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copied!<p>It all comes back to Newton's equations:</p> <pre><code>F = m * a s = s_o + v * t + a * t^2 / 2 v = v_o + a * t </code></pre> <p>In this case <code>F</code> is the force (thrust), <code>a</code> is the acceleration, and <code>m</code> is the mass of the ship. <code>s</code> is the current location, <code>s_o</code> is the original location, <code>v</code> is the velocity, and <code>t</code> is the current time.</p> <p>Of course this is along a straight line, so if you want to convert to two or three dimensions you'll have to do some math. <code>F</code>, <code>s</code>, <code>v</code>, and <code>a</code> are all vectors, meaning that their direction is equally important. Technically <code>t</code> is also a vector, but since time generally only goes one direction, we don't have to worry about that.</p> <pre><code>2d: F^2 = F_x^2 + F_y^2 (use Pythagorean theorem to split force into components) F_x = m * a_x F_y = m * a_y s_x = s_o_x + v_x * t + a_x * t^2 / 2 s_y = s_o_y + v_y * t + a_y * t^2 / 2 v_x = v_o_x + a_x * t v_y = v_o_y + a_y * t 3d: F^2 = F_x^2 + F_y^2 + F_z^2 (surprisingly, this works) F_x = m * a_x F_y = m * a_y F_z = m * a_z s_x = s_o_x + v_x * t + a_x * t^2 / 2 s_y = s_o_y + v_y * t + a_y * t^2 / 2 s_z = s_o_z + v_z * t + a_z * t^2 / 2 v_x = v_o_x + a_x * t v_y = v_o_y + a_y * t v_z = v_o_z + a_z * t </code></pre> <p>Now to adjust your velocity to the direction of the player, you've got a fixed total force (<code>F</code>) in order to change your current velocity toward the player. In physics things don't happen instantaneously, but your goal should be to minimize the time in which the change happens ('t').</p> <p>This gives you an equation in terms of your current location (<code>(s_o_x,s_o_y)</code> or <code>(s_o_x,s_o_y,s_o_z)</code>) and your opponent's current location or your target location (<code>(s_x,s_y)</code> or <code>(s_x,s_y,s_z)</code>), for your target velocity (ignores acceleration).</p> <pre><code>v_x = (s_x - s_o_x) / t v_y = (s_y - s_o_y) / t v_x = (s_x - s_o_x) / t v_y = (s_y - s_o_y) / t v_z = (s_z - z_o_y) / t </code></pre> <p>We can substitute this for our other equation:</p> <pre><code>(s_x - s_o_x) / t = v_o_x + a_x * t (s_y - s_o_y) / t = v_o_y + a_y * t (s_x - s_o_x) / t = v_o_x + a_x * t (s_y - s_o_y) / t = v_o_y + a_y * t (s_z - z_o_y) / t = v_o_z + a_z * t </code></pre> <p>We then solve for the acceleration (this is related to the force, which is what we are trying to calculate).</p> <pre><code>(s_x - s_o_x) / t^2 - v_o_x / t = a_x (s_y - s_o_y) / t^2 - v_o_y / t = a_y (s_x - s_o_x) / t^2 - v_o_x / t = a_x (s_y - s_o_y) / t^2 - v_o_y / t = a_y (s_z - z_o_y) / t^2 - v_o_z / t = a_z </code></pre> <p>Plug this into the force equation:</p> <pre><code>F_x = m * (s_x - s_o_x) / t^2 - m * v_o_x / t F_y = m * (s_y - s_o_y) / t^2 - m * v_o_y / t F_x = m * (s_x - s_o_x) / t^2 - m * v_o_x / t F_y = m * (s_y - s_o_y) / t^2 - m * v_o_y / t F_z = m * (s_z - z_o_y) / t^2 - m * v_o_z / t </code></pre> <p>Now solve for <code>t</code>:</p> <pre><code>t = (-m * v_o_x +/- sqrt(m^2 * v_o_x^2 - 4 * F_x * m * (s_x - s_o_x))) / 2 / F_x t = (-m * v_o_y +/- sqrt(m^2 * v_o_y^2 - 4 * F_y * m * (s_y - s_o_y))) / 2 / F_y t = (-m * v_o_x +/- sqrt(m^2 * v_o_x^2 - 4 * F_x * m * (s_x - s_o_x))) / 2 / F_x t = (-m * v_o_y +/- sqrt(m^2 * v_o_y^2 - 4 * F_y * m * (s_y - s_o_y))) / 2 / F_y t = (-m * v_o_z +/- sqrt(m^2 * v_o_z^2 - 4 * F_z * m * (s_z - s_o_z))) / 2 / F_z </code></pre> <p>The times should converge, so the times will be equal! This gives us a system of equations for each coordinate (plane and sphere). Note that there are multiple possible values, but some involve imaginary numbers so you'll have to eliminate those solutions:</p> <pre><code>(-m * v_o_x +/- sqrt(m^2 * v_o_x^2 - 4 * F_x * m * (s_x - s_o_x))) / 2 / F_x = (-m * v_o_y +/- sqrt(m^2 * v_o_y^2 - 4 * F_y * m * (s_y - s_o_y))) / 2 / F_y F^2 = F_x^2 + F_y^2 (-m * v_o_x +/- sqrt(m^2 * v_o_x^2 - 4 * F_x * m * (s_x - s_o_x))) / 2 / F_x = (-m * v_o_y +/- sqrt(m^2 * v_o_y^2 - 4 * F_y * m * (s_y - s_o_y))) / 2 / F_y = (-m * v_o_z +/- sqrt(m^2 * v_o_z^2 - 4 * F_z * m * (s_z - s_o_z))) / 2 / F_z F^2 = F_x^2 + F_y^2 + F_z^2 </code></pre> <p>Solve for the <code>(F_x,F_y)</code> or <code>(F_x,F_y,F_z)</code> coordinates and you've got the force you need.</p> <p>Let me know if you have any questions or if you find mistakes in my math.</p>

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