Lambda Stack - Demo 16¶
Objective¶
Remove repetition in the coordinate transformations, as previous demos had very similar transformations, especially from camera space to NDC space. Each edge of the graph of objects should only be specified once per frame.

Full Cayley graph.¶
Noticing in the previous demos that the lower parts of the transformations have a common pattern, we can create a stack of functions for later application. Before drawing geometry, we add any functions to the top of the stack, apply all of our functions in the stack to our modelspace data to get NDC data, and before we return to the parent node, we pop the functions we added off of the stack, to ensure that we return the stack to the state that the parent node gave us.
To explain in more detail —
What’s the difference between drawing paddle 1 and the square?
Here is paddle 1 code
209 GL.glColor3f(*iter(paddle1.color))
210 GL.glBegin(GL.GL_QUADS)
211 for p1_v_ms in paddle1.vertices:
212 ms_to_ndc: mu.InvertibleFunction[mu3d.Vector3D] = mu.compose(
213 [
214 # camera space to NDC
215 mu.uniform_scale(1.0 / 10.0),
216 # world space to camera space
217 mu.inverse(mu.translate(camera.position_ws)),
218 # model space to world space
219 mu.compose(
220 [
221 mu.translate(paddle1.position),
222 mu3d.rotate_z(paddle1.rotation),
223 ]
224 ),
225 ]
226 )
227
228 paddle1_vector_ndc: mu3d.Vector3D = ms_to_ndc(p1_v_ms)
229 GL.glVertex3f(
230 paddle1_vector_ndc.x, paddle1_vector_ndc.y, paddle1_vector_ndc.z
231 )
232 GL.glEnd()
Here is the square’s code:
236 # draw square
237 GL.glColor3f(0.0, 0.0, 1.0)
238 GL.glBegin(GL.GL_QUADS)
239 for ms in square:
240 ms_to_ndc: mu.InvertibleFunction[mu3d.Vector3D] = mu.compose(
241 [
242 # camera space to NDC
243 mu.uniform_scale(1.0 / 10.0),
244 # world space to camera space
245 mu.inverse(mu.translate(camera.position_ws)),
246 # model space to world space
247 mu.compose(
248 [
249 mu.translate(paddle1.position),
250 mu3d.rotate_z(paddle1.rotation),
251 ]
252 ),
253 # square space to paddle 1 space
254 mu.compose(
255 [
256 mu.translate(mu3d.Vector3D(x=0.0, y=0.0, z=-1.0)),
257 mu3d.rotate_z(rotation_around_paddle1),
258 mu.translate(mu3d.Vector3D(x=2.0, y=0.0, z=0.0)),
259 mu3d.rotate_z(square_rotation),
260 ]
261 ),
262 ]
263 )
264 square_vector_ndc: mu3d.Vector3D = ms_to_ndc(ms)
265 GL.glVertex3f(
266 square_vector_ndc.x, square_vector_ndc.y, square_vector_ndc.z
267 )
268 GL.glEnd()
The only difference is the square’s modelspace to paddle1 space. Everything else is exactly the same. In a graphics program, because the scene is a hierarchy of relative objects, it is unwise to put this much repetition in the transformation sequence. Especially if we might change how the camera operates, or from perspective to ortho. It would required a lot of code changes. And I don’t like reading from the bottom of the code up. Code doesn’t execute that way. I want to read from top to bottom.
When reading the transformation sequences in the previous demos from top down the transformation at the top is applied first, the transformation at the bottom is applied last, with the intermediate results method-chained together. (look up above for a reminder)
With a function stack, the function at the top of the stack (f5) is applied first, the result of this is then given as input to f4 (second on the stack), all the way down to f1, which was the first fn to be placed on the stack, and as such, the last to be applied. (Last In First Applied - LIFA)
|-------------------|
(MODELSPACE) | |
(x,y,z)-> | f5 |--
|-------------------| |
|
-------------------------
|
| |-------------------|
| | |
->| f4 |--
|-------------------| |
|
