伪圆柱投影的投影矩阵

Question

想象一下，在一个球体内有一组混合的3D对象，你的目标是创建整个场景的等面积柱形投影。使用OpenGL，你可以考虑从围绕中心轴旋转相机的多个渲染目标纹理（确切地说是4个），然后在后期处理着色器中纠正径向畸变，因为你将投影到一个平面而不是一个圆柱体。理想情况下，你应该能够让相机视锥体在球体的整个体积上扫过，没有任何重叠，并且每个渲染填满矩形纹理的整个像素空间（就像柱形投影那样）。

所以，只是为了清楚起见，这里有一个球形场景的可视化（其中包含对象），以及一个围绕Y轴旋转PI / 2的相机视锥体。

注意，'远'平面被缩减为一条线，与球体的Y轴共线。在视点空间中，形成“X”字形的白色相交线代表相机的原点，即(0,0,0)。这个外部面也是'近'平面，距离相机0个Z单位。

思路是球体的中心轴向外投射射线，使得所有射线都平行于Y平面（即法线为(0,1,0)的平面），并且从球体的起点发出的每条射线都以垂直角度与球体表面相交。

我的问题是：

天真地说，我认为OpenGL投影矩阵可以做到这一点 - 据我所知，我要实现的投影是线性的，因此可能吗？然而，我似乎无法正确解决方程。

让 s 是球体的半径。

因此，在视空间中，从相机的原点出发：

• 近平面的左右边缘分别位于沿X轴的-s和s单位处
• 近平面的上下边缘分别位于沿Y轴的s和-s单位处
• 远平面的左右边缘共位于沿Z轴的-s单位处（请记住，在视空间中，Z值在相机前面为负）

在OpenGL投影矩阵中：

• -w_c < x_c < w_c
• x_n = x_c / w_c

由于左右视锥面在相机前面收敛，我求解了一个将我的输入映射到预期输出的方程，并得出结论：

• x_n = x_e / (z_e + s)

这意味着x_c = x_e和w_c = z_e + s。这填补了投影矩阵中的两行：




---------- 这是我陷入困境的地方 ----------

很明显，y_n与x_e或z_e无关，并且它的方程应为：

• y_n = y_e / s

这类似于正交投影。然而，这与我在x_n方程中已经求解出的w_c存在冲突。

我通过参考this article中的步骤推导出了我的投影矩阵，该文章简明地解释了OpenGL透视和正交投影矩阵的推导过程。

看起来我可能遇到了线性变换的限制？如果这确实是非线性的话，那么我不明白为什么，并且希望能得到解释 :)

- Blake Regalia

Answer 1

要计算球形投影，必须将方位角投影到视口的Y坐标上。这是在视图空间中XZ平面上一个点的投影向量的角度。高度角必须被投影到视口的X坐标上。这是指从视图空间中的一个点到XZ平面的向量与该平面的夹角。到该点的向量的长度必须投影到深度上。

由于角度必须通过反正弦计算，因此无法通过投影矩阵完成。
投影矩阵描述了从场景的3D点到视口的2D点的映射。它将从眼睛空间转换到剪辑空间，并且剪辑空间中的坐标通过除以剪辑坐标的w分量来转换为归一化设备坐标（NDC）。 NDC范围为(-1,-1,-1)至(1,1,1)。这可以用来描述简单的有理函数，但不能描述反三角函数。
（请参见更多关于如何在现代OpenGL中使用片段着色器中的gl_FragCoord.z线性渲染深度的信息How to render depth linearly in modern OpenGL with gl_FragCoord.z in fragment shader?，而不是转换模型矩阵）

以下顶点着色器展示了如何计算球形投影，投影中心到视口的位置由视图矩阵定义：

in vec3 inPos;

uniform mat4 u_viewMat44;
uniform mat4 u_modelMat44;
uniform vec2 u_depthRange;

const float cPi = 3.141593;

void main()
{
    vec4  viewPos = u_viewMat44 * u_modelMat44 * vec4( inPos, 1.0 );
    vec2  dirXY   = normalize( vec2( -viewPos.z, viewPos.x ) );
    vec2  dirZ    = normalize( vec2( length(viewPos.xz), viewPos.y ) );
    float posX    = asin( abs( dirXY.y ) ) * 2.0 / cPi;
    float posY    = asin( abs( dirZ.y ) ) * 2.0 / cPi;

    gl_Position = vec4(
        0.5 * sign( dirXY.y ) * mix(2.0-posX, posX, step(0.0, dirXY.x) ),
        sign( dirZ.y ) * posY,
        2.0 * (length(viewPos.xyz)-u_depthRange.x) / (u_depthRange.y-u_depthRange.x) - 1.0
        1.0 );
}

