Analytical Ballistic Trajectories with Approximately Linear Drag

Algorithm 1

Specialized C++ Quartic Solver.

double GetTimeToTargetRWithMinimalInitialSpeed(double k, double vInfinity,
double rX, double rY) {
// 1. Start by getting coefficients for the function f(t) = a4t∧4 + a3t∧3
// + a1*t + a0 which is 0 at the sought time-to-target t. Solving f(t) = 0
// for t > 0 is equivalent to solving e(u) = f(1/u)u∧4 = a0u∧4 + a1*u∧3 +
// a3*u + a4 = 0 for u where u = 1 / t, but the latter is more well-behaved,
// being a strictly concave function for u > 0 for any set of valid inputs,
// so solve e(u)=0 for u instead by converging from an upper bound towards
double kVInfinity = k * vInfinity, rr = rX * rX + rY * rY; // the root and
double a0 = −rr, a1 = a0 * k, a3 = k * kVInfinity * rY; // return 1/u.
double a4 = kVInfinity * kVInfinity;
double maxInvRelError = 1.0E6; // Use an achievable inverse error bound.
double de, e, uDelta = 0;
// 2. Set u to an upper bound by solving e(u) with a3 = a1 = 0, clamped by
// the result of a Newton method’s iteration at u = 0 if positive.
double u = std::sqrt(kVInfinity / std::sqrt(rr));
if (rY < 0) u = std::min(u, −vInfinity / rY);
// 3. Let u monotonically converge to e(u)’s positive root using a modified
// Newton’s method that speeds up convergence for double roots, but is likely
// to overshoot eventually. Here, “e” = e(u) and “de” = de(u)/du.
for (int it = 0; it < 10; ++it, uDelta = e / de, u −= 1.9 * uDelta) {
de = a0 * u; e = de + a1; de = de + e; e = e * u;
de = de * u + e; e = e * u + a3; de = de * u + e; e = e * u + a4;
if (!(e < 0 && de < 0)) break; // Overshot the root.
}
u += 0.9 * uDelta; // Trace back to the unmodified Newton method’s output.
// 4. Continue to converge monotonically from the overestimated u to e(u)’s
// only positive root using Newton’s method.
for (int it = 0; uDelta * maxInvRelError > u && it < 10; ++it) {
de = a0 * u; e = de + a1; de = de + e; e = e * u;
de = de * u + e; e = e * u + a3; de = de * u + e; e = e * u + a4;
uDelta = e / de; u −= uDelta;
}
// 5. Return the solved time t to hit [rX, rY], or 0 if no solution exists.
return u > 0 1 / u : 0;
}

International Journal of Computer Games Technology

Analytical Ballistic Trajectories with Approximately Linear Drag

Algorithm 1