我设法让它工作......
所以我做了一个小的 C++/OpenGL 示例来测试它：
//---------------------------------------------------------------------------
//--- input data ------------------------------------------------------------
//---------------------------------------------------------------------------
double m[16]= // matrix
{
1.0825317547305484,-0.6249999999999999,0.0,20.0,
0.6249999999999999, 1.0825317547305484,0.0,30.0,
0.0 , 0.0 ,1.0,0.0 ,
0.0 , 0.0 ,0.0,1.0
};
double a[4][3]= // your un-transformed rectangle
{
{ 30 ,50 ,0 },
{ 30+200,50 ,0 },
{ 30+200,50+100,0 },
{ 30 ,50+100,0 },
};
//---------------------------------------------------------------------------
//--- just matrix and vector math you can ignore this -----------------------
//---------------------------------------------------------------------------
void vector_copy (double *c,double *a) { for(int i=0;i<3;i++) c[i]=a[i]; }
void matrix_mul (double *c,double *a,double *b) // c[16] = a[16]*b[16]
{
double q[16];
q[ 0]=(a[ 0]*b[ 0])+(a[ 1]*b[ 4])+(a[ 2]*b[ 8])+(a[ 3]*b[12]);
q[ 1]=(a[ 0]*b[ 1])+(a[ 1]*b[ 5])+(a[ 2]*b[ 9])+(a[ 3]*b[13]);
q[ 2]=(a[ 0]*b[ 2])+(a[ 1]*b[ 6])+(a[ 2]*b[10])+(a[ 3]*b[14]);
q[ 3]=(a[ 0]*b[ 3])+(a[ 1]*b[ 7])+(a[ 2]*b[11])+(a[ 3]*b[15]);
q[ 4]=(a[ 4]*b[ 0])+(a[ 5]*b[ 4])+(a[ 6]*b[ 8])+(a[ 7]*b[12]);
q[ 5]=(a[ 4]*b[ 1])+(a[ 5]*b[ 5])+(a[ 6]*b[ 9])+(a[ 7]*b[13]);
q[ 6]=(a[ 4]*b[ 2])+(a[ 5]*b[ 6])+(a[ 6]*b[10])+(a[ 7]*b[14]);
q[ 7]=(a[ 4]*b[ 3])+(a[ 5]*b[ 7])+(a[ 6]*b[11])+(a[ 7]*b[15]);
q[ 8]=(a[ 8]*b[ 0])+(a[ 9]*b[ 4])+(a[10]*b[ 8])+(a[11]*b[12]);
q[ 9]=(a[ 8]*b[ 1])+(a[ 9]*b[ 5])+(a[10]*b[ 9])+(a[11]*b[13]);
q[10]=(a[ 8]*b[ 2])+(a[ 9]*b[ 6])+(a[10]*b[10])+(a[11]*b[14]);
q[11]=(a[ 8]*b[ 3])+(a[ 9]*b[ 7])+(a[10]*b[11])+(a[11]*b[15]);
q[12]=(a[12]*b[ 0])+(a[13]*b[ 4])+(a[14]*b[ 8])+(a[15]*b[12]);
q[13]=(a[12]*b[ 1])+(a[13]*b[ 5])+(a[14]*b[ 9])+(a[15]*b[13]);
q[14]=(a[12]*b[ 2])+(a[13]*b[ 6])+(a[14]*b[10])+(a[15]*b[14]);
q[15]=(a[12]*b[ 3])+(a[13]*b[ 7])+(a[14]*b[11])+(a[15]*b[15]);
for(int i=0;i<16;i++) c[i]=q[i];
}
void matrix_mul_vector(double *c,double *a,double *b) // c[3] = a[16]*b[3]
{
double q[3];
q[0]=(a[ 0]*b[0])+(a[ 4]*b[1])+(a[ 8]*b[2])+(a[12]);
q[1]=(a[ 1]*b[0])+(a[ 5]*b[1])+(a[ 9]*b[2])+(a[13]);
q[2]=(a[ 2]*b[0])+(a[ 6]*b[1])+(a[10]*b[2])+(a[14]);
for(int i=0;i<3;i++) c[i]=q[i];
}
void matrix_subdet (double *c,double *a); // c[16] = all subdets of a[16]
double matrix_subdet ( double *a,int r,int s); // = subdet(r,s) of a[16]
double matrix_det ( double *a); // = det of a[16]
double matrix_det ( double *a,double *b); // = det of a[16] and subdets b[16]
void matrix_inv2 (double *c,double *a); // c[16] = a[16] ^ -1
void matrix_subdet (double *c,double *a)
{
double q[16];
int i,j;
for (i=0;i<4;i++)
for (j=0;j<4;j++)
q[j+(i<<2)]=matrix_subdet(a,i,j);
for (i=0;i<16;i++) c[i]=q[i];
}
double matrix_subdet ( double *a,int r,int s)
{