注意，这个着色器有一个大问题。如果一个基元从接近180°的方位角到接近-180°的方位角，那么该基元将无法到达视口的边界。该基元将绘制在整个视口上。

请查看以下WebGL示例，展示了该着色器和问题：

glArrayType = typeof Float32Array !="undefined" ? Float32Array : ( typeof WebGLFloatArray != "undefined" ? WebGLFloatArray : Array );

function IdentityMat44() {
  var m = new glArrayType(16);
  m[0]  = 1; m[1]  = 0; m[2]  = 0; m[3]  = 0;
  m[4]  = 0; m[5]  = 1; m[6]  = 0; m[7]  = 0;
  m[8]  = 0; m[9]  = 0; m[10] = 1; m[11] = 0;
  m[12] = 0; m[13] = 0; m[14] = 0; m[15] = 1;
  return m;
};

function RotateAxis(matA, angRad, axis) {
    var aMap = [ [1, 2], [2, 0], [0, 1] ];
    var a0 = aMap[axis][0], a1 = aMap[axis][1]; 
    var sinAng = Math.sin(angRad), cosAng = Math.cos(angRad);
    var matB = new glArrayType(16);
    for ( var i = 0; i < 16; ++ i ) matB[i] = matA[i];
    for ( var i = 0; i < 3; ++ i ) {
        matB[a0*4+i] = matA[a0*4+i] * cosAng + matA[a1*4+i] * sinAng;
        matB[a1*4+i] = matA[a0*4+i] * -sinAng + matA[a1*4+i] * cosAng;
    }
    return matB;
}

function Translate( matA, trans ) {
    var matB = new glArrayType(16);
    for ( var i = 0; i < 16; ++ i ) matB[i] = matA[i];
    for ( var i = 0; i < 3; ++ i )
        matB[12+i] = matA[i] * trans[0] + matA[4+i] * trans[1] + matA[8+i] * trans[2] + matA[12+i];
    return matB;
}

function Cross( a, b ) { return [ a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0], 0.0 ]; }
function Dot( a, b ) { return a[0]*b[0] + a[1]*b[1] + a[2]*b[2]; }
function Normalize( v ) {
    var len = Math.sqrt( v[0] * v[0] + v[1] * v[1] + v[2] * v[2] );
    return [ v[0] / len, v[1] / len, v[2] / len ];
}

var Camera = {};
Camera.create = function() {
    this.pos    = [0, 0, 0.0];
    this.target = [0, -1, 0];
    this.up     = [0, 0, 1];
    this.fov_y  = 120;
    this.vp     = [800, 600];
    this.near   = 0.5;
    this.far    = 100.0;
}
Camera.Perspective = function() {
    var fn = this.far + this.near;
    var f_n = this.far - this.near;
    var r = this.vp[0] / this.vp[1];
    var t = 1 / Math.tan( Math.PI * this.fov_y / 360 );
    var m = IdentityMat44();
    m[0]  = t/r; m[1]  = 0; m[2]  =  0;                              m[3]  = 0;
    m[4]  = 0;   m[5]  = t; m[6]  =  0;                              m[7]  = 0;
    m[8]  = 0;   m[9]  = 0; m[10] = -fn / f_n;                       m[11] = -1;
    m[12] = 0;   m[13] = 0; m[14] = -2 * this.far * this.near / f_n; m[15] =  0;
    return m;
}
Camera.LookAt = function() {
    var mz = Normalize( [ this.pos[0]-this.target[0], this.pos[1]-this.target[1], this.pos[2]-this.target[2] ] );
    var mx = Normalize( Cross( this.up, mz ) );
    var my = Normalize( Cross( mz, mx ) );
    var tx = Dot( mx, this.pos );
    var ty = Dot( my, this.pos );
    var tz = Dot( [-mz[0], -mz[1], -mz[2]], this.pos ); 
    var m = IdentityMat44();
    m[0]  = mx[0]; m[1]  = my[0]; m[2]  = mz[0]; m[3]  = 0;
    m[4]  = mx[1]; m[5]  = my[1]; m[6]  = mz[1]; m[7]  = 0;
    m[8]  = mx[2]; m[9]  = my[2]; m[10] = mz[2]; m[11] = 0;
    m[12] = tx;    m[13] = ty;    m[14] = tz;    m[15] = 1; 
    return m;
} 

