Al Jazari – ambient occlusion

A big part of the look of Minecraft comes from it’s Smooth lighting, which includes an illumination model called ambient occlusion. Ambient occlusion darkens areas of an object based on how obscured they are from a wide area light source, for example an entire sky area, as opposed to a point light source. This is often complicated to calculate in realtime, but with simplified geometric voxels it’s fairly fast to do, building on the code from the hollowing out optimisation I mentioned previously. For each point on a cube’s corners you can add up it’s surrounding empty cubes – the more empty space in the 8 voxels, the brighter the corner needs to be. In this way, we are assuming light is coming from all directions (hence ambient) and don’t need to take into account positions of lights etc.


Here is the raw ambient occlusion lighting in Al Jazari 2, rendered as unlit vertex colours on each visible cube:


Then we add standard direct lighting to the ambient occlusion:


And here is the final image with textures added:


There are a few artefacts in the lighting due to the cubes not all being the same size (in order to minimise the complexity of what is drawn), here is the world coloured based on the size of the octree leaf node – the brighter blue areas are bigger cubes:


Once the ambient occlusion is calculated we only need to recalculate it when surrounding voxels are changed. In practice the splitting/joining of the octree levels when blocks are created and destroyed seems to take care of this in most cases.

;; returns a list of samples to test
(define (occ-samples pos) 
     (vadd pos (vector  -1 -1 -1))
     (vadd pos (vector  -1 -1  0))
     (vadd pos (vector  -1  0 -1))
     (vadd pos (vector  -1  0  0))
     (vadd pos (vector   0 -1 -1))
     (vadd pos (vector   0 -1  0))
     (vadd pos (vector   0  0 -1))
     (vadd pos (vector   0  0  0))))

;; count up the empty voxels and calculate occlusion value
(define (calc-occlusion o pos)
  (let* ((samples (occ-samples pos))
         (coverage (/ (foldl
                       (lambda (p r)
                         (+ (if (octree-leaf-empty? (octree-ref o p)) 1 0) r))
                     (length samples))))
    (min 1 (* 2 coverage))))

;; helper to set a bunch of verts in one go
(define (pdata-list-set! k l v)
  (for-each (lambda (i) (pdata-set! k i v)) l))


;; uses the cube's position and size to set the vertex colour on each corner
(with-primitive my-cube
  (translate pos)
  (pdata-list-set! "c" '(3 7 19) (calc-occlusion o (vadd pos (vector 0 0 0))))
  (pdata-list-set! "c" '(10 14 21) (calc-occlusion o (vadd pos (vector size size size))))
  (pdata-list-set! "c" '(2 4 23) (calc-occlusion o (vadd pos (vector size 0 0))))
  (pdata-list-set! "c" '(9 15 17) (calc-occlusion o (vadd pos (vector 0 size size))))
  (pdata-list-set! "c" '(0 8 18) (calc-occlusion o (vadd pos (vector 0 size 0))))
  (pdata-list-set! "c" '(5 13 22) (calc-occlusion o (vadd pos (vector size 0 size))))
  (pdata-list-set! "c" '(6 12 16) (calc-occlusion o (vadd pos (vector 0 0 size))))
  (pdata-list-set! "c" '(1 11 20) (calc-occlusion o (vadd pos (vector size size 0)))))

Al Jazari – scooping out the insides of planets with scheme

Optimisation is a game where you write more code in order to do less. In Al Jazari 2 doing less means drawing less blocks. Contiguous blocks of the same type are already automatically collapsed into single larger ones with the Octree – but if we can figure out which blocks are completely surrounded by other blocks, we can save more time by not building or drawing them either.

Here is a large sphere – clipped by the world volume, showing a slice through the internal block structure:


The next version has all internal blocks removed, in this case shaving 10% off the total primitives built and drawn:


The gaps in the sphere from the clipping allow us to look inside at how the octree has optimised the structure. The gain is higher in a more normal Minecraft set up with a reasonably flat floor covering a large amount of blocks. Here is the code involved, built on top of a functional library I’m building up on to manipulate this kind of data. It maps over each Octree leaf checking all the blocks it touches on each of its six sides, taking into account that the size of the leaf block may be bigger than one.

