A coordinate measuring machine (CMM) is used to determine the distance between Surface SP and Surface SQ of an approximately cuboidal shaped part. Surface SP is declared as the datum as per the engineering drawing used for manufacturing this part. The CMM is used to measure four points P1, P2, P3, P4 on Surface SP, and four points Q1, Q2, Q3, Q4 on Surface SQ as shown. A regression procedure is used to fit the necessary planes.

The distance between the two fitted planes is ___________ mm. (Answer in integer)

Correct Answer: The distance between the two fitted planes is 5 mm.

Step-by-Step Explanation:

1. Analysis of the Datum Surface (SP):
From the given image, the coordinates of the four points measured on the datum surface SP are:
- $P_1=(4,0,3)$
- $P_2=(3,0,6)$
- $P_3=(1.5,0,7)$
- $P_4=(2,0,2)$

Since the $y$-coordinate for all four points $P_1$, $P_2$, $P_3$, and $P_4$ is exactly $0$, these points lie entirely in the $x-z$ plane.
Therefore, the regression plane fitted to Surface SP is the plane defined by:
$y=0$ (which is the $x-z$ plane).

2. Analysis of the Parallel Surface (SQ):
From the image, the coordinates of the four points measured on the surface SQ are:
- $Q_1=(4,6,3)$
- $Q_2=(1,6,3)$
- $Q_3=(1.5,4,7)$
- $Q_4=(4,4,6)$

The perpendicular distance of any point $(x,y,z)$ to the reference plane $y=0$ is simply its $y$-coordinate.
Thus, the distances of the points $Q_1$, $Q_2$, $Q_3$, and $Q_4$ from the fitted datum plane are:
- Distance of $Q_1=6\text{mm}$
- Distance of $Q_2=6\text{mm}$
- Distance of $Q_3=4\text{mm}$
- Distance of $Q_4=4\text{mm}$

3. Calculating the Distance Between the Two Fitted Planes:
A regression procedure is used to fit the best-fit plane for Surface SQ parallel to the datum plane. For planes of the form $y=d$ parallel to the $x-z$ plane, the least-squares regression parameter $d$ is the arithmetic mean of the $y$-coordinates of the measured points on Surface SQ:
$d=\frac{y_1+y_2+y_3+y_4}{4}$
$d=\frac{6+6+4+4}{4}=\frac{20}{4}=5\text{mm}$

Therefore, the distance between the two fitted planes is 5 mm.