By Kollar J., Lazarsfeld R., Morrison D. (eds.)
Read Online or Download Algebraic Geometry Santa Cruz 1995, Part 1 PDF
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Extra resources for Algebraic Geometry Santa Cruz 1995, Part 1
Matrix Lnm can be determined in accordance with Eqs. 30). The coordinate transformation from S a to S n is based on the matrix equation ρn = Lnm Lma ρa . 30). At the start of motion, coordinate system S b coincides with S a , coordinate system S n (which is rigidly connected to S b ) coincides with S m (which is rigidly connected to S a ). 44) L−1 ma Lnm Lma = Lba . 46) m3 ]T . 47) and Lnm Lma [c 1 c2 c 3 ]T = Lma Lba [c 1 c2 c 3 ]T = m = [m1 m2 Here, m is the unit vector of the axis of S m that is the axis of rotation (two components of m are equal to zero and the third is equal to one).
The radii of the circles are ρ1 and ρ2 . Point M, which is rigidly connected to circle 2, traces out an extended epicycloid in the coordinate system that is rigidly connected to circle 1 [represented in coordinate system S 1 (x1 , y1 ) as shown in Fig. 1(a)]. Mo and M are two positions of the tracing point. Equations of the extended epicycloid may be derived by using the coordinate transformation in transition from S 2 to S 1 . Coordinate systems S 1 and S 2 are shown in Fig. 1(b). 2 by Eqs. 15).
By replacing ω (2) by an equal vector that passes through point Of and moment m, we may represent the velocity v(2) as follows: v(2) = (ω (2) × r) + (R × ω (2) ). 1: Rotation about crossed axes. It is easy to prove that Eqs. 5) are equivalent by taking into account that R = Of O2 + O2 O2 and r = Of O2 + ρ. Then we obtain that v(2) = ω (2) × (r − R) = ω (2) × ρ. 7) where E = Of O2 ; r = Of M. The relative velocity v(21) of point M of body 2 with respect to point M of body 1 is v(21) = −v(12) = [(ω (2) − ω (1) ) × r] + (E × ω (2) ).