-------------------------
|
| |-------------------|
| | |
->| f3 |--
|-------------------| |
|
-------------------------
|
| |-------------------|
| | |
->| f2 |--
|-------------------| |
|
-------------------------
|
| |-------------------|
| | |
->| f1 |--> (x,y,z) NDC
|-------------------|
So, in order to ensure that the functions in a stack will execute in the same order as all of the previous demos, they need to be pushed onto the stack in reverse order.
This means that from modelspace to world space, we can now read the transformations FROM TOP TO BOTTOM!!!! SUCCESS!
Then, to draw the square relative to paddle one, those six transformations will already be on the stack, therefore only push the differences, and then apply the stack to the paddle’s modelspace data.
How to Execute¶
Load src/modelviewprojection/demo16.py in Spyder and hit the play button.
Move the Paddles using the Keyboard¶
Keyboard Input |
Action |
---|---|
w |
Move Left Paddle Up |
s |
Move Left Paddle Down |
k |
Move Right Paddle Down |
i |
Move Right Paddle Up |
d |
Increase Left Paddle’s Rotation |
a |
Decrease Left Paddle’s Rotation |
l |
Increase Right Paddle’s Rotation |
j |
Decrease Right Paddle’s Rotation |
UP |
Move the camera up, moving the objects down |
DOWN |
Move the camera down, moving the objects up |
LEFT |
Move the camera left, moving the objects right |
RIGHT |
Move the camera right, moving the objects left |
q |
Rotate the square around its center |
e |
Rotate the square around paddle 1’s center |
Description¶
Function stack. Internally it has a list, where index 0 is the bottom of the stack. In python we can store any object as a variable, and we will be storing functions which transform a vector to another vector, through the “modelspace_to_ndc” method.
277@dataclasses.dataclass
278class FunctionStack:
279 stack: typing.List[mathutils.InvertibleFunction[Vector3D]] = (
280 dataclasses.field(default_factory=lambda: [])
281 )
282
283 def push(self, o: object):
284 self.stack.append(o)
285
286 def pop(self):
287 return self.stack.pop()
288
289 def clear(self):
290 self.stack.clear()
291
292 def modelspace_to_ndc_fn(self) -> mathutils.InvertibleFunction[Vector3D]:
293 return mathutils.compose(self.stack)
294
295
296fn_stack = FunctionStack()
Define four functions, which we will compose on the stack.
Push identity onto the stack, which will will never pop off of the stack.
179def test_fn_stack():
180 def identity(x):
181 return x
182
183 mu3d.fn_stack.push(identity)
184 assert 1 == mu3d.fn_stack.modelspace_to_ndc_fn()(1)
185
186 def add_one(x):
187 return x + 1
188
189 mu3d.fn_stack.push(add_one)
190 assert 2 == mu3d.fn_stack.modelspace_to_ndc_fn()(1) # x + 1 = 2
191
192 def multiply_by_2(x):
193 return x * 2
194
195 mu3d.fn_stack.push(multiply_by_2) # (x * 2) + 1 = 3
196 assert 3 == mu3d.fn_stack.modelspace_to_ndc_fn()(1)
197
198 def add_5(x):
199 return x + 5
200
201 mu3d.fn_stack.push(add_5) # ((x + 5) * 2) + 1 = 13
202 assert 13 == mu3d.fn_stack.modelspace_to_ndc_fn()(1)
203
204 mu3d.fn_stack.pop()
205 assert 3 == mu3d.fn_stack.modelspace_to_ndc_fn()(1) # (x * 2) + 1 = 3
206
207 mu3d.fn_stack.pop()
208 assert 2 == mu3d.fn_stack.modelspace_to_ndc_fn()(1) # x + 1 = 2
209
210 mu3d.fn_stack.pop()
211 assert 1 == mu3d.fn_stack.modelspace_to_ndc_fn()(1) # x = 1
Event Loop¶
189while not glfw.window_should_close(window):
...
In previous demos, camera_space_to_ndc_space_fn was always the last function called in the method chained pipeline. Put it on the bottom of the stack, by pushing it first, so that “modelspace_to_ndc” calls this function last. Each subsequent push will add a new function to the top of the stack.

