CN104680016A - Geometric optimization approximation based minimum region fitting method for profiles of parabolas - Google Patents

Abstract

The invention relates to in-plane curve fitting methods, in particular to a geometric optimization approximation based minimum region fitting method for profiles of parabolas. The method comprises the steps: firstly, approximate coordinates of an initial focus and a vertex of a measured parabola are roughly estimated according to coordinates of measuring points, and a series of auxiliary points are arranged around the initial focus and the vertex according to certain rules; next, supposed that the auxiliary points around the initial focus and the vertex are focuses and vertexes of the parabolas to be fit, the supposed focuses and vertexes can be used for constructing a series of parabolas; then, range values from the measuring points to the parabolas are calculated, and finally, the minimum region fitting of the measured parabola is realized through comparing, judging and transforming reference points and establishing a new supposed auxiliary parabola. The method has the advantages that the fitting process is simple, the accuracy of fitting is high, and the minimum region fitting and evaluation of profile tolerance errors of the parabolas can be effectively realized.

Description

Based on the parabolic outlines Minimum Area approximating method that geometry optimization approaches

Technical field

The present invention relates to parabolical approximating method in plane, be specifically related to a kind of parabolic outlines Minimum Area approximating method approached based on geometry optimization.

Background technology

The profile tolerance identification of curve and measuring method are the gordian techniquies of computer graphics and image procossing, are especially embodied in matching and the error evaluation technical elements of the base curves such as para-curve, ellipse, hyperbolic curve.In Machine Design and manufacture field, parabolic type part also has and applies comparatively widely, as the parabolic type blade of the radial flow compressor in aero and space propulsion system, the parabolic type twist drill used in deep hole machining and the para-curve convexity etc. of bearing roller, all there is data fitting problem in design process or in manufacture, testing process in these mechanical component with parabolic outlines, the quality of fit quality directly can have influence on the quality of design and product.Parabolic fit method and the error of fitting evaluation method of GB is not also listed in current three coordinate measuring machine, reverse equipment and instrument.So the Accurate Assessment of parabolic outlines degree and fitting technique are to the precision measurement of parabolic type part and be processed with important using value.

Parabolical matching is more representational is divided into geometric distance matching and algebraic distance matching, wherein, proposed the least square Parabolic Fit based on orthogonal distance in 1996, fitting precision is higher but the iteration of algorithm is comparatively loaded down with trivial details; Sung Joon Ahn, on the basis of the least square fitting of orthogonal distance, is weighted to the matrix of fitting coefficient the iteration efficiency that process improves matching.Because para-curve just belongs to the one of non-central type quafric curve, curvature is comparatively large, and least square fitting para-curve error is out comparatively large, is difficult to meet the needs of careful design, detection and production.

Summary of the invention

The object of this invention is to provide a kind of parabolic outlines Minimum Area approximating method approached based on geometry optimization, to solve, existing approximating method error is comparatively large, method is loaded down with trivial details, the problems such as efficiency is low.

For achieving the above object, technical scheme of the present invention is: the parabolic outlines Minimum Area approximating method approached based on geometry optimization, comprises the steps:

(1), to all measurement point coordinates to average the initial reference focus obtained by parabola of fit, choose in measurement point close to para-curve vertex position a bit as initial reference summit;

(2), calculate each measurement point to the initial parabolical normal distance determined by initial reference focus and initial reference summit, and use represent the extreme difference value of each measurement point to para-curve normal distance;

(3), respectively with initial reference focus and initial reference summit for reference center's point, the extreme difference value calculated with step (2) for the length of side arranges two squares, foursquare summit is auxiliary point, then obtain four secondary foci and four auxiliary summits, and then construct 16 auxiliary para-curves, calculate each measurement point to each auxiliary parabolical normal distance, then obtain 16 extreme difference values, use represent minimum extreme difference value;

