CGAffineTransformMake 矩阵变换 的运算原理

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=cos?，b=sin?，c=-sin?，d=cos?。

[aX + cY + tx   bX + dY + ty  1] = [Xcos? - Ysin?    Xsin? + Ycos?  1] ;

可见，这个时候，?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?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;      }