanics during the forward-bending of the upper body (flexion) are well investigated by both in vivo and in vitro experiments. In both cases, the experimentally observed relative motion of vertebral bodies can be used to calculate the instantaneous center of rotation (ICR). The timely evolution of the ICR, the centrode, is widely utilized for validating computer models and is thought to serve as a criterion for distinguishing healthy and degenerative motion patterns. While in vivo motion can be induced by physiological active structures (muscles), in vitro spinal segments have to be driven by external torque-applying equipment such as spine testers. It is implicitly assumed that muscle-driven and torque-driven centrodes are similar. Here, however, we show that centrodes qualitatively depend on the impetus. Distinction is achieved by introducing confidence regions (ellipses) that comprise centrodes of seven individual multi-body simulation models, performing flexion with and without preload. Muscle-driven centrodes were generally directed superior–anterior and tail-shaped, while torque-driven centrodes were located in a comparably narrow region close to the center of mass of the caudal vertebrae. We thus argue that centrodes resulting from different experimental conditions ought to be compared with caution. Finally, the applicability of our method regarding the analysis of clinical syndromes and the assessment of surgical methods is discussed.
The purpose of this paper is to generalize the following statement: Given two points a and b in the Euclidean plane, every proper motion of the plane can be written as a product of rotations which fix either a or b. While similar statements have been proved for various plane geometries (see [1, p. 164], [4] and [5]), we consider metric vector spaces of arbitrary dimension. For the results see Theorems 1, 2, 3 and Propositions 5, 6, 7.Let V be a vector space over a field K and let q: V--+ K be a quadraticfor any asV and VI: ={vEVIf(v,x) = 0 for every x~V}. For every aeV we have a translation r,: V ~ V; x ~ x + a. If q(a) ¢ 0, we also have a reflection era:For a(sV ± the product r-ba~zb is a reflection in the affine hyperplane a ± + b. The product of two reflections fixing a common point c~V is a rotation fixing c. We set T: = {%la~V}, S: = {a, lasV and q(a)v L 0}, and $2: = {aaablaa, abeS}. The group (Sw T) generated by Sw T is the motion group, and M: = (S2w T) is the proper motion group of the metric vector space (V, q), which for char K = 2 may coincide with the motion group. Given a, bEV, we investigate the following two questions:(i) Is M generated by the rotations fixin 9 a or b? (ii) Is M generated by the rotations fixing b and the translation ra-b?Without loss of generality we may assume b = 0. In order to avoid trivial cases, we assume dim V >~ 1. M operates faithfully on V. The stabilizer of a s Vis denoted by Ma. Hence Mo = (S 2) and Ma = Tal(S 2 )~'al and our questions are the following: (i) ls M generated by M o ~ M~? (ii) Is M generated by M o ~ {r~}? REMARK. Since (MouM,) ~ (Mou {%}), (ii) has a positive answer if (i) has one. NOTATIONS. B(a): = {ap l p~ ( SZ ) } F(a): = {v e VN V ± I q(v) = q(a)} (B(a)): the additive group generated by B(a) Geometriae Dedicata 22 (1987) 22~233.
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