0 Preface ....................................................... 5
1 Line-symmetric motions with respect to reguli ................. 8
1.1 Introduction ............................................. 8
1.2 Fundamentals ............................................. 9
1.2.1 Euclidean displacements ........................... 9
1.2.2 Line geometry .................................... 12
1.3 Line-symmetric motions with respect to reguli ........... 17
1.3.1 Half turns ....................................... 17
1.3.2 Half turns with respect to reguli ................ 19
1.3.3 Line-symmetric motions given by three poses ...... 21
1.4 One-parameter motions with quadratic kinematic image .... 23
1.5 Conclusion and further work ............................. 29
2 On the contact of one-parameter motions ...................... 31
2.1 Introduction ............................................ 31
2.2 Fundamentals ............................................ 34
2.2.1 Parametrization of the special Euclidean group ... 34
2.2.2 Instantaneous kinematics ......................... 37
2.2.3 Contact of curves ................................ 39
2.3 Geometric properties and kinematic invariants ........... 40
2.3.1 First order derivative ........................... 40
2.3.2 Second order derivative .......................... 44
2.4 Canonically osculating motions .......................... 46
2.5 Numerical examples ...................................... 53
2.6 GCfc-Interpolation ...................................... 56
2.7 Conclusion and further work ............................. 61
3 Interpolation with rational motions of low degree ............ 65
3.1 Introduction ............................................ 65
3.2 Fundamentals ............................................ 67
3.2.1 Euclidean displacements and kinematic mapping .... 67
3.2.2 Instantaneous kinematics and first order
properties ....................................... 68
3.2.3 Interpolation problem ............................ 69
3.2.4 Motion interpolation using biarc construction .... 70
3.2.5 Line-symmetric motions with respect to reguli .... 74
3.3 Degrading the degree of a rational motion spline ........ 75
3.3.1 Geometric approach ............................... 75
3.3.2 Numerical implementation ......................... 78
3.3.3 Examples ......................................... 80
3.4 Conclusions ............................................. 80
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