CGAffineTransformMake(a,b,c,d,tx,ty) 矩阵运算的原理 （转载）

简记:

CGAffineTransformMake(a,b,c,d,tx,ty)

ad缩放bc旋转tx,ty位移，基础的2D矩阵

公式

x=ax+cy+tx
    y=bx+dy+ty

1.矩阵的基本知识：

struct CGAffineTransform

{
  CGFloat a, b, c, d;
  CGFloat tx, ty;
};

CGAffineTransform CGAffineTransformMake (CGFloat a,CGFloat b,CGFloat c,CGFloat d,CGFloat tx,CGFloat ty);

为了把二维图形的变化统一在一个坐标系里，引入了齐次坐标的概念，即把一个图形用一个三维矩阵表示，其中第三列总是(0,0,1)，用来作为坐标系的标准。所以所有的变化都由前两列完成。

以上参数在矩阵中的表示为：

|a    b    0|

|c    d    0|

|tx   ty   1|

运算原理：原坐标设为（X,Y,1）;

|a    b    0|

[X，Y,  1]      |c    d    0|     =     [aX + cY + tx   bX + dY + ty  1] ;

|tx    ty  1|

通过矩阵运算后的坐标[aX + cY + tx   bX + dY + ty  1]，我们对比一下可知：

第一种：设a=d=1, b=c=0.  

[aX + cY + tx   bX + dY + ty  1] = [X  + tx  Y + ty  1];

可见，这个时候，坐标是按照向量（tx，ty）进行平移，其实这也就是函数

CGAffineTransform CGAffineMakeTranslation(CGFloat tx,CGFloat ty)的计算原理。

第二种：设b=c=tx=ty=0.  

[aX + cY + tx   bX + dY + ty  1] = [aX    dY   1];

可见，这个时候，坐标X按照a进行缩放，Y按照d进行缩放，a，d就是X，Y的比例系数，其实这也就是函数

CGAffineTransform CGAffineTransformMakeScale(CGFloat sx, CGFloat sy)的计算原理。a对应于sx，d对应于sy。

第三种：设tx=ty=0，a=cosɵ，b=sinɵ，c=-sinɵ，d=cosɵ。

[aX + cY + tx   bX + dY + ty  1] = [Xcosɵ - Ysinɵ    Xsinɵ + Ycosɵ  1] ;

可见，这个时候，ɵ就是旋转的角度，逆时针为正，顺时针为负。其实这也就是函数

CGAffineTransform CGAffineTransformMakeRotation(CGFloat angle)的计算原理。angle即ɵ的弧度表示。

2.利用上面的变换写一个UIImage矩阵变换的例子：

下面是一个关于image的矩阵运算的例子,无外乎是运用以上三种变换的组合，达到所定义的效果

 //UIImageOrientation的定义，定义了如下几种变换
 typedef enum
 {
     UIImageOrientationUp,            // default orientation  
 
    UIImageOrientationDown,          // 180 deg rotation  
 
    UIImageOrientationLeft,          // 90 deg CCW  
 
     UIImageOrientationRight,         // 90 deg CW  
 
     UIImageOrientationUpMirrored,    // as above but image mirrored along other axis. horizontal flip  
 
     UIImageOrientationDownMirrored,  // horizontal flip  
 
     UIImageOrientationLeftMirrored,  // vertical flip  
 
     UIImageOrientationRightMirrored, // vertical flip  
 
 } UIImageOrientation;  
 
 //按照UIImageOrientation的定义，利用矩阵自定义实现对应的变换；  
 
 -(UIImage *)transformImage:(UIImage *)aImage    
 
 {    
 
     CGImageRef imgRef = aImage.CGImage;    
 
     CGFloat width = CGImageGetWidth(imgRef);    
 
     CGFloat height = CGImageGetHeight(imgRef);    
 
     CGAffineTransform transform = CGAffineTransformIdentity;    
 
