Question

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) 

Hint:

In CMM data analysis problems, always inspect the coordinates first. Often, the points lie perfectly on a simple plane (like \(x=k\), \(y=k\), or \(z=k\)), which simplifies finding the plane equation. For a plane fitted to points that are approximately on \(y=k\), the best-fit plane is found by averaging the y-coordinates.

Answer 1

Step 1: Understanding the Concept:
The problem requires finding the distance between two planes that are fitted to sets of 3D coordinate points. We first need to determine the equations of these two planes from the given points. The distance between two parallel planes can then be calculated.
Step 2: Detailed Explanation:
1. Analyze the points on the datum surface SP:
- P1: (4, 0, 3)
- P2: (3, 0, 6)
- P3: (1.5, 0, 7)
- P4: (2, 0, 2)
We observe that for all four points on surface SP, the y-coordinate is exactly 0. This means all these points lie on the plane defined by the equation \(y = 0\). This plane is the x-z plane. Since a perfect plane fits these points, the regression procedure will result in the plane equation: \[ \text{Plane SP: } y = 0 \] 2. Analyze the points on surface SQ:
- Q1: (4, 6, 3)
- Q2: (1, 6, 3)
- Q3: (1.5, 4, 7)
- Q4: (4, 4, 6)
The part is described as "approximately cuboidal", and SP is the datum plane \(y=0\). It is reasonable to assume that surface SQ is parallel to SP. A plane parallel to the x-z plane has the general equation \(y = k\), where \(k\) is a constant. The regression procedure (specifically, a least-squares fit) for this type of plane would find the value of \(k\) that best fits the data. This value is the average of the y-coordinates of the points. \[ k = \frac{y_{Q1} + y_{Q2} + y_{Q3} + y_{Q4}}{4} = \frac{6 + 6 + 4 + 4}{4} = \frac{20}{4} = 5 \] So, the best-fit plane for surface SQ is: \[ \text{Plane SQ: } y = 5 \] 3. Calculate the distance between the two planes:
The two fitted planes are \(y = 0\) and \(y = 5\). These are parallel planes. The distance between them is the difference in their constant y-values. \[ \text{Distance} = |5 - 0| = 5 \text{ mm} \] Step 3: Final Answer:
The distance between the two fitted planes is 5 mm.
Step 4: Why This is Correct:
The problem is simplified by observing the coordinates. The datum points perfectly define the plane \(y=0\). Assuming the opposing face of the cuboid is parallel, the best-fit plane is found by averaging the relevant coordinate (y). The distance is then the simple perpendicular distance between these two parallel planes.