var ShaderProgram = {};
ShaderProgram.Create = function( shaderList ) {
    var shaderObjs = [];
    for ( var i_sh = 0; i_sh < shaderList.length; ++ i_sh ) {
        var shderObj = this.CompileShader( shaderList[i_sh].source, shaderList[i_sh].stage );
        if ( shderObj == 0 )
            return 0;
        shaderObjs.push( shderObj );
    }
    var progObj = this.LinkProgram( shaderObjs )
    if ( progObj != 0 ) {
        progObj.attribIndex = {};
        var noOfAttributes = gl.getProgramParameter( progObj, gl.ACTIVE_ATTRIBUTES );
        for ( var i_n = 0; i_n < noOfAttributes; ++ i_n ) {
            var name = gl.getActiveAttrib( progObj, i_n ).name;
            progObj.attribIndex[name] = gl.getAttribLocation( progObj, name );
        }
        progObj.unifomLocation = {};
        var noOfUniforms = gl.getProgramParameter( progObj, gl.ACTIVE_UNIFORMS );
        for ( var i_n = 0; i_n < noOfUniforms; ++ i_n ) {
            var name = gl.getActiveUniform( progObj, i_n ).name;
            progObj.unifomLocation[name] = gl.getUniformLocation( progObj, name );
        }
    }
    return progObj;
}
ShaderProgram.AttributeIndex = function( progObj, name ) { return progObj.attribIndex[name]; } 
ShaderProgram.UniformLocation = function( progObj, name ) { return progObj.unifomLocation[name]; } 
ShaderProgram.Use = function( progObj ) { gl.useProgram( progObj ); } 
ShaderProgram.SetUniformI1  = function( progObj, name, val ) { if(progObj.unifomLocation[name]) gl.uniform1i( progObj.unifomLocation[name], val ); }
ShaderProgram.SetUniformF1  = function( progObj, name, val ) { if(progObj.unifomLocation[name]) gl.uniform1f( progObj.unifomLocation[name], val ); }
ShaderProgram.SetUniformF2  = function( progObj, name, arr ) { if(progObj.unifomLocation[name]) gl.uniform2fv( progObj.unifomLocation[name], arr ); }
ShaderProgram.SetUniformF3  = function( progObj, name, arr ) { if(progObj.unifomLocation[name]) gl.uniform3fv( progObj.unifomLocation[name], arr ); }
ShaderProgram.SetUniformF4  = function( progObj, name, arr ) { if(progObj.unifomLocation[name]) gl.uniform4fv( progObj.unifomLocation[name], arr ); }
ShaderProgram.SetUniformM33 = function( progObj, name, mat ) { if(progObj.unifomLocation[name]) gl.uniformMatrix3fv( progObj.unifomLocation[name], false, mat ); }
ShaderProgram.SetUniformM44 = function( progObj, name, mat ) { if(progObj.unifomLocation[name]) gl.uniformMatrix4fv( progObj.unifomLocation[name], false, mat ); }
ShaderProgram.CompileShader = function( source, shaderStage ) {
    var shaderScript = document.getElementById(source);
    if (shaderScript) {
      source = "";
      var node = shaderScript.firstChild;
      while (node) {
        if (node.nodeType == 3) source += node.textContent;
        node = node.nextSibling;
      }
    }
    var shaderObj = gl.createShader( shaderStage );
    gl.shaderSource( shaderObj, source );
    gl.compileShader( shaderObj );
    var status = gl.getShaderParameter( shaderObj, gl.COMPILE_STATUS );
    if ( !status ) alert(gl.getShaderInfoLog(shaderObj));
    return status ? shaderObj : 0;
} 
ShaderProgram.LinkProgram = function( shaderObjs ) {
    var prog = gl.createProgram();
    for ( var i_sh = 0; i_sh < shaderObjs.length; ++ i_sh )
        gl.attachShader( prog, shaderObjs[i_sh] );
    gl.linkProgram( prog );
    status = gl.getProgramParameter( prog, gl.LINK_STATUS );
    if ( !status ) alert("Could not initialise shaders");
    gl.useProgram( null );
    return status ? prog : 0;
}

