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    <p>Not very neat math, but I hope it works for you:</p> <h2>Input</h2> <p>Suppose you are given a </p> <ul> <li>worldViewPerspectiveMatrix <code>M</code> (Transforms world space to perspective space, orthogonal matrices should work as well!)</li> <li>a constant velocity (more specifcally a distance per Frame) in screen space <code>v_screen</code></li> <li>a (normalized) direction vector <code>V_dir</code> in world space, this will give the direction your object moves. The actual length of the vector will be calculated.</li> <li>a position <code>X</code> where your Object is placed in world space.</li> </ul> <h2>Result</h2> <ul> <li>a scaling factor <code>a</code> for moving your object 'v_screen' units in screen space along <code>V_dir</code> in world space. Thus you want to move your object actually <code>X += a*V_dir</code>.</li> </ul> <p>First we use the formula of mapping the <em>world space</em> to <em>screen space</em> by using homogenous coordinates and w-clipping. The <code>.</code> denotes the matrix multiplication operator.</p> <pre><code>X_proj := M.(x,y,z,1); X_screen := (x_proj/w_proj,y_proj/w_proj); </code></pre> <p>From there we can specify the position of the object <code>X</code> in screen space and the position after appling the directional movement <code>X+(a*V_dir)</code>.</p> <p>The length in screenspace and therefore the desired distance (per frame) is simply </p> <pre><code>Length[(X+(a*dir))_screen - X_screen] == v_screen. </code></pre> <p>Let's solve this for <code>a</code>. I used my Mathematica to calculate <code>a</code> for you. If you want further details, I may elabourate the answer. Suppose <code>M</code>, <code>X</code>, and <code>V_dir</code> is</p> <pre><code> / xp \ / vx \ / vxp \ X_proj = | yp | V_dir = | vy | V_proj = M.V_dir = | vyp | | zp | | vz | | vzp | \ wp / \ 0 / \ vwp / </code></pre> <p>And <code>v_screen</code> is your desired screen space velocity. Then <code>a</code> is:</p> <pre><code>vlen = v_screen*v_screen; a = (-dwp*vlen*wp + dxp*xp + dyp*yp + 0.5*Sqrt( Power(-2*dwp*vlen*wp + 2*dxp*xp + 2*dyp*yp,2) - 4*(dxp*dxp + dyp*dyp - dwp*dwp*vlen)*(xp*xp + yp*yp - wp*wp*vlen) ) ) / (dxp*dxp + dyp*dyp - dwp*dwp*vlen); </code></pre> <p>Remember, <code>-a</code> is also a solution. </p>
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    1. COI did not mention that I am coding in c#/XNA, maybe there is a build-in function there that already computes this? Because for me trying to code your explanation is really scary :).
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    2. COErm yes I think it is :) ... I just found the problem interessting: finding a constant screen space velocity. I try to modify my answer to XNA/C# code if you like. Another Solution I actually would propose is to define all movements in *screen space*. Then you map the screen space position to your 3d Plane by [ray casting](http://msdn.microsoft.com/de-de/library/microsoft.xna.framework.ray.aspx)
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    3. COMy game has some specific mechanics that I need be implemented as well as some visual effects. Namely I need characters to move with screen-space speed to allow for smooth aiming (the aiming will happen through movement) but also I need the characters to explode in a number of gory bits when they die :). And as the second part would most likely be implemented manually (physics engine integration would be an overkill and too difficult), I would prefer for the characters to be actually positioned on a plane in a 3d world, so I can have more flexibility creating realistically-looking gore :).
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