(4), by the minimum extreme difference value of step (3) gained the extreme difference value calculated with step (2) make comparisons: if , Ze Liangge reference center's point is constant, namely new extreme difference value constant, the length of side is reduced into original 0.5 times and resets two squares to construct new auxiliary point, again solves new minimum extreme difference value ; If , then get with corresponding auxiliary parabolical secondary foci and auxiliary summit are new reference center's point, calculate new extreme difference value , the length of side is constant resets two squares to construct new auxiliary point, again solves new minimum extreme difference value ;

(5), by the new minimum extreme difference value of gained with new extreme difference value make comparisons according to the method for step (4), again solve new minimum extreme difference value with new extreme difference value , so repeat, until when the foursquare length of side set up is less than 0.0001mm, now obtained extreme difference value with minimum extreme difference value in reckling be Minimum Area parabolic outlines degree error, with extreme difference value with minimum extreme difference value in the corresponding auxiliary summit of reckling and secondary foci be the parabolical summit and focus that meet Minimum Area matching, thus to obtain by the Minimum Area fit equation of parabola of fit and position thereof and attitude.

Beneficial effect: the inventive method proposes according to the error assessment method of geometric configuration in GB and parabolical geometrical property, it can obtain parabolic outlines degree error quickly and accurately, and obtain the parabola of fit equation meeting lowest area principal, realize parabolic outlines degree error Evaluation of Minimum Region.Whether it, when evaluation and parabola of fit, does not evenly limit measurement point, determines that the method for initial reference point is simple and clear, optimizes approximate procedure easy understand and realization, has good convergence, and is easy to programming, and fit procedure efficiency is high.

Accompanying drawing explanation

Fig. 1 is the geometry optimization approximation theory figure of Parabolic Fit of the present invention.

Be labeled as in figure: 1 represents initial para-curve, one of them auxiliary para-curve of 2 representative structures, 3 represent tested para-curve, represent initial reference summit, represent initial reference focus, D ₁, D ₂, D ₃and D ₄represent auxiliary summit, J ₁, J ₂, J ₃and J ₄represent secondary foci.

Embodiment

The parabolic outlines Minimum Area approximating method approached based on geometry optimization of the present invention, step is as follows:

Step one, choose initial reference focus and initial reference summit

If measurement point is , utilize and all measurement point coordinates averaged the initial reference focus obtained by parabola of fit, use represent, and optional a bit as initial reference summit close to the position on para-curve summit in measurement point, use represent.Its concrete defining method is:

, or (1)

Step 2, calculating initial error

If para-curve general equation is , then determined by following formula by initial parabolical each coefficient of matching:

(2)

In formula:

(3)

Assumed position for measurement point with the orthogonal points of formula (2) determined para-curve (i.e. initial para-curve), then point coordinate solved by formula (4):

(4)

Then measurement point apart from initial parabolical normal distance is:

(5)

When measurement point is positioned at outside initial para-curve get on the occasion of, otherwise get negative value; With represent the extreme difference value of each measurement point to para-curve normal distance; So, measurement point to initial parabolical extreme difference value is:

(6)

Step 3, structure auxiliary point

Respectively with initial reference focus with initial reference summit for reference center's point, the initial error extreme difference value calculated with step 2 for the length of side arranges two squares, then foursquare summit is secondary foci with auxiliary summit , according to square geometry relation, the coordinate of these auxiliary points is:

(7)

The auxiliary para-curve of step 4, structure

Two squares arranged in step 3 have 4 secondary foci and 4 auxiliary summits, and such combination of two can construct 16 auxiliary para-curves, and auxiliary parabolical principal parameter is:

(8)

Wherein, for auxiliary parabolical burnt parameter, for the angle of auxiliary para-curve axis of symmetry and X-axis.

