Warp: Difference between revisions

==Time taken to warp==

Warp consists of 3 stages:

Calculating the time taken to warp is done by calculating the time spent in each of these phases and adding them together. This requires calculating acceleration and deceleration time first, followed by cruise time.

<math>\pagecolor{Black}\color{White}t_{total} = t_{accel} + t_{decel} + t_{cruise}</math>

=Long warps=

==Acceleration==

===The formulae===

CCP provided formulae for both distance traveled and velocity reached after ''t'' seconds of acceleration. If ''d'' is distance in meters, ''v'' is speed in meters per second, ''k'' is a (sort of) constant defined as the warp speed (in AU/s) and a = 149,597,870,700 meters (1 AU).

<math>\pagecolor{Black}\color{White}

d & = e^{kt} \\

v_{warp} & = k * a\\

<math>\pagecolor{Black}\color{White}

d & = e^{kt} \\

\therefore d & = \frac{v}{k}

The distance covered while accelerating to v_warp is  

<math>\pagecolor{Black}\color{White}

d_{accel} & = \frac{v_{warp}}{k}

& = \frac{k*a}{k}

===Time===

<math>\pagecolor{Black}\color{White}

kt & = \ln{(\frac{v}{k})}\\

t & =\frac{\ln{(\frac{v}{k})}}{k}\\

<math>\pagecolor{Black}\color{White}

v_{warp} & = k * a\\

t_{accel} & = \frac{\ln{(\frac{v_{warp}}{k})}}{k}\\

This formula can be simplified further to <math>\pagecolor{Black}\color{White}\frac{\ln{a}}{k}</math>, although you may choose not to do this for implementation reasons.

==Deceleration==

Deceleration is calculated slightly differently. Instead of using ''k'' to calculate distance and velocity, it uses ''j'', which is defined as <math>\pagecolor{Black}\color{White}\min(\frac{k}{3},2)</math>. A different rate of deceleration is used to prevent ships suddenly transitioning from "many, many AU away" to "on grid and out of warp" more rapidly than other pilots (or the server / client) can keep up with.

<math>\pagecolor{Black}\color{White}s = \min(100, v_{subwarp}/2)</math>

Where v_subwarp is the maximum subwarp velocity of the ship; this varies greatly depending on the ship hull and pilot skills.

===Distance===

<math>\pagecolor{Black}\color{White}

d_{decel} & = \int_{0}^{\infty}k*e^{jt}\,dt = \frac{k*e^{jt}}{j}\\

The distance covered while decelerating from v_warp is

<math>\pagecolor{Black}\color{White}

d_{decel} & = \frac{v_{warp}}{j}

= \frac{k*a}{j}

Note that for ships that travel at up to 6 AU/s, k / j = k / (k/3) = 3, so these ships cover 3 AU while decelerating. The complication of not stopping warp at 0 can be safely ignored for distance calculations, because the distance that would be covered while decelerating from 100 m/s is insignificant compared to the ~450 billion meters it takes to decelerate from warp speed to warp drop speed.

===Time===

<math>\pagecolor{Black}\color{White}

t & = \frac{\ln{(\frac{v}{k})}}{j}

<math>\pagecolor{Black}\color{White}

t_{decel} & = t_{decel\_warp} - t_{decel\_s}\\

& = \frac{\ln{(\frac{v_{warp}}{k})}}{j} - \frac{\ln{(\frac{s}{k})}}{j}\\

& = \frac{\ln{(\frac{v_{warp}}{k})} - \ln{(\frac{s}{k})}}{j}\\

& = \frac{\ln{v_{warp}} - \ln{k} - \ln{s} + \ln{k}}{j}\\

& = \frac{\ln{v_{warp}} - \ln{s}}{j}\\

& = \frac{\ln{(\frac{v_{warp}}{s})}}{j}

==Cruising==

===Distance===

<math>\pagecolor{Black}\color{White}d_{cruise} = d_{total} - d_{accel} - d_{decel}</math>

===Time===

This one is easy. Time spent cruising is simply

<math>\pagecolor{Black}\color{White}t_{cruise} = \frac{d_{cruise}}{v_{warp}}</math>

=Short Warps=

The above calculations work as long as some time is spent at maximum warp speed. If the warp is short enough that the ship never reaches top speed, a different set of calculations are needed.

<math>\pagecolor{Black}\color{White}

d_{accel} & = \frac{v_{max}}{k}, d_{decel} = \frac{v_{max}}{j}\\

d_{total} & = d_{accel} + d_{decel} = v_{max}(\frac{1}{k} + \frac{1}{j})\\

v_{max} & = \frac{d_{total}*k*j}{k + j}

This enables the calculation of new acceleration and deceleration times using the formulae described in the previous sections, but substituting in the new v_max

<math>\pagecolor{Black}\color{White}

t_{accel} & = \frac{\ln{(\frac{v_{max}}{k})}}{k}\\

t_{decel} & = \frac{\ln{(\frac{v_{max}}{s})}}{j}\\

t_{total} & = t_{accel} + t_{decel}

=Implementation=

The following python code is one possible implementation of the above. It attempts to generate the same data as presented by CCP in the forums. It matches with their numbers, except for 50 AU titan warps, which are one second out. Note that the sub warp speed of the ship is fixed at 200 m/s. This is because the CCP-produced tables assume that every ship drops out of warp at 100 m/s. If trying to run calculations for actual ships, this value will need to be replaced with a more appropriate one. The output values are also passed through the ceil() function, as this is what seems to match the rounding mode that CCP used.