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math.html
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math.html
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<title>How to go from device coordinates back to worldspace in OpenGL (with explanation)</title>
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<h1>How to go from device coordinates back to worldspace in OpenGL (with explanation)</h1>
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<p>So you’re writing a 3D engine or somesuch in OpenGL (modern or legacy), and you’re not a beginner so you know that the worldspace coordinates get transformed using your modelview/projection matrix (which we’ll treat as one for the purpose of this article) into clip space, which gets transformed by the perspective division into normalized device coordinates.</p>
<p>If you don’t, go read <a href="http://unspecified.wordpress.com/2012/06/21/calculating-the-gluperspective-matrix-and-other-opengl-matrix-maths/">http://unspecified.wordpress.com/2012/06/21/calculating-the-gluperspective-matrix-and-other-opengl-matrix-maths/</a>, it’s really good.</p>
<p>So you end up with <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAKCAIAAAACUFjqAAAABmJLR0QA/wD/AP+gvaeTAAAA/UlEQVQYlU3QscqCYBjF8WO+g4ggOCV4A+5CbS2CNxCEU4Ogd+Dg7C20Bl1CS6uEo6OjoGNDNOSLiujr8w3xRb/xv50D+jdNE+dcCEFE4zhyzomIAQBQluX5fAbweDx838/znHPOGAMRPZ/POI6XZSGiw+Hged4wDGEY7nY7BuByuQRBIEkSgNfrtd1uFUVJkgQA6Edd147j3G63b1nhx/1+l2V5s9l8CwPQNI1hGLquF0WxXq8NwwCQZdn7/V51Xef7/ul06vu+qipN0wAIIa7Xq+u60jzPx+PRNE1VVff7fZqmlmUJIaIosm0bRLQsS9u2n0OEEG3bfkYS0R/iuqlK1A2WLwAAAABJRU5ErkJggg==" style="vertical-align:middle" alt="x" title="x" />, <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAANCAIAAAD0YtNRAAAABmJLR0QA/wD/AP+gvaeTAAABHUlEQVQYlU2QP+uCcBDGT/1NtlhNLoLxBWeXQHoFtomvxCnKMXB3dxRFdPUFOARNgqtjU5EaRYH/4Bq+Er/bjs9zd889DCICwDiOfd/zPA//CxHrut7v94ZhxHGMiIh4uVxM02QBwHGcw+EgSdLpdKIDSZI0TcP2fT+fz1mWLYpCURTKiqIghABdEgTBer1+PB601TTNdV2WCrMsI4QIggAAeZ63bavr+sSu1ysh5KcTBGG1Wk1MFMXn8wkAZVmmaSrLMsdxEzsej+/327KsKIo+n4+qqtN/5/N5t9tRF2EYbjabqqoQ8Q8AfN9/vV4AcL/fgyCwbXu5XAIAg4h5nnuet1gsbrebYRjb7ZYeYmieXdcNwzCbzRiG+cX5BSN6m36WZ6cRAAAAAElFTkSuQmCC" style="vertical-align:middle" alt="y" title="y" /> and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAKCAIAAADpZ+PpAAAABmJLR0QA/wD/AP+gvaeTAAAAwElEQVQYlW2PIQqEUBiEZ8FLiMEDGC3iFTyJVRCTwWIUfF7CYjOaTTaz4AVEUJH3i/pveOyuYSfNMMMHA/7ovu91Xa/rUpGINAAA9n33fb/ve9d1hRBVVZ3n+WJmAEKIeZ4dx6nrelkWwzDSNIUiDMOgzDiOcRwTETODH2rbNkkSKaWKv65pmqIonlP8LaIoYmYNQNd1eZ7rup5lmWVZZVmapgkAUsogCIhomibP82zbDsNQvYS6qVDHcWzb9iW/AWFRzzGLtvnzAAAAAElFTkSuQmCC" style="vertical-align:middle" alt="z" title="z" /> (depth buffer) values; that is, device coordinates. But for some occasions, like shadow mapping or deferred lighting, it’s useful to do this transformation backwards - going from device coordinates back to worldspace. In this article, I’m going to walk you through the math of it.</p>
