Quick Reference (Formulas)

This page provides a quick lookup for essential formulas and constants used throughout bot development. For detailed explanations, see the linked pages.

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Physics Constants

Bot Dimensions

Both games render the bot as a 36×36 unit square visually, centered at the bot's position.

Collision hitbox:

• Classic Robocode: Axis-aligned 36×36 square (does not rotate with bot heading)
  • Diagonal: ~50.9 units
• Tank Royale: Circle with radius 18 units (diameter 36; always circular)

Movement Limits

Both Classic Robocode and Tank Royale use identical movement physics:

Property	Value	Units
Max velocity ($v_{\text{max}}$)	8	units/turn
Max acceleration	1	units/turn²
Max deceleration	2	units/turn²
Max turn rate (at rest)	10	degrees/turn
Max turn rate (at max speed)	4	degrees/turn

Linear velocity has acceleration/deceleration limits, but **rotation (body, gun, radar) has no

acceleration/deceleration**—turn rate changes instantly up to the maximum limit.

Notation:

• $v$: current speed (units/turn)
• $v_{\text{max}}$: maximum speed ($=8$ units/turn)
• $v_{\text{rest}}$: rest ($=0$ units/turn)

Turn rate formula:

$\text{turnRate} = 10 - \frac{3}{4} \times |v|$

$\text{maxTurnRate}_{\text{max}} = 10 - \frac{3}{4} \times |v_{\text{max}}| = 10 - \frac{3}{4} \times 8 = 10 - 6 = 4^\circ$

$\text{maxTurnRate}_{\text{rest}} = 10 - \frac{3}{4} \times |v_{\text{rest}}| = 10 - \frac{3}{4} \times 0 = 10 - 0 = 10^\circ$

Rotation Limits

Component	Max Rate	Units
Body	10°	degrees/turn
Gun (relative to body)	20°	degrees/turn
Radar (relative to gun)	45°	degrees/turn

Maximum combined rates:

• Gun relative to battlefield: 30°/turn (body 10° + gun 20°)
• Radar relative to battlefield: 75°/turn (body 10° + gun 20° + radar 45°)

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Bullet Physics

Bullet Speed

$\text{bulletSpeed} = 20 - 3 \times \text{bulletPower}$

Power	Speed	Units/turn
0.1	19.7	units/turn
1.0	17.0	units/turn
2.0	14.0	units/turn
3.0	11.0	units/turn

Bullet Damage

$\text{damage} = \begin{cases} 4 \times \text{bulletPower} & \text{if } \text{bulletPower} \leq 1 \\ 4 \times \text{bulletPower} + 2 \times (\text{bulletPower} - 1) & \text{if } \text{bulletPower} > 1 \end{cases}$

Power	Damage	Bonus	Total
0.1	0.4	0	0.4
1.0	4.0	0	4.0
2.0	8.0	2.0	10.0
3.0	12.0	4.0	16.0

Gun Heat

$\text{gunHeat} = 1 + \frac{\text{bulletPower}}{5}$

$\text{coolingRate} = 0.1 \text{ per turn}$

Power	Heat	Cool Time
0.1	1.02	11 turns
1.0	1.20	12 turns
2.0	1.40	14 turns
3.0	1.60	16 turns

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Angle & Distance Calculations

Coordinate Systems

Classic Robocode:

• Origin: bottom-left (0, 0)
• Heading 0°: North (up)
• Angles: clockwise from north

Tank Royale:

• Origin: bottom-left (0, 0)
• Heading 0°: East (right)
• Angles: counterclockwise from east

Absolute Bearing

Angle from your position to a target:

dx = target.x - my.x
dy = target.y - my.y
absoluteBearing = atan2(dx, dy)  // Classic: atan2(x, y)
absoluteBearing = atan2(dy, dx)  // Tank Royale: atan2(y, x)

Relative Bearing

Angle from your heading to a target:

$\text{relativeBearing} = \text{absoluteBearing} - \text{myHeading}$

$\text{relativeBearing} = \text{normalizeAngle}(\text{relativeBearing})$ (wrap to $[-180°, +180°]$)

Distance

$\text{distance} = \sqrt{(\text{target.x} - \text{my.x})^2 + (\text{target.y} - \text{my.y})^2}$