double c,q[9];
int i,j,k;
k=0; // q = sub matrix
for (j=0;j<4;j++)
if (j!=s)
for (i=0;i<4;i++)
if (i!=r)
{
q[k]=a[i+(j<<2)];
k++;
}
c=0;
c+=q[0]*q[4]*q[8];
c+=q[1]*q[5]*q[6];
c+=q[2]*q[3]*q[7];
c-=q[0]*q[5]*q[7];
c-=q[1]*q[3]*q[8];
c-=q[2]*q[4]*q[6];
if (int((r+s)&1)) c=-c; // add signum
return c;
}
double matrix_det ( double *a)
{
double c=0;
c+=a[ 0]*matrix_subdet(a,0,0);
c+=a[ 4]*matrix_subdet(a,0,1);
c+=a[ 8]*matrix_subdet(a,0,2);
c+=a[12]*matrix_subdet(a,0,3);
return c;
}
double matrix_det ( double *a,double *b)
{
double c=0;
c+=a[ 0]*b[ 0];
c+=a[ 4]*b[ 1];
c+=a[ 8]*b[ 2];
c+=a[12]*b[ 3];
return c;
}
void matrix_inv(double *c,double *a)
{
double d[16],D;
matrix_subdet(d,a);
D=matrix_det(a,d);
if (D) D=1.0/D;
for (int i=0;i<16;i++) c[i]=d[i]*D;
}
//---------------------------------------------------------------------------
//--- render ----------------------------------------------------------------
//---------------------------------------------------------------------------
void TForm1::draw()
{
int i;
double xs=ClientWidth,ys=ClientHeight; // just my GL view resolution
double p[3],im[16]; // some temp point and inverse matrix
double b[4][3]; // your rectangle
matrix_inv(im,m); // im=Inverse(m)
glClear(GL_COLOR_BUFFER_BIT);
glDisable(GL_CULL_FACE);
glDisable(GL_DEPTH_TEST);
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
glTranslated(-0.5,+0.5,0.0);
glScaled(2.0/xs,-2.0/ys,1.0);
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
// render transformed a and compute its BBOX into b
glColor3d(0.0,0.5,0.5); // aqua
glBegin(GL_LINE_LOOP);
for (i=0;i<4;i++)
{
// transform and render rectangle a
matrix_mul_vector(p,im,a[i]); // p = inverse(m)*a[i]
glVertex2dv(p);
// compute transformed BBOX b[0]=min(p) b[2]=max(p)
if (i==0)
{
vector_copy(b[0],p);
vector_copy(b[2],p);
}
if (b[0][0]>p[0]) b[0][0]=p[0];
if (b[2][0]<p[0]) b[2][0]=p[0];
if (b[0][1]>p[1]) b[0][1]=p[1];
if (b[2][1]<p[1]) b[2][1]=p[1];
}
glEnd();
// convert BBOX b[0],b[2] to rectangle
b[1][0]=b[2][0];
b[1][1]=b[0][1];
b[3][0]=b[0][0];
b[3][1]=b[2][1];
// render transformed b
glColor3d(0.8,0.0,0.0); // red
glBegin(GL_LINE_LOOP);
for (i=0;i<4;i++) glVertex2dv(b[i]); glEnd();
// untransform b to rectangle local coordinates
for (i=0;i<4;i++) matrix_mul_vector(b[i],m,b[i]); // b[i] = m*b[i]
// render a,b in local coordiantes (untransformed)
glColor3d(0.0,0.25,0.25); // aqua
glBegin(GL_LINE_LOOP);
for (i=0;i<4;i++) glVertex2dv(a[i]);
glEnd();
glColor3d(0.4,0.0,0.0); // red
glBegin(GL_LINE_LOOP);
for (i=0;i<4;i++) glVertex2dv(b[i]);
glEnd();
glFlush();
SwapBuffers(hdc);
}
//---------------------------------------------------------------------------
并调整了你的转换符号，直到它与你的预览相匹配：
较浅的颜色对应于由m 转换的东西，而较深的颜色则没有转换（矩形局部）。
我发现您将p 转换为画布坐标的符号是：
p' = Inverse(m)*p
所以要转换回来，你通常会完成：
p = m*p`
但是，如果您的逆矩阵只是伪逆矩阵（如 this），您需要将逆矩阵除以比例。
因此，如果您获得了 BBOX 的积分，您只需将 m 乘以它们，结果就是您所寻求的。