<p>Let’s start with what we have. We start out with our world space position, <img src="data:image/png;base64,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" style="vertical-align:middle" alt="world" title="world" />. <strong>Note that <img src="data:image/png;base64,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" style="vertical-align:middle" alt="world_w" title="world_w" /> is assumed to be 1</strong> - for the rationale, see the article linked above.</p>
<p>We transform it to clip space by multiplying it with our projection/modelview matrix.</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip = Matrix\text{ }world" title="clip = Matrix\text{ }world" /><br /></p>
<p>Then move on to device coordinates by dividing with <img src="data:image/png;base64,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" style="vertical-align:middle" alt="w" title="w" />.</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="device = clip_{xyz} / clip_w" title="device = clip_{xyz} / clip_w" /><br /></p>
<p>So the problem we face is: given <img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip = Matrix\text{ }world" title="clip = Matrix\text{ }world" />, <img src="data:image/png;base64,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" style="vertical-align:middle" alt="device = clip_{xyz} / clip_w" title="device = clip_{xyz} / clip_w" />, <img src="data:image/png;base64,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" style="vertical-align:middle" alt="world_w = 1" title="world_w = 1" />,</p>
<p>and given <img src="data:image/png;base64,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" style="vertical-align:middle" alt="device" title="device" /> as an input and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADkAAAAOCAIAAAAaBSPHAAAABmJLR0QA/wD/AP+gvaeTAAAFB0lEQVRIibWWX0hTbxjHn7lDzm1gkODmWG5Ujkhxf8gQdDpXF3XRnwsRkUAwG+QKQSNQ8saosbxQEyOimzCUshX2j/yDzEbDbcx1QFy5NnQOd2Uqx72bds77u3h/HE8FP/394Pe9et7n/bzP85znPOfwijDGAAAAHMelUikAEIlEMpkM/hDGeGtri9hSqTQrK+tP5n/Vbr6lpSWHw3Hu3LmqqqpwOPwn+vDhQ7PZ3NTU5HA4lpaW9gz98+fPHz9+7LOO27dvd3R07LdWrVbb3d19+PBhiqK+f//+GxePx91ut0gk6urqunPnjlar3TP94OBgfX39PmsFAKPR+M8AJVwkk8n8/PxEIhGJRIR+juP6+/s1Gk0ymdTpdPvM7fP5NBrNPuHu7u49mV9m7v3792VlZXl5eYuLi0L/69ev9Xp9OBzWaDQURcE+lE6no9Honq36V/olcSAQcDgcHz9+3NjY4J3JZNLv99+8ebO/v99isfx23u/3Dw0NsSxbWVkJAJcuXWIY5u7duwzDpFKpubm59vb2xsZGlUrV2dmZTqc7Ozs/fPgQjUYBwOl0UhQ1MjIyPT1dXFx8/fp1EtPlcnk8ns3NzZaWFq/XG41Gd3Z27t27B1igK1euYIx7e3urq6t5Z1dXVyKRePfunclkomlayA8PD589ezaRSGCMb9y4YTQaecDpdFqtVp50Op00TV+4cMFqtc7Ozr59+9ZoNAaDweHh4VevXr148eLUqVOEdLlcz549wxiPjY1VVFT09fV9+vTJZDJNTU3tzkA8Hj948CAASCQSlmW3t7cBYGZmRqlUFhQUeL1eqVRaUlIi5AcGBurq6goKCgBAp9PJZDIeoGn66NGjPLyyslJSUsIwjMViKSsrMxgMLS0ter3e6/VevHjR5/ORIADgdrvJF7mxscFxnM1mO378eHNzs9ls3u3rkydPRkZGMMaBQECv18diMYZh2tradnZ2MMbnz5+/fPmysKl2u728vDydTpNlY2MjD2QymfLy8kePHvHw+vp6JBLR6/U+n08YZH19nWVZs9nc09PDe4hx7dq12tpaIbzb1y9fvlRVVQGASqUSi8XhcLinp+fq1asURaVSqUQiYTKZhJO6sLBQWFiYnZ1NlouLiwaDgdihUAghZLVaeTg3N3dyclIqlZaWlgqD5ObmPn/+HCFks9l4DzG+fv1aXFwshHdr3draUigUAKBQKLKzsz0ej0wmKyoqIu8FY1xTUyM8mclk8vLyiD0/P48QOn369P379wl/6NChI0eOAADxkAfQarUHDhyAX/XmzZuTJ0/K5XJCxmIxAFheXt7c3OQ/ZbL1d60vX74UiUT8eZVKRdN0a2srWX7+/Fk4i0QnTpwgBkKor69PIpEUFRWtrKwAwNramkqlAoCxsTGlUsk3/remEsVisYqKCpLi8ePH9fX1CKGJiQmO49RqNQA8ffqUNFjc0dFx69atqakphmGCwWB1dbVYLA4GgzabTa1Wz8zMPHjw4Nu3bzk5OYFAQCKRFBYWkhylpaWjo6OhUGh8fPzMmTPz8/OhUKimpkan02UyGY/HQ9M0x3HNzc0AsLq6OjQ0ZLfb+dJ5eb3eeDy+urra2tqKEIpGo5FIRC6XHzt2bHR0dG5uLisrq6GhAQBE/N3lP4hlWYQQuceQX0dOTg7Z2t7e5jhOIpHwMEKI3xWK3Jnkcjl/kGVZQiKExGIxPzZ/Aaer3ZP3NmMvAAAAAElFTkSuQmCC" style="vertical-align:middle" alt="Matrix" title="Matrix" /> as a constant, calculate <img src="data:image/png;base64,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" style="vertical-align:middle" alt="world" title="world" />.</p>
<p>Let’s walk through it. Invert the first step:</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="Matrix^{-1}\text{ }clip = Matrix^{-1}\text{ }Matrix\text{ }world" title="Matrix^{-1}\text{ }clip = Matrix^{-1}\text{ }Matrix\text{ }world" /><br /> <br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="Matrix^{-1}\text{ }clip = world" title="Matrix^{-1}\text{ }clip = world" /><br /></p>
<p>Now let’s see what we can do with the second equation.</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="device = clip_{xyz} / clip_w" title="device = clip_{xyz} / clip_w" /><br /></p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip_w\text{ }device = clip_{xyz}" title="clip_w\text{ }device = clip_{xyz}" /><br /></p>
<p>Let’s use this syntax to indicate a 4-vector formed by combining a 3-vector and a fourth number:</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip = clip_{xyzw} = (clip_{xyz}, clip_w)" title="clip = clip_{xyzw} = (clip_{xyz}, clip_w)" /><br /></p>
<p>substitute <img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip_{xyz}" title="clip_{xyz}" /></p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip = (clip_w\text{ }device, clip_w)" title="clip = (clip_w\text{ }device, clip_w)" /><br /></p>
<p>insert into our earlier equation</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="Matrix^{-1}\text{ }clip = world" title="Matrix^{-1}\text{ }clip = world" /><br /> <br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="Matrix^{-1}\text{ }(clip_w\text{ }device, clip_w) = world" title="Matrix^{-1}\text{ }(clip_w\text{ }device, clip_w) = world" /><br /> <br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="Matrix^{-1}\text{ }clip_w\text{ }(device, 1) = world" title="Matrix^{-1}\text{ }clip_w\text{ }(device, 1) = world" /><br /></p>
<p>And note that since matrices are linear transforms, we can pull that <img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip_w" title="clip_w" /> in front of the matrix multiply:</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip_w\text{ }Matrix^{-1}\text{ }(device, 1) = world" title="clip_w\text{ }Matrix^{-1}\text{ }(device, 1) = world" /><br /></p>
