Automated Generation of Kempe Linkages for ... - Alexander Kobel

List of Figures
2.1 Watt‘s linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Peaucellier-Lipkin cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 The Peaucellier cell on slot . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Two translators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 The distance copier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 The angle multiplicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 The parallelogram defined by P . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 The final translations of Kempe‘s construction . . . . . . . . . . . . . . . 19
3.3 Three possible configurations of a simple translator linkage . . . . . . . 23
3.4 A braced parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 The perpendicular bisector of a segment . . . . . . . . . . . . . . . . . . 26
4.2 The intersection of two parallel lines at infinity . . . . . . . . . . . . . . . 27
4.3 The angular bisector in continuous and deterministic geometry systems 27
4.4 Automatic theorem checking in Cinderella . . . . . . . . . . . . . . . . . . 29
4.5 Multiplication and addition of angles . . . . . . . . . . . . . . . . . . . . 34
4.6 Multiplication of lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 Division of lengths by application of the intercept theorem . . . . . . . . 35
4.8 The graphical user interface of our implementation . . . . . . . . . . . . 37
4.9 Static screenshots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.10 Animation stills of a Kempe linkage simulation of y 2 = x 2 − x 3 . . . . . 39
4.11 Animation stills of a Kempe linkage simulation of y 2 = x 2 − x 3 (cont‘d) 40
4.12 Some complicated examples . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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