Converting Isometric Tile Coordinates To Screen Coordinates

The Core Problem

Isometric video games and simulations utilize a specialized coordinate system to render angled tile-based worlds that appear three-dimensional. However, most game engines and graphics APIs expect standard Cartesian coordinates for positioning, mapping textures, and rendering sprites. This necessitates a conversion between the isometric tile coordinates and the screen space coordinates.

An isometric tile represents a single unit within an isometric grid layout where the x and y position encodes both the spatial location and depth. Typically, the isometric y increases from bottom to top of the screen and x from left to right. Each tile coordinate maps to a specific place within a staggered grid. This creates the illusion of depth without the computational complexity of a true 3D environment.

To translate the isometric tiles into a renderable scene, we need to project the tile coordinates into the 2D screen space recognized by game engines and APIs. This involves applying conversion formulas and transformations to map the isometric x,y coordinates into equivalent screen space x,y positions.

Table of Contents

Mapping Isometric Coordinates to Screens

The mathematical conversion from isometric to screen space centers around deriving projections where the depth component gets transferred into the vertical axis. By applying these formulas, we can plot tiles to appear layered on top of each other within a 2D viewing plane instead of along an isometric z-axis.

The key formulas for performing this coordinate translation are:

ScreenX = (IsoX – IsoY) * TileWidth / 2
ScreenY = (IsoX + IsoY) * TileHeight / 2

Where TileWidth and TileHeight refer to the rendered size dimensions of each isometric tile. These formulas effectively convert the diagonal isometric coordinates into cartesian screen positions, preserving depth order in the process.

Here is an example Python function to translate isometric to screen coordinates:

def iso_to_screen(iso_x, iso_y, tile_width, tile_height):
  screen_x = (iso_x - iso_y) * (tile_width / 2) 
  screen_y = (iso_x + iso_y) * (tile_height / 2)
  
  return screen_x, screen_y

This applies the math formulas above to take an isometric X,Y value and compute the equivalent screen space coordinates after factoring in the tile dimensions. The output can then directly feed into a rendering engine.

Handling Tile Layers

A key technique for building depth and verticality in isometric games is to stack multiple tile layers atop one another. For instance, ground tiles on the bottom layer, buildings in the middle layers, and roofs at the top. To properly composite these layers into a final scene, we need to process each layer’s isometric coordinates separately.

The same isometric to screen space formulas apply equally to all layers. The z-order stacking gets introduced once the layers get rendered to the screen one-by-one. This mimics the depth position along the isometric z-axis. The order translates to nearer tiles obscuring tiles behind them.

Here is Python code for rendering 3 isometric tile layers:

for y in range(map_height):
  for x range(map_width):
    # Ground tiles 
    screen_x, screen_y = iso_to_screen(x, y, 64, 32)
    draw_tile(ground_tiles[x][y], screen_x, screen_y)

    # Building tiles
    screen_x, screen_y = iso_to_screen(x, y, 64, 32) 
    draw_tile(building_tiles[x][y], screen_x, screen_y)
    
    # Roof tiles
    screen_x, screen_y = iso_to_screen(x, y, 64, 32)
    draw_tile(roof_tiles[x][y], screen_x, screen_y)

This loops through each isometric map location, converting it into screen coordinates before passing it into a tile drawing function. By separating the layers into distinct draw calls, tiles automatically occlusion sort from back-to-front.

Sprite and Decal Rendering

In addition to environment tiles, many isometric games rely heavily on sprites and decals. These include character models, vegetation like trees and rocks, and directional textures like arrows or footprints.

To properly overlay sprites and decals within an isometric scene, we need to position them based on the equivalent screen coordinates instead of raw isometric values. This keeps the sprites aligned with the surrounding tile geometry.

The difference comes down to converting the sprite’s intended isometric coordinate into screen space first, before passing that translated position to the rendering function:

sprite_iso_x = 10 
sprite_iso_y = 5

screen_x, screen_y = iso_to_screen(sprite_iso_x, sprite_iso_y, 64, 32)

draw_sprite(tree_sprite, screen_x, screen_y)

This ensures tree sprites, arrow markers, footstep decals, and other non-tile content layers cleanly atop the isometric tile artwork according to the scene’s depth and perspective rules.

Optimizing Draw Order

Performance and rendering efficiency for isometric graphics depends heavily on properly sorting elements before drawing. Arranging tiles front-to-back prior to rendering reduces wasteful overdraw and unnecessary fragment shading calculations.

Because the isometric math directly translates spatial depth into vertical screen position, we can leverage this to sort by y-position before submitting draw calls. Elements with lower screen space y values appear further towards the back and should draw first.

iso_tiles.sort(key=lambda t: t.screen_y) 

for tile in iso_tiles:
  draw_tile(tile.image, tile.screen_x, tile.screen_y)

This optimized approach processes tiles from furthest in the back to closest in the front. By sorting on the converted screen y coordinate, we visualize the isometric scene using the minimum number of rendered pixels at each point. This improves rendering throughput and reduces fill rate bound scenarios.

Additional optimizations like culling fully obstructed lower layers can build upon this sorted draw order to further improve runtime performance of isometric graphics.