     CGRect bounds = CGRectMake(, , width, height);    
 
     CGFloat scaleRatio = ;    
 
     CGFloat boundHeight;    
 
     UIImageOrientation orient = aImage.imageOrientation;    
 
     switch(UIImageOrientationLeftMirrored)    
 
     {    
 
         case UIImageOrientationUp:    
 
             transform = CGAffineTransformIdentity;  
 
             break;    
 
         case UIImageOrientationUpMirrored:   
 
             transform = CGAffineTransformMakeTranslation(width, 0.0);    
 
             transform = CGAffineTransformScale(transform, -1.0, 1.0); //沿y轴向左翻   
 
             break;    
 
         case UIImageOrientationDown:
             transform = CGAffineTransformMakeTranslation(width, height);    
 
             transform = CGAffineTransformRotate(transform, M_PI);    
 
             break;    
 
         case UIImageOrientationDownMirrored:   
 
             transform = CGAffineTransformMakeTranslation(0.0, height);    
 
             transform = CGAffineTransformScale(transform, 1.0, -1.0);    
 
             break;    
 
         case UIImageOrientationLeft:     
 
             boundHeight = bounds.size.height;    
 
             bounds.size.height = bounds.size.width;    
 
             bounds.size.width = boundHeight;    
 
             transform = CGAffineTransformMakeTranslation(0.0, width);    
 
             transform = CGAffineTransformRotate(transform, 3.0 * M_PI / 2.0);    
 
             break;  
 
         case UIImageOrientationLeftMirrored:     
 
             boundHeight = bounds.size.height;    
 
             bounds.size.height = bounds.size.width;    
 
             bounds.size.width = boundHeight;    
 
             transform = CGAffineTransformMakeTranslation(height, width);    
 
             transform = CGAffineTransformScale(transform, -1.0, 1.0);    
 
             transform = CGAffineTransformRotate(transform, 3.0 * M_PI / 2.0);    
 
             break;    
 
         case UIImageOrientationRight: //EXIF = 8    
 
             boundHeight = bounds.size.height;    
 
             bounds.size.height = bounds.size.width;    
 
             bounds.size.width = boundHeight;    
 
             transform = CGAffineTransformMakeTranslation(height, 0.0);    
 
             transform = CGAffineTransformRotate(transform, M_PI / 2.0);    
 
             break;  
 
         case UIImageOrientationRightMirrored:   
 
             boundHeight = bounds.size.height;    
 
             bounds.size.height = bounds.size.width;    
 
             bounds.size.width = boundHeight;    
 
             transform = CGAffineTransformMakeScale(-1.0, 1.0);    
 
             transform = CGAffineTransformRotate(transform, M_PI / 2.0);    
 
             break;    
 
         default:    
 
             [NSException raise:NSInternalInconsistencyException format:@"Invalid image orientation"];    
 
     }    
 
     UIGraphicsBeginImageContext(bounds.size);    
 
     CGContextRef context = UIGraphicsGetCurrentContext();    
 
     if (orient == UIImageOrientationRight || orient == UIImageOrientationLeft) {    
 
         CGContextScaleCTM(context, -scaleRatio, scaleRatio);    
 
         CGContextTranslateCTM(context, -height, );    
 
     }    
 
     else {    
 
         CGContextScaleCTM(context, scaleRatio, -scaleRatio);    
 
         CGContextTranslateCTM(context, , -height);    
 
     }    
 
     CGContextConcatCTM(context, transform);    
 
     CGContextDrawImage(UIGraphicsGetCurrentContext(), CGRectMake(, , width, height), imgRef);    
 
     UIImage *imageCopy = UIGraphicsGetImageFromCurrentImageContext();  
 
     UIGraphicsEndImageContext();    
 
     return imageCopy;    
 
 } 

借鉴： https://developer.apple.com/library/ios/documentation/GraphicsImaging/Conceptual/drawingwithquartz2d/dq_affine/dq_affine.html