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    <p>I wrote an aiming subroutine for <a href="http://quozl.netrek.org/xtank/" rel="noreferrer">xtank</a> a while back. I'll try to lay out how I did it.</p> <p><strong>Disclaimer:</strong> I may have made one or more silly mistakes anywhere in here; I'm just trying to reconstruct the reasoning with my rusty math skills. However, I'll cut to the chase first, since this is a programming Q&amp;A instead of a math class :-)</p> <h3>How to do it</h3> <p>It boils down to solving a quadratic equation of the form: </p> <pre><code>a * sqr(x) + b * x + c == 0 </code></pre> <p>Note that by <code>sqr</code> I mean square, as opposed to square root. Use the following values:</p> <pre><code>a := sqr(target.velocityX) + sqr(target.velocityY) - sqr(projectile_speed) b := 2 * (target.velocityX * (target.startX - cannon.X) + target.velocityY * (target.startY - cannon.Y)) c := sqr(target.startX - cannon.X) + sqr(target.startY - cannon.Y) </code></pre> <p>Now we can look at the discriminant to determine if we have a possible solution.</p> <pre><code>disc := sqr(b) - 4 * a * c </code></pre> <p>If the discriminant is less than 0, forget about hitting your target -- your projectile can never get there in time. Otherwise, look at two candidate solutions:</p> <pre><code>t1 := (-b + sqrt(disc)) / (2 * a) t2 := (-b - sqrt(disc)) / (2 * a) </code></pre> <p>Note that if <code>disc == 0</code> then <code>t1</code> and <code>t2</code> are equal.</p> <p>If there are no other considerations such as intervening obstacles, simply choose the smaller positive value. (Negative <em>t</em> values would require firing backward in time to use!)</p> <p>Substitute the chosen <code>t</code> value back into the target's position equations to get the coordinates of the leading point you should be aiming at:</p> <pre><code>aim.X := t * target.velocityX + target.startX aim.Y := t * target.velocityY + target.startY </code></pre> <h3>Derivation</h3> <p>At time T, the projectile must be a (Euclidean) distance from the cannon equal to the elapsed time multiplied by the projectile speed. This gives an equation for a circle, parametric in elapsed time.</p> <pre><code>sqr(projectile.X - cannon.X) + sqr(projectile.Y - cannon.Y) == sqr(t * projectile_speed) </code></pre> <p>Similarly, at time T, the target has moved along its vector by time multiplied by its velocity:</p> <pre><code>target.X == t * target.velocityX + target.startX target.Y == t * target.velocityY + target.startY </code></pre> <p>The projectile can hit the target when its distance from the cannon matches the projectile's distance.</p> <pre><code>sqr(projectile.X - cannon.X) + sqr(projectile.Y - cannon.Y) == sqr(target.X - cannon.X) + sqr(target.Y - cannon.Y) </code></pre> <p>Wonderful! Substituting the expressions for target.X and target.Y gives</p> <pre><code>sqr(projectile.X - cannon.X) + sqr(projectile.Y - cannon.Y) == sqr((t * target.velocityX + target.startX) - cannon.X) + sqr((t * target.velocityY + target.startY) - cannon.Y) </code></pre> <p>Substituting the other side of the equation gives this:</p> <pre><code>sqr(t * projectile_speed) == sqr((t * target.velocityX + target.startX) - cannon.X) + sqr((t * target.velocityY + target.startY) - cannon.Y) </code></pre> <p>... subtracting <code>sqr(t * projectile_speed)</code> from both sides and flipping it around:</p> <pre><code>sqr((t * target.velocityX) + (target.startX - cannon.X)) + sqr((t * target.velocityY) + (target.startY - cannon.Y)) - sqr(t * projectile_speed) == 0 </code></pre> <p>... now resolve the results of squaring the subexpressions ...</p> <pre><code>sqr(target.velocityX) * sqr(t) + 2 * t * target.velocityX * (target.startX - cannon.X) + sqr(target.startX - cannon.X) + sqr(target.velocityY) * sqr(t) + 2 * t * target.velocityY * (target.startY - cannon.Y) + sqr(target.startY - cannon.Y) - sqr(projectile_speed) * sqr(t) == 0 </code></pre> <p>... and group similar terms ...</p> <pre><code>sqr(target.velocityX) * sqr(t) + sqr(target.velocityY) * sqr(t) - sqr(projectile_speed) * sqr(t) + 2 * t * target.velocityX * (target.startX - cannon.X) + 2 * t * target.velocityY * (target.startY - cannon.Y) + sqr(target.startX - cannon.X) + sqr(target.startY - cannon.Y) == 0 </code></pre> <p>... then combine them ...</p> <pre><code>(sqr(target.velocityX) + sqr(target.velocityY) - sqr(projectile_speed)) * sqr(t) + 2 * (target.velocityX * (target.startX - cannon.X) + target.velocityY * (target.startY - cannon.Y)) * t + sqr(target.startX - cannon.X) + sqr(target.startY - cannon.Y) == 0 </code></pre> <p>... giving a standard quadratic equation in <em>t</em>. Finding the positive real zeros of this equation gives the (zero, one, or two) possible hit locations, which can be done with the quadratic formula:</p> <pre><code>a * sqr(x) + b * x + c == 0 x == (-b ± sqrt(sqr(b) - 4 * a * c)) / (2 * a) </code></pre>
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    1. COThis is great stuff ::- D. Completely saved my sorry ass.
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    2. CONote that this approach will fail whenever target and projectile have the same speed, independent of direction, as a==0 and causes division by zero issues. See Broofa's answer which handles this and another potential problem case.
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    3. CO@e100 I did leave out some of the corner cases, such as dealing with zero coefficients; I may have to go back and add them sometime.
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