(define (octree-check-edge f o pos size)
  (define (do-x x y)
     ((eq? x -1) #f)
     ((octree-leaf-empty? (octree-ref o (vadd pos (f x y)))) #t)
     (else (do-x (- x 1) y))))
  (define (do-y y)
     ((eq? y -1) #f)
     ((do-x size y) #t)
     (else (do-y (- y 1)))))
  (do-y size))

(define (octree-is-visible? o pos size)
   (octree-check-edge (lambda (x y) (vector size x y)) o pos size)
   (octree-check-edge (lambda (x y) (vector -1 x y)) o pos size)
   (octree-check-edge (lambda (x y) (vector x size y)) o pos size)
   (octree-check-edge (lambda (x y) (vector x -1 y)) o pos size)
   (octree-check-edge (lambda (x y) (vector x y size)) o pos size)
   (octree-check-edge (lambda (x y) (vector x y -1)) o pos size)))

(define (octree-calc-viz o)
   (lambda (v pos size depth)
       o pos size)
      (octree-leaf-value v)))

Building pyramids with code composition

The Al Jazari 2 bots currently have six basic actions – move forward/backwards, turn 90 degrees left or right, pick up the block underneath them or drop the block to the space they are currently sitting on. Given these instructions, how do we procedurally build pyramids (of any given size) like this in their minecraft-esque world?

The Pyramids of Guimar, Canary Islands, Spain
The Pyramids of Guimar, Canary Islands, Spain

1. A pyramid can be built as a series of plateaus layered on top of each other, the plateaus can be built from material mined from nearby:


2. A single plateau can be built as a series of single block wide ridges next to each other, mined from a series of trenches. This is an example ridge/trench of size 3:


3. We need a gap between the ridge and the trench in order to place the plateau in the correct place in the pyramid (also the bots can only climb a single block at a time, otherwise they get stuck).

So in order to build a complete pyramid, we write the code to build a ridge/trench of any size and figure out the steps in-between to get the robot into the right position for the next one. The simplest ridge/trench is a single block long, so lets try writing some code to do that:


The lambda, and bot-sequence etc are scheme code required to get the bot language working, we’re just interested in the contents of the “seq”. After running these instructions we’re in the right place for the next block. Note that the majority of the actions are involved with positioning the bot after doing it’s work. To place the next cube we copy the code and add some more ‘forward’s (as we have to travel a bit further going back and forth):


This is already getting pretty long – we could do with a way to do repetition, so I’ve added a ‘repeat’ form to the language which takes a count, a name bound to the current iteration number (like a ‘for’ loop) and a list of instructions. This is the complete ridge/trench definition for any size, including gaps of any size:


The majority of the code is the maths to get the bot picking up and placing blocks further and further apart including the gap parameter. When collapsed into a function and run we get this:


With the bot ending up in the same place as it started. In order to create a square plateau we call this function, move sideways and repeat, and then move back to where we started again when we’re done:


Going from a plateau function to a pyramid is even shorter, and involves moving inwards diagonally and building smaller plateaus each time. Of course it also mines out a negative pyramid at the same time:


Here is a time lapse of a massive 8×8 pyramid being built, the code ‘compiles’ to 3224 low level instructions:

So this is a kind of programming that encourages solving problems through composition of abstractions – from the low level instructions, simple loops, primitive building constructs, up to complete structures. I’m not sure why in educational languages such as Scratch this is somehow sidelined (interestingly it’s not in Logo, it’s predecessor). Whether this is due to the ubiquity of imperative programming that leads to a focus on manipulation of state, or this kind of programming being considered as too advanced – but for me it’s fundamental, and I’m pretty sure it wouldn’t be that challenging for the kids in the CodeClub I’m running either.

Al Jazari 2 re-arranger robot program

Onboard a robot, who is following a sequence of instructions (in the spirit of the original Al Jazari), to pick up, carry and drops blocks in order to re-organise it’s environment. Halfway through we switch to a robot which is running code that makes it player controllable, so we can look around. This is all more than slightly influenced by teaching Scratch at Code Club and feeling the need for something 3D for them to play with. The scheme bricks coding interface is next…