Normalize Angle to [-180°, +180°]

$\text{while } \text{angle} > 180°: \quad \text{angle} \mathrel{-}= 360°$

$\text{while } \text{angle} < -180°: \quad \text{angle} \mathrel{+}= 360°$

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Targeting Formulas

Head-On Targeting

Aim directly at the current enemy position:

$\text{gunTurnRequired} = \text{normalizeAngle}(\text{absoluteBearing} - \text{gunHeading})$

Linear Targeting

Predict enemy position assuming constant velocity:

// Solve for bullet flight time (iterative)
bulletSpeed = 20 - 3 * bulletPower
time = distance / bulletSpeed  // Initial guess

for iteration in 1..5:
  futureX = enemy.x + enemy.velocityX * time
  futureY = enemy.y + enemy.velocityY * time
  distance = sqrt((futureX - my.x)² + (futureY - my.y)²)
  time = distance / bulletSpeed

aimAngle = atan2(futureX - my.x, futureY - my.y)

Circular Targeting

Predict enemy position assuming a constant turn rate:

// Estimate angular velocity
angleToEnemy = atan2(enemy.x - my.x, enemy.y - my.y)
lateralVelocity = enemy.velocity * sin(enemy.heading - angleToEnemy)
angularVelocity = lateralVelocity / distance

// Solve for time iteratively
bulletSpeed = 20 - 3 * bulletPower
time = distance / bulletSpeed

for iteration in 1..5:
  futureHeading = enemy.heading + angularVelocity * time
  futureX = enemy.x + enemy.velocity * sin(futureHeading) * time
  futureY = enemy.y + enemy.velocity * cos(futureHeading) * time
  distance = sqrt((futureX - my.x)² + (futureY - my.y)²)
  time = distance / bulletSpeed

aimAngle = atan2(futureX - my.x, futureY - my.y)

GuessFactor Formula

$\text{lateralDirection} = \text{sign}(\text{lateralVelocity})$

$\text{maxEscapeAngle} = \arcsin\left(\frac{8}{\text{bulletSpeed}}\right)$

$\text{angleOffset} = \text{actualBearingAtHit} - \text{bearingAtFire}$

$\text{guessFactor} = \frac{\text{angleOffset}}{\text{maxEscapeAngle}}$

GuessFactor ranges from -1.0 (maximum clockwise dodge) to +1.0 (maximum counter-clockwise dodge).

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Movement Calculations

Lateral Velocity

Velocity perpendicular to the enemy line of fire:

$\text{angleToEnemy} = \text{atan2}(\text{enemy.x} - \text{my.x}, \text{enemy.y} - \text{my.y})$

$\text{lateralVelocity} = \text{my.velocity} \times \sin(\text{my.heading} - \text{angleToEnemy})$

Sign:

• Positive: moving counter-clockwise (left) around an enemy
• Negative: moving clockwise (right) around an enemy

Advancing Velocity

Velocity toward/away from enemy:

$\text{advancingVelocity} = \text{my.velocity} \times \cos(\text{my.heading} - \text{angleToEnemy})$

Sign:

• Positive: moving toward an enemy
• Negative: moving away from an enemy

Orbital Movement

Maintain a constant distance while orbiting:

$\text{orbitAngle} = 90° \text{ (perpendicular to enemy)}$

$\text{orbitDirection} = +1 \text{ or } -1 \text{ (counter-clockwise or clockwise)}$

$\text{desiredHeading} = \text{angleToEnemy} + \text{orbitDirection} \times \text{orbitAngle}$

$\text{turnRequired} = \text{normalizeAngle}(\text{desiredHeading} - \text{my.heading})$

Wall Distance

Minimum distance to any wall:

$\text{wallDistance} = \min(\text{my.x}, \text{my.y}, \text{fieldWidth} - \text{my.x}, \text{fieldHeight} - \text{my.y})$

Wall Smoothing (Basic)

Adjust heading to avoid hitting walls. For each wall, calculate push force proportional to proximity:

$\text{margin} = 150 \text{ (safety margin in units)}$

$\text{adjustHeading} \mathrel{+}= \begin{cases} ({\text{margin} - \text{my.x}}) \times 0.1 & \text{if } \text{my.x} < \text{margin} \\ 0 & \text{otherwise} \end{cases}$

Similar calculations apply for the right wall, top wall, and bottom wall.

