Contributions to Four-Position Theory with Relative Rotations

We consider the geometry of four spatial displacements, arranged in cyclic order, such that the relative motion between neighbouring displacements is a pure rotation. We compute the locus of points whose homologous images lie on a circle, the locus of oriented planes whose homologous images are tangent to a cone of revolution, and the locus of oriented lines whose homologous images form a skew quadrilateral on a hyperboloid of revolution

The Kinematic Image of 2R Dyads and Exact Synthesis of 5R Linkages

We characterise the kinematic image of the constraint variety of a 2R dyad as a regular ruled quadric in a 3-space that contains a "null quadrilateral". Three prescribed poses determine, in general, two such quadrics. This allows us to modify a recent algorithm for the synthesis of 6R linkages in such a way that two consecutive revolute axes coincide, thus producing a 5R linkage. Using the classical geometry of twisted cubics on a quadric, we explain some of the peculiar properties of the the resulting synthesis procedure for 5R linkages.Comment: Accepted for publication in the proceedings of the IMA Conference on Mathematics of Robotics, Oxford, 201

Ortologni tetraedri s bridovima koji se sijeku

Two tetrahedra are called orthologic if the lines through vertices of one and perpendicular to corresponding faces of the other are intersecting. This is equivalent to the orthogonality of non-corresponding edges. We prove that the additional assumption of intersecting non-corresponding edges (“orthosecting tetrahedra”) implies that the six intersection points lie on a sphere. To a given tetrahedron there exists generally a one-parametric family of orthosecting tetrahedra. The orthographic projection of the locus of one vertex onto the corresponding face plane of the given tetrahedron is a curve which remains fixed under isogonal conjugation. This allows the construction of pairs of conjugate orthosecting tetrahedra to a given tetrahedron.Dva tetraedra nazivamo ortolognim ako se pravci koji prolaze vrhovima jednog i okomiti su na odgovarajuće stranice drugog međusobno sijeku. Ovo je ekvivalentno ortogonalnosti ne-odgovarajućih bridova. Mi dokazujemo kako dodatna pretpostavka da se ne-odgovarajući bridovi sijeku (”ortopresječni tetraedar”) povlači da šest sjecišta leži na jednoj kugli. Za dani tetraedar postoji općenito jednoparametarska familija ortopresječnih tetraedara. Ortogonalna projekcija geometrijskog mjesta jednog vrha na pripadajuću ravninu danog tetraedra je krivulja koja ostaje fiksnom pod djelovanjem izogonalne konjugacije. Ovo dopušta konstrukciju parova konjugiranih ortopresječnih tetraedara za dani tetraedar