Least Squares Parameters

In general, the PC-DMIS Blade least squares bestfit algorithm aligns input points ${\overrightarrow{p}}_i$ to a curve. The bestfit algorithm finds a transformation (consisting of an offset and a rotation) that minimizes a least-squares objective function. The algorithm works as follows:

1. Set $T$ to the identity transformation (no offset and no rotation)

2. Find the points on the curve ${\overrightarrow{c}}_i$ that are the nearest points on the curve to the transformed input points $T({\overrightarrow{p}}_i)$

3. Find a new transformation $T$ that best aligns the transformed input points $T({\overrightarrow{p}}_i)$ to the curve points ${\overrightarrow{c}}_i$, by minimizing an objective function described below

4. Repeat MAXITERS times by going back to step 2, finding new points ${\overrightarrow{c}}_i$, and so forth

The first iteration yields an approximate bestfit $T$. On the second iteration, step 2 produces improved closest-on-the-curve points ${\overrightarrow{c}}_i$ because $T$ is better than the initial identity transformation. Therefore, step 3 of the second iteration can produce a further-improved estimate of $T$, and so forth. By including enough iterations, the result will typically converge to a very high-accuracy alignment $T$ between the input points and the curve.

Step 3 minimizes the following objective function:

$$\mathcal{L}=\sum _i{w}_i{r}_i^2$$

In the above:

• $w_i$ is the weight associated with point $i$

• $r_i$ is the distance between ${\overrightarrow{c}}_i$ and $T({\overrightarrow{p}}_i)$

In most cases, it is a simple Euclidean distance, but in the case of the Vector Fit, it is projected onto the surface normal of the curve at ${\overrightarrow{c}}_i$. Using the Vector Fit typically promotes faster convergence and higher accuracy, since it more nearly approximates the distance from the point to the curve.

Additional Parameters

These parameters further define how PC-DMIS Blade performs the least squares bestfit:

• Use in Fit - If you select the check boxes, PC-DMIS Blade includes that side or edge in the bestfit. In the Flavor file example below, the least squares algorithm is performed on both edges (LE and TE) and both sides (CC and CV).

These parameters further define how PC-DMIS Blade performs the least squares bestfit:

BESTFIT1 LSQ CC CV LE TE



• Weight - Provides the weight that is given to the associated edge or side to determine the bestfit. These weights are the values $w_i$ described above; all points associated with a given edge or side have the same weight. In the Flavor file example below, the four values represent the weight that is given to first, the concave side; second, the convex side; third, the leading edge; and fourth, the trailing edge.

WEIGHTS 2 1 3 4



• Max Iterations - A bestfit can be constrained to a maximum number of iterations. The Flavor file example below states that the fit can be rerun 15 times. Increasing this number can sometimes improve the fit and always slows down the processing of the blade.

MAXITERS 15



• Meas Pts to Nom Curve and Meas Curve to Nom Pts - From the list, select whether measured points are bestfit to the nominal curve (use a value of 0), or the measured curve is bestfit to the nominal points (use a value of 1). The Flavor file example below measures the curve to the nominal points:

USENOMINALS 1



• Vector Fit (ONLY for Full Blade Least Squares) - Determines whether the Full Blade Least squares algorithm looks at the vectors of the points on the spline as with a Vector Least Squares algorithm. This alters the distances $r_i$ as described above, speeding convergence and improving accuracy.