Isometric Projection

Overview

Isometric projection displays three-dimensional scenes from a fixed angle where all three axes are equally foreshortened. On 8-bit computers, this created the impression of explorable 3D space while using only 2D sprite and tile techniques. Knight Lore, Marble Madness, and Solstice defined the look.

The mathematics

Standard isometric

True isometric uses ~30° angles:

X-axis: 30° from horizontal
Y-axis: 90° (vertical)
Z-axis: 30° from horizontal (opposite direction)

Pixel isometric (2:1)

Games often used 2:1 ratio:

For every 2 pixels horizontal, 1 pixel vertical
Simpler maths for 8-bit processors
Visually similar to true isometric

Coordinate conversion

Converting 3D to screen coordinates:

screen_x = (tile_x - tile_z) × tile_width / 2
screen_y = (tile_x + tile_z) × tile_height / 2 - height

Rendering order

Objects must be drawn back-to-front (painter’s algorithm):

for z from far to near:
    for x from left to right:
        draw_tile(x, z)

Or sort by depth:

depth = tile_x + tile_z

Collision detection

Separate world coordinates from screen:

Store position in 3D (x, y, z)
Calculate screen position for display
Collide in 3D space, not screen space

The Filmation engine

Ultimate’s approach (Knight Lore):

Feature	Implementation
Solid objects	Z-ordered sprite masking
Room-based	Each room calculated fresh
Object interaction	Physics in isometric space
Memory efficiency	Room data, not full map

Tile-based isometric

Common in strategy games:

Element	Structure
Ground tiles	Diamond-shaped
Objects	Placed at tile positions
Height	Stack tiles or offset

Challenges

Depth sorting

Objects can overlap in complex ways:

Characters moving between objects
Multi-tile objects
Different heights

Click detection

Converting screen clicks to tiles:

tile_x = (screen_x / (tile_width/2) + screen_y / (tile_height/2)) / 2
tile_z = (screen_y / (tile_height/2) - screen_x / (tile_width/2)) / 2

Movement

Eight-direction movement maps to 3D:

“Up” moves toward back corner
“Left” moves toward left corner
Diagonal movement along axes

Notable games

Game	Year	Platform
Zaxxon	1982	Arcade
Knight Lore	1984	Spectrum
Marble Madness	1984	Arcade
Batman	1986	Spectrum
Head Over Heels	1987	Spectrum
Solstice	1990	NES

Modern use

Isometric persists in:

City builders (SimCity 2000)
Tactical RPGs (Final Fantasy Tactics)
Action RPGs (Diablo)
Indie games (Hades)

3D from 2D