var VertexBuffer = {};
VertexBuffer.Create = function( attributes, indices ) {
    var buffer = {};
    buffer.buf = [];
    buffer.attr = []
    for ( var i = 0; i < attributes.length; ++ i ) {
        buffer.buf.push( gl.createBuffer() );
        buffer.attr.push( { size : attributes[i].attrSize, loc : attributes[i].attrLoc } );
        gl.bindBuffer( gl.ARRAY_BUFFER, buffer.buf[i] );
        gl.bufferData( gl.ARRAY_BUFFER, new Float32Array( attributes[i].data ), gl.STATIC_DRAW );
    }
    buffer.inx = gl.createBuffer();
    gl.bindBuffer( gl.ELEMENT_ARRAY_BUFFER, buffer.inx );
    gl.bufferData( gl.ELEMENT_ARRAY_BUFFER, new Uint16Array( indices ), gl.STATIC_DRAW );
    buffer.inxLen = indices.length;
    gl.bindBuffer( gl.ARRAY_BUFFER, null );
    gl.bindBuffer( gl.ELEMENT_ARRAY_BUFFER, null );
    return buffer;
}
VertexBuffer.Draw = function( bufObj ) {
  for ( var i = 0; i < bufObj.buf.length; ++ i ) {
        gl.bindBuffer( gl.ARRAY_BUFFER, bufObj.buf[i] );
        gl.vertexAttribPointer( bufObj.attr[i].loc, bufObj.attr[i].size, gl.FLOAT, false, 0, 0 );
        gl.enableVertexAttribArray( bufObj.attr[i].loc );
    }
    gl.bindBuffer( gl.ELEMENT_ARRAY_BUFFER, bufObj.inx );
    gl.drawElements( gl.TRIANGLES, bufObj.inxLen, gl.UNSIGNED_SHORT, 0 );
    for ( var i = 0; i < bufObj.buf.length; ++ i )
       gl.disableVertexAttribArray( bufObj.attr[i].loc );
    gl.bindBuffer( gl.ARRAY_BUFFER, null );
    gl.bindBuffer( gl.ELEMENT_ARRAY_BUFFER, null );
}

        
function drawScene(){

    var projection = document.getElementById( "projection" ).value;

    var canvas = document.getElementById( "glow-canvas" );
    Camera.create();
    Camera.vp = [canvas.width, canvas.height];
    var currentTime = Date.now();   
    var deltaMS = currentTime - startTime;
        
    gl.viewport( 0, 0, canvas.width, canvas.height );
    gl.enable( gl.DEPTH_TEST );
    gl.clearColor( 0.0, 0.0, 0.0, 1.0 );
    gl.clear( gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT );
    gl.enable( gl.CULL_FACE );
    gl.cullFace( gl.BACK );
    gl.frontFace( gl.CCW );
    
    // set up draw shader
    ShaderProgram.Use( progDraw );
    ShaderProgram.SetUniformM44( progDraw, "u_projectionMat44", Camera.Perspective() );
    ShaderProgram.SetUniformM44( progDraw, "u_viewMat44", Camera.LookAt() );
    ShaderProgram.SetUniformF2( progDraw, "u_depthRange", [ Camera.near, Camera.far ] );
    ShaderProgram.SetUniformF1( progDraw, "u_projection", projection )
    ShaderProgram.SetUniformF3( progDraw, "u_lightDir", [-1.0, -0.5, -2.0] );
    ShaderProgram.SetUniformF1( progDraw, "u_ambient", 0.2 );
    ShaderProgram.SetUniformF1( progDraw, "u_diffuse", 0.7 );
    ShaderProgram.SetUniformF1( progDraw, "u_specular", 0.8 );
    ShaderProgram.SetUniformF1( progDraw, "u_shininess", 10.0 );
    var modelMat = IdentityMat44()
    modelMat = RotateAxis( modelMat, CalcAng( currentTime, 10.0 ), 2 );
    modelMat = Translate( modelMat, [0.0, -2.5, 0.0] );
    modelMat = RotateAxis( modelMat, CalcAng( currentTime, 13.0 ), 0 );
    modelMat = RotateAxis( modelMat, CalcAng( currentTime, 17.0 ), 1 );
    ShaderProgram.SetUniformM44( progDraw, "u_modelMat44", modelMat );
    
