Mathematics for Game Physics

# CHAPTER 2

Mathematics for Game Physics #

1. Introduction #

2. Learning Objectives #

• Understand 2D and 3D Coordinate Systems (X, Y, Z).

• Define a Vector and differentiate between its Magnitude and Direction.

• Calculate the distance between two points in space.

• Understand the hierarchical relationship between Position, Velocity, and Acceleration.

• Use Vector Normalization to create pure directional arrows.

3. Coordinates and Space #

• 2D Games: Use an X/Y grid. X is left/right. Y is up/down.

• 3D Games: Add a Z-axis. Z represents depth (forward/backward).

4. What is a Vector? #

• Imagine a physical arrow pointing in 3D space.

• Which way is it pointing? (Direction).

• How long is the arrow? (Magnitude).

5. Vector Math Basics #

• Adding Vectors: If you are walking Forward (0,0,1) and the wind is blowing you to the Right (1,0,0), your resulting movement vector is diagonal: (1,0,1).

• Finding Direction (Target - Source): To find the arrow pointing from an Enemy to the Player, you subtract the Enemy's position from the Player's position.

• Normalization: If you only care about the *direction* and want to strip away the length, you "Normalize" the vector. This forces the arrow's length to be exactly 1.0.

6. Position, Velocity, and Acceleration #

1. 1. Acceleration: How fast your speed is changing. (e.g., pressing the gas pedal adds 5m/s to your speed every second).

1. 2. Velocity: Your current speed and direction. (e.g., traveling 60 mph North).

1. 3. Position: Your actual location.

// The Core Math Loop of Physics Engines
void UpdatePhysics()
{
    // 1. Acceleration modifies Velocity
    velocity += acceleration * Time.deltaTime;

    // 2. Velocity modifies Position
    position += velocity * Time.deltaTime;
}

7. Visual Learning: Vector Addition #

     (Wind Vector: 1,0)
       ----->
      |      \
      |       \  (Resulting Vector: 1,1)
      v        v
(Walk Vector: 0,-1)

8. Best Practices #

• Use Built-in Engine Math: Do not write your own Pythagorean theorem code to calculate distance. Every engine has highly optimized C++ math libraries. Use Vector3.Distance(a, b) or Vector3.Magnitude(v). Let the engine do the heavy lifting!

9. Common Mistakes #

• Forgetting to Normalize Directions: If you calculate Vector3 dir = player.pos - enemy.pos; and the player is 50 meters away, the length of dir is 50. If you then say enemy.position += dir * speed, the enemy will warp 50 meters instantly! Always use dir.normalized to ensure the direction vector has a length of exactly 1 before applying speed.

10. Mini Project: Calculate and Move #

class ChaserEnemy
{
    public Transform target;
    public float speed = 5.0f;

    void Update()
    {
        // 1. Calculate the exact distance
        float distance = Vector3.Distance(transform.position, target.position);
        Console.WriteLine("Distance to target: " + distance);

        if (distance > 1.0f)
        {
            // 2. Calculate the Direction Vector (Target - Me)
            Vector3 direction = target.position - transform.position;

            // 3. Normalize the vector (force length to 1)
            direction = direction.normalized;

            // 4. Move! (Direction * Speed * Time)
            transform.position += direction * speed * Time.deltaTime;
        }
    }
}

11. Practice Exercises #

1. 1. What is the standard mathematical formula used in code to find a Vector pointing directly from Object A to Object B?

1. 2. If you have a Vector representing a character's velocity, what does "Normalizing" that Vector achieve?

12. MCQs with Answers #

13. Interview Questions #

• Q: Explain the mathematical relationship between Position, Velocity, and Acceleration. How does DeltaTime factor into this relationship within a Game Loop?

• Q: Describe the concept of Vector Normalization. Provide a concrete gameplay programming example where failing to normalize a vector results in a critical movement bug.

• Q: If you have an object's current Velocity vector, how do you use the Pythagorean theorem (or engine magnitude functions) to determine the object's exact speed in meters per second?

14. FAQs #

15. Summary #

16. Next Chapter Recommendation #