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Energy & Scoring

Energy Bonuses

Bullet hit:

$\text{energyGained} = 3 \times \text{bulletPower}$

NOTE

Neither Classic Robocode nor Tank Royale awards energy bonuses for killing enemies. Energy is only gained from bullet hits (3× bullet power). Kills award scoring points (see Damage Scoring below), not energy.

Damage Scoring

Bullet hit damage:

$\text{score} \mathrel{+}= \text{damage}$

Bullet kill bonus:

When a bot kills an enemy with a bullet, it scores an additional bonus:

Both Classic Robocode and Tank Royale:

$\text{bulletDamageBonus} = 20\% \text{ of all bullet damage dealt to that enemy}$

Or equivalently: $\text{bulletDamageBonus} = 0.20 \times \sum \text{bullet damage}$

Ram damage:

$\text{damage} = 0.6 \text{ per turn of contact}$

$\text{score} \mathrel{+}= 2 \times \text{ram damage}$

Ram kill bonus:

When a bot kills an enemy by ramming, it scores an additional bonus:

Both Classic Robocode and Tank Royale:

$\text{ramDamageBonus} = 30\% \text{ of all ram damage dealt to that enemy}$

Or equivalently: $\text{ramDamageBonus} = 0.30 \times \sum \text{ram damage}$

Survival score:

$\text{survivalScore} = 50 \times \text{number of bots that died while you survived}$

Last survivor bonus:

$\text{lastSurvivorBonus} = 10 \times \text{number of opponent bots}$

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Wave Calculations

Max Escape Angle

Maximum angle the enemy can reach by moving perpendicular at full speed:

$\text{maxEscapeAngle} = \arcsin\left(\frac{8}{\text{bulletSpeed}}\right)$

Bullet Power	Speed	Max Escape Angle
0.1	19.7	~24°
1.0	17.0	~28°
2.0	14.0	~35°
3.0	11.0	~47°

Wave Intersection

Time until a wave intersects bot position:

$\text{distanceFromWaveOrigin} = \sqrt{(\text{my.x} - \text{wave.x})^2 + (\text{my.y} - \text{wave.y})^2}$

$\text{waveFront} = (\text{currentTime} - \text{wave.fireTime}) \times \text{wave.bulletSpeed}$

$\text{timeUntilHit} = \frac{\text{distanceFromWaveOrigin} - \text{waveFront}}{\text{wave.bulletSpeed}}$

If $\text{timeUntilHit} \leq 0$, wave has already passed.

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Turn Radius

Minimum turn radius at a given velocity:

$turnRate = 10 - \frac{3}{4} \times |velocity|$

$radius = \frac{|velocity|}{\sin(turnRate \times \frac{π}{180})}$

Velocity	Turn Rate	Radius
0	10°	0
4	7°	~33 units
8	4°	~115 units

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Common Trigonometry

Radians ↔ Degrees:

$\text{radians} = \text{degrees} \times \frac{\pi}{180}$

$\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Pythagorean theorem:

$\text{hypotenuse} = \sqrt{a^2 + b^2}$

Sine law:

$\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$

Small angle approximation ($|\theta| < 15°$):

$\sin(\theta) \approx \theta$

$\cos(\theta) \approx 1$

$\tan(\theta) \approx \theta$

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Platform-Specific API Equivalents

Classic Robocode	Tank Royale	Purpose
getX()	getX()	Bot X position
getY()	getY()	Bot Y position
getHeading()	getDirection()	Body heading (note: 0° differs!)
getGunHeading()	getGunDirection()	Gun absolute heading
getRadarHeading()	getRadarDirection()	Radar absolute heading
getVelocity()	getSpeed()	Current velocity
getEnergy()	getEnergy()	Current energy
getBattleFieldWidth()	getArenaWidth()	Arena width
getBattleFieldHeight()	getArenaHeight()	Arena height

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Further Reading

• Coordinate Systems & Angles — Detailed angle conventions
• Movement Constraints & Bot Physics — Physics deep dive
• Bullet Travel & Bullet Physics — Bullet mechanics
• GuessFactor Targeting — GuessFactor formula explained
• Wave Surfing Introduction — Wave math in action