#空間直角坐標(biāo)轉(zhuǎn)換之仿射變換

1、空間直角坐標(biāo)轉(zhuǎn)換之仿射變換一、仿射變換仿射變換是空間直角坐標(biāo)變換的一種，它是一種二維坐標(biāo)到二維坐標(biāo)之間的線性變換，保持二維圖形的“平直線”和“平行性”，其可以通過(guò)一系列的原子變換的復(fù)合來(lái)實(shí)現(xiàn)，包括平移(Translation)、縮放(Seale、翻轉(zhuǎn)(Flip)、旋轉(zhuǎn)(Rotation)和剪切(Shear，此類變換可以用一個(gè)3X3的矩陣來(lái)表示，其最后一行為(0,0,1)。該變換矩陣將原坐標(biāo)(x,y)變換為新坐標(biāo)(x,y),這里原坐標(biāo)和新坐標(biāo)皆視為最末一行為(1)的三維列向量，原列向量左乘變換矩陣得到新的列向量：xm00m01m02xm00*x+m01*y+m02y=m10m11m12y=m10

2、*x+m11*y+m12100111如果將它寫(xiě)成按旋轉(zhuǎn)、縮放、平移三個(gè)分量的復(fù)合形式，則其代數(shù)式如下：x=m00*x+m01*y+m02;y=m10*x+m11*y+m12;其示意圖如下:ro幾種典型的仿射變換：1 .publicstaticAffineTransformgetTranslateInstance（doubletx,doublety）平移變換，將每一點(diǎn)移動(dòng)到（x+tx,y+ty）,變換矩陣為：10tx01ty001（譯注：平移變換是一種“剛體變換，rigid-bodytransformation,中學(xué)學(xué)過(guò)的物理，都知道啥叫“剛體”吧，就是不會(huì)產(chǎn)生形變的理想物體，平移當(dāng)然不會(huì)改變二

3、維圖形的形狀。同理，下面的“旋轉(zhuǎn)變換”也是剛體變換，而“縮放”“錯(cuò)切”都是會(huì)改變圖形形狀的。）2 .publicstaticAffineTransformgetScaleInstance(doublesx,doublesy)縮放變換，將每一點(diǎn)的橫坐標(biāo)放大（縮?。﹕x倍，縱坐標(biāo)放大（縮?。┲羢y倍，變換矩陣為:sx000sy03 .publicstaticAffineTransformgetShearInstance(doubleshx,doubleshy)剪切變換，變換矩陣為:1shx0shy10001相當(dāng)于一個(gè)橫向剪切與一個(gè)縱向剪切的復(fù)合1001shx0shy10010001001（譯注：“

4、剪切變換”又稱“錯(cuò)切變換”，指的是類似于四邊形不穩(wěn)定性那種性質(zhì)，街邊小商店那種鐵拉門都見(jiàn)過(guò)吧？想象一下上面鐵條構(gòu)成的菱形拉動(dòng)的過(guò)程，那就是“錯(cuò)切”的過(guò)程。）4 .publicstaticAffineTransformgetRotateInstance(doubletheta)旋轉(zhuǎn)變換，目標(biāo)圖形圍繞原點(diǎn)順時(shí)針旋轉(zhuǎn)theta弧度，變換矩陣為:cos(theta)-sin(theta)0sin(theta)cos(theta)00015 .publicstaticAffineTransformgetRotateInstance(doubletheta,doublex,doubley)旋轉(zhuǎn)變換，目標(biāo)圖

5、形以(x,y)為軸心順時(shí)針旋轉(zhuǎn)theta弧度，變換矩陣為:cos(theta)-sin(theta)x-x*cos+y*sinsin(theta)cos(theta)y-x*sin-y*cos相當(dāng)于兩次平移變換與一次原點(diǎn)旋轉(zhuǎn)變換的復(fù)合:10-xcos(theta)01-ysin(theta)0010、仿射變換四參數(shù)求解-sin(theta)010xcos(theta)001y01001A、C#自定義函數(shù)實(shí)現(xiàn)求解:1、求解旋轉(zhuǎn)參數(shù)Rotaion:1日日/summary234567891011121314/獲取旋轉(zhuǎn)角度/summary/paramname=fromCoordPoint1/源點(diǎn)1目標(biāo)

