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How to calculate angles for 3D space exploration games with FOV

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SpaceMath_Alex

Posted on July 19, 2025 • Advanced

🚀 3D Space Angle Calculation Problem

I’m working on a semi-3D space exploration game and I’m stuck on the trigonometry! Here’s what I’m trying to achieve:

  • 🛸 Spaceship can rotate and move in 3D space
  • 🌍 Planets need to appear when they’re within the ship’s field of view (FOV)
  • 📐 Need to calculate the angle from spaceship to each planet
  • 🎯 Angle must update as ship moves and rotates
  • 📺 Convert angles to screen positions (FOV 60°, screen 480px)

The math is getting really complex with the 3D rotations and perspective calculations. Can someone help me break this down? 🤯

3D

TrigMaster_Emma

Replied 2 hours later • ⭐ Best Answer

Great question @SpaceMath_Alex! 3D angle calculations can be tricky. Let me break this down step by step:

🧭 Understanding the 3D Coordinate System

First, let’s establish our coordinate system and what we’re calculating:

flowchart TD A[🛸 Spaceship Position<br/>X, Y, Z] --> B[🌍 Planet Position<br/>Px, Py, Pz] B --> C[📐 Calculate Vector<br/>dx = Px - X<br/>dy = Py - Y<br/>dz = Pz - Z] C --> D[🔄 Apply Ship Rotation<br/>Account for ship's orientation] D --> E[📏 Calculate Angles<br/>Horizontal & Vertical] E --> F{Within FOV?} F -->|Yes| G[📺 Convert to Screen Position] F -->|No| H[🚫 Hide Planet] G --> I[🎮 Display Planet] style A fill:#e1f5fe style B fill:#f3e5f5 style G fill:#e8f5e8 style I fill:#fff3e0

🔢 Step 1: Basic Vector Calculation

Calculate the vector from spaceship to planet:

    // Calculate vector from ship to planet
set [dx v] to ((planet x) - (ship x))
set [dy v] to ((planet y) - (ship y))
set [dz v] to ((planet z) - (ship z))

// Calculate distance (for scaling)
set [distance v] to ([sqrt v] of (((dx) * (dx)) + ((dy) * (dy)) + ((dz) * (dz))))

🔄 Step 2: Apply Ship Rotation

Transform the vector based on ship’s orientation:

    // Ship rotation angles (yaw, pitch, roll)
set [ship yaw v] to [ship direction] // Horizontal rotation
set [ship pitch v] to [ship pitch angle] // Vertical tilt

// Rotate vector to ship's local coordinate system
// Yaw rotation (around Y axis)
set [temp x v] to (((dx) * ([cos v] of (ship yaw))) + ((dz) * ([sin v] of (ship yaw))))
set [temp z v] to (((dz) * ([cos v] of (ship yaw))) - ((dx) * ([sin v] of (ship yaw))))
set [dx v] to (temp x)
set [dz v] to (temp z)

// Pitch rotation (around X axis)
set [temp y v] to (((dy) * ([cos v] of (ship pitch))) - ((dz) * ([sin v] of (ship pitch))))
set [temp z v] to (((dy) * ([sin v] of (ship pitch))) + ((dz) * ([cos v] of (ship pitch))))
set [dy v] to (temp y)
set [dz v] to (temp z)

📐 Step 3: Calculate View Angles

Convert 3D vector to horizontal and vertical angles:

    // Calculate horizontal angle (left/right)
if <(dz) = [0]> then
if <(dx) > [0]> then
set [horizontal angle v] to [90]
else
set [horizontal angle v] to [-90]
end
else
set [horizontal angle v] to ([atan v] of ((dx) / (dz)))
if <(dz) < [0]> then
change [horizontal angle v] by [180]
end
end

// Calculate vertical angle (up/down)
set [horizontal distance v] to ([sqrt v] of (((dx) * (dx)) + ((dz) * (dz))))
if <(horizontal distance) = [0]> then
if <(dy) > [0]> then
set [vertical angle v] to [90]
else
set [vertical angle v] to [-90]
end
else
set [vertical angle v] to ([atan v] of ((dy) / (horizontal distance)))
end

