Tiny Planet Vehicle Physics

TODO: collapasable, copyable, download, etc

I want to make a game where the player controls a vehicle on a tiny planet. In the viewport, you can see the curve of the horizon and it is clearly a small sphere. What math techniques would be needed to implement this mechanic?Creating a game where the player controls a vehicle on a tiny planet involves several mathematical and programming techniques to achieve the desired mechanics and visuals. Here are the key steps and techniques you would need:

1. Coordinate Systems and Transforms

• Spherical Coordinates: Use spherical coordinates to position the vehicle on the surface of the planet. The coordinates are typically represented as ((r, \theta, \phi)), where (r) is the radius of the sphere, (\theta) is the azimuthal angle (longitude), and (\phi) is the polar angle (latitude).

• Conversion between Spherical and Cartesian Coordinates: Convert spherical coordinates to Cartesian coordinates for rendering and physics calculations: [ x = r \sin(\phi) \cos(\theta) ] [ y = r \sin(\phi) \sin(\theta) ] [ z = r \cos(\phi) ]

2. Vehicle Movement on a Sphere

• Local Tangent Plane: At any point on the sphere, the vehicle should move along the local tangent plane. This involves calculating the tangent vectors at the vehicle's position:

  • Tangent vector in the (\theta) direction: [ \mathbf{t_\theta} = \left( -r \sin(\phi) \sin(\theta), r \sin(\phi) \cos(\theta), 0 \right) ]
  • Tangent vector in the (\phi) direction: [ \mathbf{t_\phi} = \left( r \cos(\phi) \cos(\theta), r \cos(\phi) \sin(\theta), -r \sin(\phi) \right) ]

• Updating Position: When the player moves the vehicle, update the (\theta) and (\phi) angles based on the input, and convert back to Cartesian coordinates for rendering.

3. Gravity and Physics

• Gravity Vector: The gravity vector always points towards the center of the sphere. For a point ((x, y, z)), the gravity vector is: [ \mathbf{g} = -\frac{G M}{r^2} \hat{\mathbf{r}} ] where (\hat{\mathbf{r}}) is the normalized position vector ((x, y, z)/r).

• Vehicle Orientation: The vehicle needs to be oriented correctly on the surface. Align the vehicle's "down" vector with the gravity vector and adjust the "forward" vector based on movement direction.

4. Rendering the Horizon

• Curved Horizon: To render the curved horizon, ensure that the camera is positioned such that it emphasizes the curvature of the planet. The camera should be relatively close to the surface of the sphere. Use a perspective projection to exaggerate the curvature.

5. Collision Detection

• Sphere Collision: Check for collisions between the vehicle and the surface of the sphere or other objects by calculating the distance between the vehicle's position and the center of the sphere.

6. Implementing the Game Mechanics

• Input Handling: Capture player input to adjust the (\theta) and (\phi) angles, resulting in movement around the sphere.
• Smooth Movement: Implement smooth interpolation for the movement and camera transitions to ensure fluid gameplay.

Example Pseudocode for Vehicle Movement:

# Initial spherical coordinates for the vehicle
theta = initial_theta
phi = initial_phi
r = planet_radius

# Convert to Cartesian coordinates for rendering
x, y, z = spherical_to_cartesian(r, theta, phi)

# Handle input to update theta and phi
if input_left:
    theta -= delta_theta
if input_right:
    theta += delta_theta
if input_up:
    phi -= delta_phi
if input_down:
    phi += delta_phi

# Convert updated spherical coordinates to Cartesian
x, y, z = spherical_to_cartesian(r, theta, phi)

# Update vehicle position and orientation
vehicle.position = (x, y, z)
vehicle.orientation = calculate_orientation(theta, phi)

Conclusion

By combining these mathematical techniques with game development tools and frameworks (like Unity or Unreal Engine), you can create a compelling game where the player controls a vehicle on a tiny planet with a clearly visible curved horizon.

Tiny Planet Vehicle Physics created at Tue Jan 20 1970 21:10:06 GMT+0000 (Coordinated Universal Time)