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;

caseUIImageOrientationRightMirrored:

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:

[NSExceptionraise: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;

}