<p>So it seems we run into a wall. <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACkAAAAVCAIAAABUulB3AAAABmJLR0QA/wD/AP+gvaeTAAAEN0lEQVRIibWUS0gybRTHz6uToxXdocA0DDOKwgiyAkHoYnYxuhBYURFtIipoISQF3bFFtS4XEXSBVoUFgYvCRQ1CRRlGFyIYC7GkzMrRMZxvMR/DyxevRn3vbzXzf86Z/5zzHM4viqIgJD6fDwC4XG7osG/ACnF2f3/f399fXl7e09PDiA6Ho76+fn9//+968/n8sbExn88nEokY0e12s9lsgUDwc2+gQmI0GqVS6d7eXuiw7xGqbgDY39/n8XiFhYX/Q5WfQJgngiDW1taOj49jY2M7OzszMjIA4Pz8XCgU8ng8ACBJcm5u7vX1VavVxsXFMYmrq6sHBwfNzc35+fkLCws3NzfFxcWtra3hzenyn5+f6+rqJicnX15ebDZbfX09RVEul0smk42Pj9Mxg4ODOI7X1tYODw8zfXM6nX19fQaDQS6XDw4OYhhGEERZWdnS0tKXeu52u9vb26VS6dDQUExMzPz8vNvtBoCDgwOSJBUKBQDs7u6mpqYKBIKnp6fIyEjm141Go0qlcrlcXq+3u7u7qKiIy+WmpKSYTKYv9Xx2dvb+/n5xcZGWRkZGAoEAAGAYxuVyZTIZAHA4nJaWlp2dHYIgmpqamHw+n69UKpeXl/l8flpaGi1+fHyw2eyw3qxgMGixWAQCQVJSEi0lJiampKQAgM1mEwgE9FaRy+Xx8fHr6+tisVgsFjP5lZWVfr/fbrdnZ2fTCkmSOI5/ydvj8Tw+PtKT9Tsul8vpdObm5vp8PqvVCgB2u91mszU0NHg8nouLCyby9PT0/f2dbg8AmEym//Tmj94xMTF0lQwkSU5NTd3d3fn9foVCsbW1dXZ2BgAbGxsoilZVVa2uruI4zsSbzWYWi6VSqejX7e1tiURSU1MT3pvFYnV0dFxeXrrd7kAgYDQae3t7NRoNiqIREREikQjDsMrKSgB4e3tLTk5+eXm5uroqKytjPmG1WhEEMRgMJElOTk46nc7p6emwxgDwi6IoANja2trZ2UEQpKioqKGhgb7jlZWVo6OjlpaWgoICAPD5fDqdLjo6emBgICEhgc73er0VFRUymSw5ORnH8by8vLa2NhRFGYPr62sMwzQaDYfDoSgKx/G0tDSLxRIREfGv97exWCzd3d3Dw8ONjY2fT9/e3jY3N6+vr0mS1Ov1MzMz5+fni4uLExMTHo8nzE4Ny97eHgD86XbNZrNarT45OZFIJABwdHSUk5MDAF1dXenp6T/y1uv1h4eH8fHxWq3298lnqK6ufn5+fnh4UKvVwWDw9va2pKQEAKxWa2lpKfI54evodLqwMZubm0KhMCkpCcMwDoeTlZUFAA6HQ6lU/rTnYfF4PNHR0QDw8PCAoiiKogRBIAjCYrHYo6Ojf9U7MzPTbDYHAgF63yEIcnZ21tTUhCDIT+f8KwSDQa/XGxUVFQwGCYLg8Xj0xv0H+b6BBzgneqEAAAAASUVORK5CYII=" style="vertical-align:middle" alt="clip_w" title="clip_w" /> is lost, right? Don’t give up hope: we haven’t used the third of our initial givens yet.</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="world_w = 1" title="world_w = 1" /><br /></p>
<p>So let’s look at just the <img src="data:image/png;base64,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" style="vertical-align:middle" alt="w" title="w" /> component of that last equation there:</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip_w\text{ }\left(Matrix^{-1}\text{ }(device, 1)\right)_w = world_w = 1" title="clip_w\text{ }\left(Matrix^{-1}\text{ }(device, 1)\right)_w = world_w = 1" /><br /></p>
<p>Divide:</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="clip_w = \frac 1 {\left(Matrix^{-1}\text{ }(device, 1)\right)_w}" title="clip_w = \frac 1 {\left(Matrix^{-1}\text{ }(device, 1)\right)_w}" /><br /></p>
<p>And insert into the equation that previously gave us trouble:</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="\frac{Matrix^{-1}\text{ }(device, 1)}{\left(Matrix^{-1}\text{ }(device, 1)\right)_w} = world" title="\frac{Matrix^{-1}\text{ }(device, 1)}{\left(Matrix^{-1}\text{ }(device, 1)\right)_w} = world" /><br /></p>
<p>Or in other words:</p>
<p><br /><img src="data:image/png;base64,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" style="vertical-align:middle" alt="\left(Matrix^{-1}\text{ }(device, 1)\right)_{xyz/w} = world\text{ ... and done.}" title="\left(Matrix^{-1}\text{ }(device, 1)\right)_{xyz/w} = world\text{ ... and done.}" /><br /></p>
<p>You’ve probably seen that equation in shader code before. Well, now you know why.</p>
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