Determining the Center and Diameter of a Drilled Hole
Three point locations on the surface of a drilled hole have been measured by a CMM in the x–y axes. The three coordinates are (34.41, 21.07), (55.19, 30.50), and (50.10, 13.18) millimeters. The given coordinates have been corrected for probe radius. Determine (a) coordinates of the hole center and (b) hole diameter, as they would be computed by the CMM software.

Question

Accepted Answer

To determine the coordinates of the hole center, three equations are set up based on Equation (22A.3):

[latex](x - a)^{2} + (y - b)^{2} = R^{2}[/latex]

[latex](34.41 - a)^{2} + (21.07 - b)^{2} = R^{2}[/latex] (i)

[latex](55.19 - a)^{2} + (30.50 - b)^{2} = R^{2}[/latex] (ii)

[latex](50.10 - a)^{2} + (13.18 - b)^{2} = R^{2}[/latex] (iii)

Expanding each of the equations,
[latex]1,184.0481 - 68.82a + a^{2} + 443.9449 - 42.14b + b^{2} = R^{2}[/latex] (i)

[latex]3,045.9361 - 110.38a + a^{2} + 930.25 - 61b + b^{2} = R^{2}[/latex] (ii)

[latex]2,510.01 - 100.2a + a^{2} + 173.7124 - 26.36b + b^{2} = R^{2}[/latex] (iii)

Simultaneous solution of the three equations yields the following values:
a = 45.66 mm, b = 23.89 mm, and R = 11.60 mm. Thus, the center of the hole is located at x = 45.66 mm and y = 23.78 mm, and the hole diameter is D = 23.20 mm.

Question 22A.2: Determining the Center and Diameter of a Drilled Hole Three ...

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