    // draw scene
    VertexBuffer.Draw( bufTorus );
}

var startTime;
function Fract( val ) { 
    return val - Math.trunc( val );
}
function CalcAng( currentTime, intervall ) {
    return Fract( (currentTime - startTime) / (1000*intervall) ) * 2.0 * Math.PI;
}
function CalcMove( currentTime, intervall, range ) {
    var pos = self.Fract( (currentTime - startTime) / (1000*intervall) ) * 2.0
    var pos = pos < 1.0 ? pos : (2.0-pos)
    return range[0] + (range[1] - range[0]) * pos;
}    
function EllipticalPosition( a, b, angRag ) {
    var a_b = a * a - b * b
    var ea = (a_b <= 0) ? 0 : Math.sqrt( a_b );
    var eb = (a_b >= 0) ? 0 : Math.sqrt( -a_b );
    return [ a * Math.sin( angRag ) - ea, b * Math.cos( angRag ) - eb, 0 ];
}

var sliderScale = 100.0
var gl;
var progDraw;
var bufCube = {};
var bufTorus = {};
function sceneStart() {

    document.getElementById( "projection" ).value = 0;
    
    var canvas = document.getElementById( "glow-canvas");
    var vp = [canvas.width, canvas.height];
    gl = canvas.getContext( "experimental-webgl" );
    if ( !gl )
      return;

    progDraw = ShaderProgram.Create( 
      [ { source : "draw-shader-vs", stage : gl.VERTEX_SHADER },
        { source : "draw-shader-fs", stage : gl.FRAGMENT_SHADER }
      ],
      [ "u_projectionMat44", "u_viewMat44", "u_modelMat44", 
        "u_lightDir", "u_ambient", "u_diffuse", "u_specular", "u_shininess", ] );
    progDraw.inPos = gl.getAttribLocation( progDraw, "inPos" );
    progDraw.inNV  = gl.getAttribLocation( progDraw, "inNV" );
    progDraw.inCol = gl.getAttribLocation( progDraw, "inCol" );
    if ( progDraw == 0 )
        return;

    // create torus
    var circum_size = 32, tube_size = 32;
    var rad_circum = 1.0;
    var rad_tube = 0.5;
    var torus_pts = [];
    var torus_nv = [];
    var torus_col = [];
    var torus_inx = [];
    var col = [1, 0.5, 0.0];
    for ( var i_c = 0; i_c < circum_size; ++ i_c ) {
        var center = [
            Math.cos(2 * Math.PI * i_c / circum_size),
            Math.sin(2 * Math.PI * i_c / circum_size) ]
        for ( var i_t = 0; i_t < tube_size; ++ i_t ) {
            var tubeX = Math.cos(2 * Math.PI * i_t / tube_size)
            var tubeY = Math.sin(2 * Math.PI * i_t / tube_size)
            var pt = [
                center[0] * ( rad_circum + tubeX * rad_tube ),
                center[1] * ( rad_circum + tubeX * rad_tube ),
                tubeY * rad_tube ]
            var nv = [ pt[0] - center[0] * rad_tube, pt[1] - center[1] * rad_tube, tubeY * rad_tube ]
            torus_pts.push( pt[0], pt[1], pt[2] );
            torus_nv.push( nv[0], nv[1], nv[2] );
            torus_col.push( col[0], col[1], col[2] );
            var i_cn = (i_c+1) % circum_size
            var i_tn = (i_t+1) % tube_size
            var i_c0 = i_c * tube_size; 
            var i_c1 = i_cn * tube_size; 
            torus_inx.push( i_c0+i_tn, i_c0+i_t, i_c1+i_t, i_c0+i_tn, i_c1+i_t, i_c1+i_tn )
        }
    }
    bufTorus = VertexBuffer.Create(
      [ { data : torus_pts, attrSize : 3, attrLoc : progDraw.inPos },
        { data : torus_nv,  attrSize : 3, attrLoc : progDraw.inNV },
        { data : torus_col, attrSize : 3, attrLoc : progDraw.inCol } ],
        torus_inx
    );

