CGAffineTransformMake 矩阵变换 的运算原理(转)

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(0, 0, width, height);    
         
    CGFloat scaleRatio = 1;    
         
    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);    
              
            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, 0);    
          
    }    
      
    else {    
          
        CGContextScaleCTM(context, scaleRatio, -scaleRatio);    
          
        CGContextTranslateCTM(context, 0, -height);    
          
    }    
         
    CGContextConcatCTM(context, transform);    
         
    CGContextDrawImage(UIGraphicsGetCurrentContext(), CGRectMake(0, 0, width, height), imgRef);    
      
    UIImage *imageCopy = UIGraphicsGetImageFromCurrentImageContext();  
         
    UIGraphicsEndImageContext();    
         
    return imageCopy;    
      
}