Application of four times EB spline curve and surface based on coordinate measuring machine (2)

1.3 Characteristics of four EB splines ^[2]

(1) Since a geometrically significant parameter K is introduced in the EB spline equation, the designer can intuitively adjust the shape of the spline curve (surface), so that the difference of the feature polygon can be large or small.

(2) Four EB splines can be applied to a given set of special feature vertices to generate curves containing straight lines and sharp points.

(3) The tangent at the end of the quadruple EB spline curve segment is only related to the basic curve segment, but not to the blending curve segment.

(4) When using four times EB spline curve (surface) as the interpolation curve (surface), the design method and calculation method are unchanged, and only need to take the parameter K to zero, which is very useful for the preparation of the calculation program.

(5) When K=1/6, the four EB spline curves (surfaces) become B-spline curves (surfaces). It can be seen that the four-time EB spline curve (surface) is an extension of the B-spline curve (surface).

2 Error measurement of the measured data fitting surface

　　 When assessing the error between the fitted surface of the coordinate measuring machine and the measured data, the distance from the measured point of the space to the fitted surface is calculated. This paper uses a fast iterative convergence algorithm ^{[3, 4] to} find the distance from any point in the space to the fitted surface.

Let P(x, y, z) be the outer space of the surface and find the distance from point P to the parametric surface ∑ s(u, w). Where Q _i (u _i , w _i ) is any point on the surface, as the mapping point of the P point on the surface, it is the initial point of the iteration. Q _i+1 (u _i+1 , w _i+1 ) is the foot of the P point on the curved surface. ∑s(u,w) is the tangent plane of the Q _i point, and S is the intersection of PQ _i+1 on the tangent plane. When Q _i coincides with Q _i+1 , PQ _i+1 is the distance from the point to the surface. Q ^u _i , Q ^w _i are two tangent vectors of Q _i in the u, w direction. So there is

If PQ _i+1 is the distance from point to surface, then S must coincide with Q _i+1 , and S is also the foot of the point on the surface. According to the condition that the distance from the differential geometric point to the plane is the minimum distance

The search method is to obtain Δu, Δw in the formula (11). When (Δu, Δw) > ε (ε is any given small positive number), Q _{i+1 is used} instead of Q _i for the next round of calculation. Iterative stop when (Δu, Δw) < ε, at this time

PQ _i+1 = PQ _i+1 = PQ _i = PS= d

Since the specified surface normal vector points to the outside of the entity, P is d>0 when it is outside the entity, and d<0 when P is inside the entity.

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