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

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(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); //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, 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;

}