Abstract
In this paper, J. Jacobi’s Theorems [9] have been considered for the spherical curves drawn on the unit dual sphere during the closed space motions. The integral invariants of the ruled surfaee corresponding, in the line space, to the spherical curve dtawn by a fixed point on the moving unit dual sphere during the one-parameter closed motion were calculated with a different approach from the area vector used by H. R. Müller [11], In addition, the ruled surfaces corresponding to the curves drawn by the unit tangent vector, principal normal vector or a unit vector on the osculating plane of the mentioned curve, were seen to be cones with this approach.
Highlights
Jacobi [9] showed that the indicator of tangents of any real closed spherical curve divides the surfaee area of a unit sphere into two equal parts. He showed that the indicator of the Principal normais of any closed curve divides the surfaee area of the unit sphere into two equal parts
Fenchel [2] and V.G, Avaqumovic [1], using Jacobi Theorems, showed that “A necessary and sufficient condition for a closed spherical indicator of the principal normais of another spatial curve is that the spherical curve divides the surfaee of the sphere into two equal parts” and “The spherical indicator of principal normais of a closed spherical curve divides the surfaee area of the unit sphere into equal parts”
Yapar [16] showed that the spherical indicator of each unit vector lying in the osculating plane of a closed spherical curve which is fixed to the curve divides the surfaee area of the unit sphere into two equal parts
Summary
J. Jacobi [9] showed that the indicator of tangents of any real closed spherical curve divides the surfaee area of a unit sphere into two equal parts. He showed that the indicator of the Principal normais of any closed curve divides the surfaee area of the unit sphere into two equal parts. The angle of pitch and length of pitch, which are the real integral invariants of a closed ruled surface, are very important to study the geometry of lines from the perspectives of instantaneous space kinematics and mechanisms. The integral invariants of the ruled surface are studied with a different method considering the Jacobi Theorems with the help of area vector. We hope that the presented results would bring new perspectives to the spatial kinematics
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