最小二乘法平面拟合word精品文档17页.docx

最小二乘法平面拟合word精品文档17页.docx

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最小二乘法平面拟合word精品文档17页

最小二乘法平面拟合

在介绍平面拟合之前，我先给大家介绍一下有关平面的相关知识（相关介绍来自QVPak3D,日本三丰）

DefinitionofthePlaneFeature

Aplanefeatureisreportedastheprojectionofthecentroidofthepointsusedtofittheplane,whichisprojectedontotheplanefeature,ameasurementofthedirectionmeasuredasanangle,ameasurementoftheflatnessoftheplaneandameasurementoftheparallelismoftheplane. 

IfmeasuredinaCartesiancoordinatesystem,thecoordinatesoftheplane'scentroidarereportedasfollows:

X:

Thedistancefromtheorigintothecentroid,asmeasuredalongthex-axis. 

Y:

Thedistancefromtheorigintothecentroid,asmeasuredalongthey-axis. 

Z:

Thedistancefromtheorigintothecentroid,asmeasuredalongthez-axis. 

IfmeasuredinaCylindricalcoordinatesystem,thecoordinatesoftheplane'scentroidarereportedasfollows:

R:

Thedistancefromthez-axisofthecoordinatesystemtothecentroid,asmeasuredwithinaplanewhichcontainsthecentroidandisorthogonaltothez-axisofthecoordinatesystem. 

A:

Thedirection,measuredasanangle,betweenareferenceradiusvectorandaradiusvectorthatcontainsthecentroidandisprojectedontothexy-plane.Thereferenceradiusvectormaybeconsideredtobethex-axis. 

Z:

Theheightfromtheorigintothecentroidinthecylindricalcoordinatesystem,asmeasuredalongthez-axis. 

Theotherattributesoftheplanefeatureare:

Angle:

Theanglebetweentheprojectionoftheplane’snormalvectorontothexy-planeandthex-axisofthecurrentcoordinatesystem. 

X-angle:

Theanglebetweentheplane’snormalvectorandthex-axisofthecurrentcoordinatesystem.（X-Angle=arccosinek）.Thex-angleisapositivenumberbetween0and180degrees. 

Y-angle:

Theanglebetweentheplane’snormalvectorandthey-axisofthecurrentcoordinatesystem.（Y-Angle=arccosinel）.They-angleisapositivenumberbetween0and180degrees. 

Z-angle:

Theanglebetweentheplane’snormalvectorandthez-axisofthecurrentcoordinatesystem.（Z-Angle=arccosinem）.Thez-angleisapositivenumberbetween0and180degrees. 

Flatness:

Flatnessisaconditionforwhichanelementofasurfaceisinaplane. 

Flatnessisreportedasthewidthofthezoneformedbytwoclosestparallelplanesthatfullycontainthepointsetusedtofittheplanefeature.Avalueofzeroindicatesperfectflatness. 

Flatness（minimum）:

Thedistancefromthefittedplanetothemeasuredpointfarthestbelowthefittedplaininthepointset.Aboveandbelowaredeterminedbythedirectionoftheplanevector.SeeExplanationofMax/Mindistanceindifferentcases. 

Flatness（maximum）:

Thedistancefromthefittedplanetothemeasuredpointfarthestabovethefittedplaininthepointset.Aboveandbelowaredeterminedbythedirectionoftheplanevector.SeeExplanationofMax/Mindistanceindifferentcases. 

Parallelism:

Theconditionofafeature,projectedtoacertainplane,beingequidistantatallelementsfromadatum（reference）.Quantitatively,parallelismisdefinedastheabsolutedistantdifferencebetweenthefarthestandclosestpointsfromthedatum. 

Parallelismisevaluatedrelativetoareferencelineorxy-plane.Whenevaluatingasetofpointswithareferenceline,parallelismusestheprojectionsofthepointsandreferencelineontothexy-planeinthecurrentcoordinatesystem,orz/refplanefeature,andisspecifiedasazonetolerance.Thez/refplanefeatureisaplaneincludingthereferencelineandparallelto（orincluding）thez-axis.Whenevaluatingasetofpointswithaxy-plane,parallelismiscalculatedinthree-dimensionalspace. 

Parallelism（minimum）:

Thedistancefromthereferencedlineorplanetothepointinthepointsetwiththeleastvalue（leastpositivevalueifallevaluatedpointsarepositive,ormostnegativevalueifevaluatedpointsincludenegativevalues）.SeeExplanationofMax/Mindistanceindifferentcases. 

Parallelism（maximum）:

Thedistancefromthereferencelineorplanetothepointinthepointsetwiththegreatestvalue（mostpositivevalueifevaluatedpointsincludepositivevalues,orleastnegativevalueifallevaluatedpointsarenegative）.SeeExplanationofMax/Mindistanceindifferentcases.

平面相关知识：

算法如下：

VB源代码：

OptionExplicit

PublicConstPI=3.1415926535897

PublicTypetagPoint

  xAsDouble

  yAsDouble

  zAsDouble

EndType

PublicTypetagLine2D

  kAsDouble     'Slope,Kis the"K"of"y=kx+b"

  bAsDouble     'intercept,Bis the"B"of"y=kx+b"

  AngleAsDouble 'arctg（k） '0to180deg

  StraightnessAsDouble

  RSQAsDouble

EndType

PublicTypetagLine3D

'3Dline'sformulaisshowingasfollowing.

'1）|Ax+By+Z+D=0

   ' |A1x+B1y+z+D1=0

   '2）（x-x0）/m=（y-y0）/n=（z-z0）/p----（x-x0）/m=（y-y0）/n=z/1

   '3）x=mt+x0,y=nt+y0,z=pt+z0

   'Onlypoint'scoordinateis（a+b*Z,c+d*Z,Z）,sothe