您应该指定您真正想要的投影。曲面有很多方法（不仅仅是球体）。你的问题是这种变换的逆，所以首先是直接投影（平面->球面）。我使用这两个（都用于特定目的）：
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球体上区域中心的距离与平面上的距离相匹配
这用于校正曲面上的纹理，例如眼镜上的装饰品等...
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球体上到视轴的垂直距离与平面上的距离相匹配
因此，如果您从视轴看，您会在球体和平面上看到相同的图像，只需设置坐标系，Z 轴是观察方向，x,y 轴对应于您的 2D 平面轴.然后只需计算 z 坐标以匹配球面
我想你想要第一个选项
所以计算中间点 (x0,y0) 作为边界框的中心或均匀间隔的点平均点。通过atan2 计算每个点的ang 和坐标（从中间点）弧度！！！
然后计算dx,dy 并计算二维坐标为(x,y)=(x0+dx,y0+dy)
这里的结果示例（我将其用于任何类型的曲率）：
[备注]
还有其他基于光线投射的方法，可能还有更多...
[edit1] C++ 示例
为您准备的小型 C++ 类：
//---------------------------------------------------------------------------
#include <Math.h>
class sphere_projection
{
public:
float x0,y0,z0,r0; // 3D sphere
float u0,v0; // mid point of 2D image
float m; // scale 2D image
int mode; // which projection type
sphere_projection()
{
x0=0.0; y0=0.0; z0=0.0; r0=1.0;
u0=0.0; v0=0.0; m=1.0;
mode=1;
}
void uv2xyz(float &x,float &y,float &z,float u,float v)
{
if (mode==1)
{
float a,b;
// 2D position scaled around midpoint and converted from arclength to angle
u=(u-u0)*m/r0;
v=(v-v0)*m/r0;
// correct on radius distrotion in both axises
a=u/cos(v);
b=v/cos(u);
// compute the 3D cartesian point on surface
z=z0+(r0*cos(b)*cos(a));
x=x0+(r0*cos(b)*sin(a));
y=y0+(r0*sin(b));
}
if (mode==2)
{
// 2D position scaled around midpoint
x=(u-u0)*m;
y=(v-v0)*m;
// compute the 3D cartesian point on surface
x=x0+x;
y=y0+y;
z=z0+sqrt(r0*r0-x*x-y*y);
}
}
void uv2xy (float &x,float &y, float u,float v)
{
if (mode==1)
{
float a,b,z;
// 2D position scaled around midpoint and converted from arclength to angle
a=(u-u0)*m/r0;
b=(v-v0)*m/r0;
// correct on radius distrotion in both axises and convert back to 2D position
x=u0+(a*r0/(m*cos(b)));
y=v0+(b*r0/(m*cos(a)));
}
if (mode==2)
{
float z;
// 2D position scaled around midpoint + Z axis
x=(u-u0)*m;
y=(v-v0)*m;
z=sqrt(r0*r0-x*x-y*y);
// compute arclengths and convert back to 2D position
x=u0+(r0*atan2(x,z)/m);
y=v0+(r0*atan2(y,z)/m);
}
}
};
//---------------------------------------------------------------------------
这是如何使用它（在 OpenGL 中渲染）：
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
glMatrixMode(GL_MODELVIEW);
glPushMatrix();
glTranslatef(0.0,+2.5,-20.0);
static float ang=0.0; ang+=2.5;
float x,y,z,u,v,d=0.2;
sphere_projection sp;
sp.x0=0.0;
sp.y0=0.0;
sp.z0=0.0;
sp.r0=1.5;
sp.u0=0.0;
sp.v0=0.0;
sp.m =0.5;
for (sp.mode=1;sp.mode<=2;sp.mode++)
{
// original 2D grid
glMatrixMode(GL_MODELVIEW);
glTranslatef(-5.0,0.0,0.0);
glColor3f(1.0f, 1.0f, 1.0f);
for (u=d-1.0;u<=1.0;u+=d)
for (v=d-1.0;v<=1.0;v+=d)
{
glBegin(GL_LINE_LOOP);
glVertex3f(u-d,v-d,0.0);
glVertex3f(u-d,v ,0.0);
glVertex3f(u ,v ,0.0);
glVertex3f(u ,v-d,0.0);
glEnd();
}
// sphere mapped corrected
glMatrixMode(GL_MODELVIEW);
glTranslatef(+5.0,0.0,0.0);
glPushMatrix();
glRotatef(ang,0.0,1.0,0.0);
glColor3f(1.0f, 0.0f, 0.0f);
for (u=d-1.0;u<=1.0;u+=d)
for (v=d-1.0;v<=1.0;v+=d)
{
glBegin(GL_LINE_LOOP);
sp.uv2xyz(x,y,z,u-d,v-d); glVertex3f(x,y,z);
sp.uv2xyz(x,y,z,u-d,v ); glVertex3f(x,y,z);
sp.uv2xyz(x,y,z,u ,v ); glVertex3f(x,y,z);
sp.uv2xyz(x,y,z,u ,v-d); glVertex3f(x,y,z);
glEnd();
}
glMatrixMode(GL_MODELVIEW);
glPopMatrix();
// sphere mapped corrected
glMatrixMode(GL_MODELVIEW);
glTranslatef(+5.0,0.0,0.0);
glColor3f(0.0f, 0.0f, 1.0f);
for (u=d-1.0;u<=1.0;u+=d)
for (v=d-1.0;v<=1.0;v+=d)
{
glBegin(GL_LINE_LOOP);
sp.uv2xy(x,y,u-d,v-d); glVertex3f(x,y,0.0);
sp.uv2xy(x,y,u-d,v ); glVertex3f(x,y,0.0);
sp.uv2xy(x,y,u ,v ); glVertex3f(x,y,0.0);
sp.uv2xy(x,y,u ,v-d); glVertex3f(x,y,0.0);
glEnd();
}
glTranslatef(-5.0,-5.0,0.0);
}
glMatrixMode(GL_MODELVIEW);
glPopMatrix();
glFlush();
SwapBuffers(hdc);
这是结果：
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sp.uv2xy 将 2D (u,v) 图像坐标转换为经过投影校正的 2D (x,y) 坐标（图像）
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sp.uv2xyz 将 2D (u,v) 图像坐标转换为经过投影校正的 3D (x,y,x) 坐标（x,y 轴对应于屏幕 x,y 轴的球面）
-
sp.mode {1,2} 选择您要使用的投影类型
-
sp.u0,v0,m选择投影图像中点和比例
-
sp.x0,y0,z0,r0 定义了您要投影的球体
[edit2] Sphere EquirectangularProjection
这个二维u,v坐标不需要校正直接转换成球角a=long,b=lat所以u,v在<0,+1>范围内：
a=x*2.0*M_PI; b=(y-0.5)*M_PI;
那么3D坐标就是球面变换：
x=x0+(r0*cos(b)*cos(a));
y=y0+(r0*cos(b)*sin(a));
z=z0+(r0*sin(b));
如果你想要反向变换谷歌球坐标系