🎯 Step 4: Check Field of View

Determine if planet is within the ship’s FOV:

    // FOV settings
set [FOV horizontal v] to [60] // degrees
set [FOV vertical v] to [45] // degrees

// Check if planet is in view
if <<([abs v] of (horizontal angle)) < ((FOV horizontal) / [2])> and <([abs v] of (vertical angle)) < ((FOV vertical) / [2])>> then
// Planet is visible
set [planet visible v] to [true]

// Convert to screen coordinates
set [screen x v] to (((horizontal angle) / (FOV horizontal)) * [480])
set [screen y v] to (((vertical angle) / (FOV vertical)) * [360])

// Scale based on distance
set [planet size v] to ([max v] of ([min v] of ((1000) / (distance)) [100]) [10])
else
set [planet visible v] to [false]
end

🎮 Step 5: Complete Implementation

Put it all together in a reusable custom block:

    // Custom block: calculate planet angle (planet sprite, ship sprite)
define calculate planet angle (planet) (ship)
// Get positions
set [px v] to ([x position v] of (planet))
set [py v] to ([y position v] of (planet))
set [pz v] to (item (3) of [planet positions v]) // Z coordinate from list

set [sx v] to ([x position v] of (ship))
set [sy v] to ([y position v] of (ship))
set [sz v] to (item (3) of [ship positions v])

// Calculate vector
set [dx v] to ((px) - (sx))
set [dy v] to ((py) - (sy))
set [dz v] to ((pz) - (sz))

// Apply ship rotation (use the rotation code above)
apply ship rotation

// Calculate angles
calculate view angles

// Check FOV and set screen position
check field of view

// Return results in variables or lists
replace item (1) of [planet screen pos v] with (screen x)
replace item (2) of [planet screen pos v] with (screen y)
replace item (3) of [planet screen pos v] with (planet visible)

⚡ Step 6: Optimization Tips

For better performance in your space game:

    // Only calculate for nearby planets
if <(distance) < [5000]> then
calculate planet angle (planet) (ship)
else
set [planet visible v] to [false]
end

// Use simplified calculations for distant objects
if <(distance) > [2000]> then
// Skip pitch rotation for distant objects
set [vertical angle v] to [0]
end

// Cache calculations that don't change often
if <not <(ship moved) or (ship rotated)>> then
// Skip recalculation if ship hasn't moved
else
// Recalculate all planet positions
end

This system will give you proper 3D angle calculations with FOV! The key is breaking it down into vector math, rotation transforms, and angle calculations. 🚀

SpaceMath_Alex

Replied 3 hours later

@TrigMaster_Emma This is absolutely incredible! 🤩 The step-by-step breakdown makes so much sense now!

I implemented the vector rotation and FOV checking - it works perfectly! One question: how do I handle the case where the ship rolls (rotates around its forward axis)? Do I need another rotation step?

3D

TrigMaster_Emma

Replied 30 minutes later

@SpaceMath_Alex Great question about roll! Yes, you need a third rotation step:

    // Roll rotation (around Z axis) - add this after pitch
set [ship roll v] to [ship roll angle]

// Roll rotation
set [temp x v] to (((dx) * ([cos v] of (ship roll))) - ((dy) * ([sin v] of (ship roll))))
set [temp y v] to (((dx) * ([sin v] of (ship roll))) + ((dy) * ([cos v] of (ship roll))))
set [dx v] to (temp x)
set [dy v] to (temp y)

// Now dz stays the same for roll

The order matters: Yaw → Pitch → Roll for proper 3D rotation! 🔄

GP

GamePhysics_Pro

Replied 1 hour later

Excellent explanation @TrigMaster_Emma! For anyone wanting to take this further, here are some advanced optimizations:

  • 🎯 Frustum Culling: Pre-calculate which planets are potentially visible
  • ⚡ Level of Detail: Use simpler calculations for distant objects
  • 🔄 Matrix Math: Combine all rotations into a single transformation matrix
  • 📊 Spatial Partitioning: Only check nearby planets using octrees or similar

These techniques can handle hundreds of planets smoothly! 🌌

VB

Vibelf_Community

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