الفهرس 
ACKNOWLEDGMENTS
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Abstract
In this thesis, we concerned mainly with the evolute and the involute curves br as well as their geometric properties in hyperbolic and de Sitter spaces. The br evolute of a regular curve is a classical object -#102;-#114;-#111;-#109; the viewpoint of differential br geometry. The evolute and involute curves have the most important positions and br applications in the study of design problems in spatial mechanisms and physics, br kinematics, computer aided design (CAD) and computer aided geometric design br (CAGD), it is one of the most important topics of differential geometry. Because of br this, geometers have studied it in Euclidean, Minkowski, hyperbolic, and de Sitter br spaces and they have investigated many properties it. The evolute of a spherical br curve is defined to be the locus of the center of its osculating spheres. br Therefore, the evolute of a regular curve without inflection points is given by br not only the locus of all its centers of curvature but also the envelope of its br normal lines.