If the auxiliary general quadratic equation of para-curve is represent, then its each coefficient is drawn by following formula:

(9)

Step 5, computation and measurement point is to auxiliary parabolical distance extreme difference

Assumed position for measurement point with auxiliary parabolical orthogonal points, then point coordinate figure can be solved by system of equations (10):

(10)

Orthogonal points is solved by above formula after coordinate, calculate all measurement points to auxiliary parabolical distance by formula (11):

(11)

When measurement point is positioned at outside para-curve get on the occasion of, otherwise get negative value; Measurement point can be obtained to 16 auxiliary parabolical distance extreme difference values by formula (12):

(12)

With represent minimum extreme difference value, then the reckling in these 16 extreme difference values is:

(13)

Step 6, approach search

By the above-mentioned minimum extreme difference value calculated with initial error extreme difference value make comparisons: if , then with reference to focus and datum vertex constant, Ji Liangge reference center's point is constant, namely new extreme difference value constant, the length of side is reduced into 0.5 times of original new square of structure and auxiliary point, again solves new minimum extreme difference value ; If , then get with corresponding auxiliary parabolical secondary foci and auxiliary summit are new reference center's point, calculate new extreme difference value , the square of the position that the constant structure of the length of side is new, namely constructs new auxiliary point, again solves new minimum extreme difference value .

Described new extreme difference value refer to last round of with after making comparisons, each measurement point to the extreme difference value of the auxiliary parabolical normal distance constructed with redefine two new reference center's points, so, if redefine two reference points are constant, then new extreme difference value with last round of extreme difference value identical, also constant; If redefine two reference points change, then new extreme difference value with last round of extreme difference value different.

Then by the new minimum extreme difference value of gained with new extreme difference value make comparisons according to the method described above, again again solve new minimum extreme difference value with new extreme difference value circulation like this, until when the foursquare length of side constructed is very little, generally get≤0.0001mm, can know, the parabolical focus of institute's matching and summit are in the scope of 0.0001mm in the length of side, can conclude the auxiliary para-curve closely actual ideal parabolic searched, stop search, the extreme difference value now obtained with minimum extreme difference value in reckling be Minimum Area parabolic outlines degree error , that is: , now with with in the auxiliary parabolical focus corresponding to reckling and summit be the parabolical focus and summit that meet Minimum Area matching, if focus now and summit are used respectively with represent, parabolic equation general equation represent, then each coefficient is drawn by formula (14):

(14)

In formula:

So far, obtain by the Minimum Area fit equation of parabola of fit and position thereof and attitude.

Claims (1)

1., based on the parabolic outlines Minimum Area approximating method that geometry optimization approaches, it is characterized in that: comprise the steps:

(1), to all measurement point coordinates to average the initial reference focus obtained by parabola of fit, choose in measurement point close to para-curve vertex position a bit as initial reference summit;

(2), calculate each measurement point to the initial parabolical normal distance determined by initial reference focus and initial reference summit, and use represent the extreme difference value of each measurement point to para-curve normal distance;

(3), respectively with initial reference focus and initial reference summit for reference center's point, the extreme difference value calculated with step (2) for the length of side arranges two squares, foursquare summit is auxiliary point, then obtain four secondary foci and four auxiliary summits, and then construct 16 auxiliary para-curves, calculate each measurement point to each auxiliary parabolical normal distance, then obtain 16 extreme difference values, use represent minimum extreme difference value;

(4), by the minimum extreme difference value of step (3) gained the extreme difference value calculated with step (2) make comparisons: if , Ze Liangge reference center's point is constant, namely new extreme difference value constant, the length of side is reduced into original 0.5 times and resets two squares to construct new auxiliary point, again solves new minimum extreme difference value ; If , then get with corresponding auxiliary parabolical secondary foci and auxiliary summit are new reference center's point, calculate new extreme difference value , the length of side is constant resets two squares to construct new auxiliary point, again solves new minimum extreme difference value ;

(5), by the new minimum extreme difference value of gained with new extreme difference value make comparisons according to the method for step (4), again solve new minimum extreme difference value with new extreme difference value , so repeat, until when the foursquare length of side set up is less than 0.0001mm, now obtained extreme difference value with minimum extreme difference value in reckling be Minimum Area parabolic outlines degree error, with extreme difference value with minimum extreme difference value in the corresponding auxiliary summit of reckling and secondary foci be the parabolical summit and focus that meet Minimum Area matching, thus to obtain by the Minimum Area fit equation of parabola of fit and position thereof and attitude.