    startTime = Date.now();
    setInterval(drawScene, 50);
}

<script id="draw-shader-vs" type="x-shader/x-vertex">
precision mediump float;

attribute vec3 inPos;
attribute vec3 inNV;
attribute vec3 inCol;

varying vec3 vertPos;
varying vec3 vertNV;
varying vec3 vertCol;

uniform mat4  u_projectionMat44;
uniform mat4  u_viewMat44;
uniform mat4  u_modelMat44;
uniform vec2  u_depthRange;
uniform float u_projection;

const float cPi = 3.141593;

void main()
{
    vec3 modelNV  = mat3( u_modelMat44 ) * normalize( inNV );
    vertNV        = mat3( u_viewMat44 ) * modelNV;
    vertCol       = inCol;
    vec4 modelPos = u_modelMat44 * vec4( inPos, 1.0 );
    vec4 viewPos  = u_viewMat44 * modelPos;
    vertPos       = viewPos.xyz / viewPos.w;
    
    vec2 dirXY    = normalize( vec2( -viewPos.z, viewPos.x ) );
    vec2 dirZ     = normalize( vec2( length(viewPos.xz), viewPos.y ) );
    float posX    = asin( abs( dirXY.y ) ) * 2.0 / cPi;
    float posY    = asin( abs( dirZ.y ) ) * 2.0 / cPi;
    vec3 prjPos = vec3(
        0.5 * sign( dirXY.y ) * mix(2.0-posX, posX, step(0.0, dirXY.x) ),
        sign( dirZ.y ) * posY,
        2.0 * (length(viewPos.xyz)-u_depthRange.x) / (u_depthRange.y-u_depthRange.x) - 1.0 );
    gl_Position   = mix( vec4( prjPos.xyz, 1.0 ), u_projectionMat44 * viewPos, u_projection );
}
</script>

<script id="draw-shader-fs" type="x-shader/x-fragment">
precision mediump float;

varying vec3 vertPos;
varying vec3 vertNV;
varying vec3 vertCol;

uniform vec3  u_lightDir;
uniform float u_ambient;
uniform float u_diffuse;
uniform float u_specular;
uniform float u_shininess;

void main()
{
    vec3 color      = vertCol;
    vec3 lightCol   = u_ambient * color;
    vec3  normalV   = normalize( vertNV );
    vec3  lightV    = normalize( -u_lightDir );
    float NdotL     = max( 0.0, dot( normalV, lightV ) );
    lightCol       += NdotL * u_diffuse * color;
    vec3  eyeV      = normalize( -vertPos );
    vec3  halfV     = normalize( eyeV + lightV );
    float NdotH     = max( 0.0, dot( normalV, halfV ) );
    float kSpecular = ( u_shininess + 2.0 ) * pow( NdotH, u_shininess ) / ( 2.0 * 3.14159265 );
    lightCol       += kSpecular * u_specular * color;
    gl_FragColor    = vec4( lightCol.rgb, 1.0 );
}
</script>

<body onload="sceneStart();">
    <div style="margin-left: 520px;">
        <div style="float: right; width: 100%; background-color: #CCF;">
            <form name="inputs">
                <table>
                    <tr> <td> projection </td> <td>
                        <select id="projection">>
                            <option value="0">spherical</option>
                            <option value="1">perspectiv</option>
                        </select>
                    </td> </tr>
                </table>
            </form>
        </div>
        <div style="float: right; width: 520px; margin-left: -520px;">
            <canvas id="glow-canvas" style="border: none;" width="512" height="256"></canvas>
        </div>
        <div style="clear: both;"></div>
    </div>
</body>

- Rabbid76

Answer 2

由于您的投影需要 y = sin(phi)，因此映射是非线性的。

理论上，使用4x4矩阵（以及齐次坐标），您可以描述仿射变换，这比线性变换更具表现力。仿射变换的形式为 a * x + b，其中a和x是向量，b是标量值。无法用它来表达三角函数。

- BDL