209 # camera space to NDC
210 mu3d.fn_stack.push(mu.uniform_scale(1.0 / 10.0))
Unlike in previous demos in which we read the transformations from modelspace to world space backwards; this time because the transformations are on a stack, the fns on the model stack can be read forwards, where each operation translates/rotates/scales the current space
The camera’s position and orientation are defined relative to world space like so, read top to bottom:

But, since we need to transform world-space to camera space, they must be inverted by reversing the order, and negating the arguments
Therefore the transformations to put the world space into camera space are.

214 # world space to camera space
215 mu3d.fn_stack.push(mu.inverse(mu.translate(camera.position_ws)))
draw paddle 1¶
Unlike in previous demos in which we read the transformations from modelspace to world space backwards; because the transformations are on a stack, the fns on the model stack can be read forwards, where each operation translates/rotates/scales the current space

219 # paddle 1 model space to world space
220 mu3d.fn_stack.push(
221 mu.compose(
222 [mu.translate(paddle1.position), mu3d.rotate_z(paddle1.rotation)]
223 )
224 )
for each of the modelspace coordinates, apply all of the procedures on the stack from top to bottom this results in coordinate data in NDC space, which we can pass to glVertex3f
228 GL.glColor3f(*iter(paddle1.color))
229 GL.glBegin(GL.GL_QUADS)
230 for p1_v_ms in paddle1.vertices:
231 paddle1_vector_ndc = mu3d.fn_stack.modelspace_to_ndc_fn()(p1_v_ms)
232 GL.glVertex3f(
233 paddle1_vector_ndc.x,
234 paddle1_vector_ndc.y,
235 paddle1_vector_ndc.z,
236 )
237 GL.glEnd()
draw the square¶
since the modelstack is already in paddle1’s space, and since the blue square is defined relative to paddle1, just add the transformations relative to it before the blue square is drawn. Draw the square, and then remove these 4 transformations from the stack (done below)

241 mu3d.fn_stack.push(
242 mu.compose(
243 [
244 mu.translate(mu3d.Vector3D(x=0.0, y=0.0, z=-1.0)),
245 mu3d.rotate_z(rotation_around_paddle1),
246 mu.translate(mu3d.Vector3D(x=2.0, y=0.0, z=0.0)),
247 mu3d.rotate_z(square_rotation),
248 ]
249 )
250 )
253 GL.glColor3f(0.0, 0.0, 1.0)
254 GL.glBegin(GL.GL_QUADS)
255 for ms in square:
256 square_vector_ndc = mu3d.fn_stack.modelspace_to_ndc_fn()(ms)
257 GL.glVertex3f(
258 square_vector_ndc.x,
259 square_vector_ndc.y,
260 square_vector_ndc.z,
261 )
262 GL.glEnd()
Now we need to remove fns from the stack so that the lambda stack will convert from world space to NDC. This will allow us to just add the transformations from world space to paddle2 space on the stack.
266 mu3d.fn_stack.pop() # pop off square space to paddle 1 space
267 # current space is paddle 1 space
268 mu3d.fn_stack.pop() # # pop off paddle 1 model space to world space
269 # current space is world space
since paddle2’s modelspace is independent of paddle 1’s space, only leave the view and projection fns (1) - (4)
draw paddle 2¶

273 mu3d.fn_stack.push(
274 mu.compose(
275 [mu.translate(paddle2.position), mu3d.rotate_z(paddle2.rotation)]
276 )
277 )
281 # draw paddle 2
282 GL.glColor3f(*iter(paddle2.color))
283 GL.glBegin(GL.GL_QUADS)
284 for p2_v_ms in paddle2.vertices:
285 paddle2_vector_ndc = mu3d.fn_stack.modelspace_to_ndc_fn()(p2_v_ms)
286 GL.glVertex3f(
287 paddle2_vector_ndc.x,
288 paddle2_vector_ndc.y,
289 paddle2_vector_ndc.z,
290 )
291 GL.glEnd()
remove all fns from the function stack, as the next frame will set them clear makes the list empty, as the list (stack) will be repopulated the next iteration of the event loop.
295 # done rendering everything for this frame, just go ahead and clear all functions
296 # off of the stack, back to NDC as current space
297 mu3d.fn_stack.clear()
298
Swap buffers and execute another iteration of the event loop
302 glfw.swap_buffers(window)
Notice in the above code, adding functions to the stack is creating a shared context for transformations, and before we call “glVertex3f”, we always call “modelspace_to_ndc” on the modelspace vector. In Demo 19, we will be using OpenGL 2.1 matrix stacks. Although we don’t have the code for the OpenGL driver, given that you’ll see that we pass modelspace data directly to “glVertex3f”, it should be clear that the OpenGL implementation must fetch the modelspace to NDC transformations from the ModelView and Projection matrix stacks.