Section 4.3 (four-bar linkages)

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Consider using isotropic coordinates for solving the nine-point path synthesis problem as presented in [MSW]. (See Four-bar linkages using alphaCertified for more details.) Using Newton-invariant sets, one is able to determine which solutions are "real" in isotropic coordinates.

The first instance shows that there are 64 "real" four-bar linkages through the following 9 real points:

(0.25, 0.00),	(0.52, 0.10),	(0.80, 0.70),
(1.20, 1.00),	(1.40, 1.30),	(1.10, 1.48),
(0.70, 1.40),	(0.20, 1.00),	(0.02, 0.40).

The second instance shows that there are 51 "real" four-bar linkages through the following 9 real points:

(0.000, 0.000),	(0.250, 1.330),	(0.500, 1.740),
(0.750, 1.930),	(1.000, 2.010),	(1.250, 1.950),
(1.500, 1.730),	(1.750, 1.330),	(2.000, -0.007).

The remainder of this page documents this computation.

Directions
Before you begin, you will need to have a working version of alphaCertified.

Save the files config and invariantSet in the same directory.

For the first instance:

Save the files system and points in the same directory.
Execute the command
>> alphaCertified system points config
from the directory.
Here is the output displayed to the screen. This shows that 64 "real" four-bar linkages.

For the second instance:

Save the files system and points in the same directory.
Execute the command
>> alphaCertified system points config
from the directory.
Here is the output displayed to the screen. This shows that 51 "real" four-bar linkages.

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[MSW]

A.P. Morgan, A.J. Sommese, and C.W. Wampler II, Complete solution of the nine-point path synthesis problem for four-bar linkages, ASME J. Mech. Des., 114 (1992), 153–159.