6、點(diǎn)1源點(diǎn)2目標(biāo)點(diǎn)215/returns返回旋轉(zhuǎn)角度/returns1617privatedoubleGetRotation(CoordPointfromPoint1,CoordPointtoPoint1,CoordPointfromPoint2,CoordPointtoPoint2)1819日日(2021doublea=(toPoint2.Y-toPoint1.Y)*(fromPoint2.X-fromPoint1.X)-(toPoint2.X-toPoint1.X)*(fromPoint2.Y-fromPoint1.Y);2223Idoubleb=(toPoint2.X-toPoint1.X

7、)*(fromPoint2.X-fromPoint1.X)+(toPoint2.Y-toPoint1.Y)*(fromPoint2.Y-fromPoint1.Y);242526272829303132if(Math.Abs(b)0)returnMath.Tan(a/b);else33returnMath.Tan(0);2345678910111213141516173435L)2、求解縮放比例參數(shù)(Scale):1日U/獲取縮放比例因子/源點(diǎn)1/目標(biāo)點(diǎn)1/源點(diǎn)2/目標(biāo)點(diǎn)2/旋轉(zhuǎn)角度/返回旋轉(zhuǎn)因子1819privatedoubleGetScale(CoordPointfromPoint1,Coo

8、rdPointtoPoint1,CoordPointfromPoint2,CoordPointtoPoint2,doublerotation)2021日日(2223 doublea=toPoint2.X-toPoint1.X;2425 Idoubleb=(fromPoint2.X-fromPoint1.X)*Math.Cos(rotation)-(fromPoint2.Y-fromPoint1.Y)*Math.Sin(rotation);2627 if(Math.Abs(b)0)2829 returna/b;3031 else3233 return0;3435 L)3、求解X方向偏移距離參數(shù)(

9、XTranslate:1/*/23/得到X方向偏移量45/67/源點(diǎn)189/目標(biāo)點(diǎn)11011/旋轉(zhuǎn)角度1213/縮放因子1415/返回X方向偏移量1617privatedoubleGetXTranslation(CoordPointfromPoint1,CoordPointtoPoint1,doublerotation,doublescale)18192021return(toPoint1.X-scale*(fromPoint1.X*Math.Cos(rotation)-fromPoint1.Y*Math.Sin(rotation);222324254、求解Y方向偏移距離參數(shù)(YTransla

10、te):1/*/23/得到Y(jié)方向偏移量45/67/源點(diǎn)189/目標(biāo)點(diǎn)11011/旋轉(zhuǎn)角度1213/縮放因子1415/返回丫方向偏移量1617privatedoubleGetYTranslation(CoordPointfromPoint1,CoordPointtoPoint1,doublerotation,doublescale)18192021return(toPoint1.Y-scale*(fromPoint1.X*Math.Sin(rotation)+fromPoint1.Y*Math.Cos(rotation);2223EkC#+AE)ftW：1LJL_lIIIIII從控制點(diǎn)定義仿射變

11、換程式IIIIII源控制點(diǎn)III目標(biāo)控制點(diǎn)1011III返回變換定義1213privateITransformationGetAffineTransformation(IPointpFromPoints,IPointpToPoints)1415口日1617II實(shí)例化仿射變換對(duì)象1819)；IAffineTransformation2D3GENtAffineTransformation=newAffineTransformation2DClass(21II從源控制點(diǎn)定義參數(shù)202223tAffineTransformation.DefineFromControlPoints(refpFromPo

12、ints,refpToPoints);2425查詢引用接口2627ITransformationtTransformation=tAffineTransformationasITransformation;2829returntTransformation;3031L三、空間對(duì)象轉(zhuǎn)換求出參數(shù)后，再利用公式對(duì)相應(yīng)坐標(biāo)點(diǎn)進(jìn)行轉(zhuǎn)換是一件相對(duì)簡(jiǎn)單的事件了。示例代碼:/轉(zhuǎn)換空間點(diǎn)/點(diǎn)/返回轉(zhuǎn)換后的點(diǎn)privateIGeometryTransformPoint(IPointpPoint)/*/說(shuō)明：采用相似變換模型（四參數(shù)變換模型）/X=ax+by+c/Y=-bx+ay+d/*doubleA=this.m_Scale*Math.Cos(doubleB=this.m_Scale*Math.Sin(IPointtPoint=newPointClass();tPoint.X=A*pPoint.X+B*pPoint.Y+this.m_RotationAngle);this.m_RotationAngle);this.m_DX;tPoint.Y=B*(-1.0)*