In the manufacturing industry, we often have to redesign a product.
Usually a new design requires a completely different fixture design,
which costs both money and time. This tempts us to consider the
following problem: given a planar polygonal part, and its fixture
which consists of three locators and one clamp, is it possible to
specify how much the part shape can be modified such that the
redesigned part can still use the same set of locators? Moreover, we
want to have an interactive user interface so that the designer can be
immediately notified if the modification is too great to re-use the
same fixture.
Figure 4: How much can we modify the third edge such that
we can still achieve three point contact?
We assume that the designer will redesign the part by modifying one edge at a time. Clearly, the designer can modify the non-contacting edges at will, without affecting the fixture. The real question is: given three locators and their corresponding contacting edges, if we keep two of the edges, how much we can modify the third edge while still being able to achieve three point contact? (Figure 4) Note that the dashed curves in Figure 4 represent all the edges connecting the three edges in consideration. Without loss of generality, we expand the part by the radius of locators using a Minkowski sum operation. This allows us to consider each locator as an ideal point. The original part contacts a locator if and only if the corresponding edge of the expanded part passes through the point. In the rest of this section, we simply refer to the expanded part as the part.