By Heiberg J.L. (ed.)

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Geometry, mechanics, and dynamics: volume in honor of the 60th birthday of J.E. Marsden

Jerry Marsden, one of many world’s pre-eminent mechanicians and utilized mathematicians, celebrated his sixtieth birthday in August 2002. the development used to be marked by means of a workshop on “Geometry, Mechanics, and Dynamics”at the Fields Institute for study within the Mathematical Sciences, of which he wasthefoundingDirector.

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We could use the normalization of a homogeneous vector with its length for an alternative 22 2 Representation of Geometric Entities and Transformations definition of the equivalence class for two homogeneous vectors We sometimes denote the spherical normalization by The definition of the equivalence class is ambiguous as the orientation of the normalized vectors can be positive or negative. In other words, a projective point from has two representations on a sphere To overcome this ambiguity one may use oriented projective geometry, as proposed by STOLFI 1991, where every element has an intrinsic orientation.

Fixing the plane adds only one additional independent parameter to the existing two of the first part since and are constrained to be perpendicular to each other, This constraint is called the Plücker constraint of a 3D line, see below for more details. In order to uniquely fix the 3D line we need the minimal distance of the line to the origin as the fourth parameter of the 3D-line. e. if and only if the line intersects the origin. Fig. 4. A 3D line may be determined by two points which intersect with two planes of the coordinate system; the two points have 2 degrees of freedom since they are fixed to planes (left).

78. HARTLEY AND ZISSERMAN 2000 take a similar approach, see p. 3f. In the following, we assume that the correct interpretation is clear from the context. 1(b). The index in the homogeneous vector indicates that the vector is normalized and thus lies on the unit sphere. We could use the normalization of a homogeneous vector with its length for an alternative 22 2 Representation of Geometric Entities and Transformations definition of the equivalence class for two homogeneous vectors We sometimes denote the spherical normalization by The definition of the equivalence class is ambiguous as the orientation of the normalized vectors can be positive or negative.