最小二乘法平面拟合.docx

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最小二乘法平面拟合.docx

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最小二乘法平面拟合.docx

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:

Thedistancefromtheorigintothecentroid,asmeasuredalongthex-axis.

Thedistancefromtheorigintothecentroid,asmeasuredalongthey-axis.

Thedistancefromtheorigintothecentroid,asmeasuredalongthez-axis.

IfmeasuredinaCylindricalcoordinatesystem,thecoordinatesoftheplane'scentroidarereportedasfollows:

Thedistancefromthez-axisofthecoordinatesystemtothecentroid,asmeasuredwithinaplanewhichcontainsthecentroidandisorthogonaltothez-axisofthecoordinatesystem.

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

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）,sotheline'svectoris{b,d,1}

xAsDouble

yAsDouble

zAsDouble

x0AsDouble'AddedonJul.1st,2009

y0AsDouble'AddedonJul.1st,2009

z0AsDouble

mAsDouble '（x-x0）/m=（y-y0）/n=（z-z0）/p,sameas:

z=a+bx,z=c+dy

nAsDouble

pAsDouble 'vectors={m,n,p}

AngleAsDouble

xAnAsDouble

yAnAsDouble

zAnAsDouble

StraightnessAsDouble

MinStAsDouble

MaxStAsDouble

EndType

PublicTypetagCircle

xAsDouble

yAsDouble

zAsDouble

rAsDouble

dAsDouble

CircularAsDouble

EndType

PublicTypetagVector

aAsDouble

bAsDouble

cAsDouble

EndType

TypetagPlane

xAsDouble 'Thedistancefromtheorigintothecentroid,asmeasuredalongthex-axis.

yAsDouble 'Thedistancefromtheorigintothecentroid,asmeasuredalongthey-axis

zAsDouble 'Thedistancefromtheorigintothecentroid,asmeasuredalongthez-axis

'Z+A*x+B*y+C=0 z'scoefficientisjust1

axAsDouble 'coefficient ofX

byAsDouble 'coefficient ofY

czAsDouble

dAsDouble 'ConstantC

AngleAsDouble

xAnAsDouble

YAnasDouble

zAnAsDouble

FlatAsDouble

MinFlatAsDouble

EndType

'*************************************************************

' 函数名：

PlaneSet

' 功能：

求拟合平面（相关参数）

' 参数：

dataRaw -tagPoint型自定义点类型（x,y,z），数组

' PlaneSet–tagPlane 其值返回为平面圆相关参数

' 返回值：

Long型，失败为0，成功为-1

'*************************************************************

PublicFunctionPlaneSet（dataRaw（）AstagPoint,PlaneAstagPlane）AsLong'z+Ax+BY+C=0

PlaneSet=0

DimlbAsLong,ubAsLong,MaxFAsDouble,MinFAsDouble,tmpAsDouble

lb=LBound（dataRaw）:

ub=UBound（dataRaw）

Ifub-lb+1<4ThenExitFunction

DimiAsLong,nAsLong

n=ub-lb+1

DimxAsDouble,yAsDouble,zAsDouble

DimXYAsDouble,XZAsDouble,YZAsDouble

DimX2AsDouble,Y2AsDouble

DimaAsDouble,bAsDouble,cAsDouble,dAsDouble

Dima1AsDouble,b1AsDouble,z1AsDouble

Dima2AsDouble,b2AsDouble,z2AsDouble

Dimn1AstagVector'{.ax,by,1} s1

Dimn2AstagVector'{0,0,N}XYplane s2

Dimn3AstagVector'lineprojectedplane

DimSLineAstagVector

DimxLineAstagVector'{1,0,0}

DimyLineAstagVector'{1,0,0}

DimzLineAstagVector'{1,0,0}

DimVectorPlaneAstagVector

xLine.a=1:

xLine.b=0:

xLine.c=0

yLine.a=0:

yLine.b=1:

yLine.c=0

zLine.a=0:

zLine.b=0:

zLine.c=1

Fori=lbToub

WithdataRaw（i）

x=x+.x

y=y+.y

z=z+.z

XY=XY+.x*.y

XZ=XZ+.x*.z

YZ=YZ+.y*.z

X2=X2+.x^2

Y2=Y2+.y^2

EndWith

Nexti

z1=n*XZ-x*z 'e=z-Ax-By-C z=Ax+By+D

a1=n*X2-x^2

b1=n*XY-x*y

z2=n*YZ-y*z

a2=n*XY-x*y

b2=n*Y2-y^2

a=（z1*b2-z2*b1）/（a1*b2-a2*b1）

b=（a1*z2-a2*z1）/（a1*b2-a2*b1）

c=1

d=（z-a*x-b*y）/n

WithPlane

.x=x/（ub-lb+1）

.y=y/（ub-lb+1）

.z=z/（ub-lb+1）

'sum（Mi*Ri）/sum（Mi）,Miismass. here Miissetedtobe1and.zisjusttheaverageofz

.Ax=-a

.By=-b

.Cz=1

.d=-d 'z=Ax+By+D-----Ax+By+Z+D=0

VectorPlane.a=.Ax:

VectorPlane.b=.By:

VectorPlane.c=1

.xAn=Intersect（VectorPlane,xLine）

.yAn=Intersect（VectorPlane,yLine）

.zAn=Intersect（VectorPlane,zLine）

n1.a=.Ax:

n1.b=.By:

n1.c=1

SLine.a=.Ax:

SLine.b=.By:

SLine.c=0

.Angle=Intersect（xLine,SLine） '（xLine.A*SLine.A+xLine.A*SLine.B+xLine.C*SLine.C）

IfSLine.b<0Then.Angle=360-.Angle

Fori=lbToub

PointToPlanedataRaw（i）,Plane,tmp,0

Ifi=lbThen

MaxF=tmp:

MinF=tmp

Else

IfMaxF

IfMinF>tmpThenMinF=tmp

EndIf

Nexti

.MaxFlat=MaxF

.MinFlat=MinF

.Flat=MaxF-MinF

EndWith

PlaneSet=-1

EndFunction

' 函数名：

VectorMulti

' 功能：

求两向量的积,结果存放于参数rtV3中

' 参数：

v1 -tagVector

' v2 -tagVector

' rtV3 -tagVector

'*************************************************************

PublicFunctionVectorMulti（v1AstagVector,v2AstagVector,rtv3AstagVector）AsLong

'rtV3=v1*v2=|i j k |

'|v1.A v1.B v1.C|

'|v2.A v2.B v2.C|

'rtv3.A=（v1.B*v2.c-v2.B*v1.C） 'i

'rtV3.B=-（v1.A*v2.C-v2.A*v1.C）'j

'rtV3.C=（v1.A*v2.B-v2.A*V1.B） 'k

rtv3.a=（v1.b*v2.c-v2.b*v1.c）

rtv3.b=-（

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