3D Perspective - Demo 17¶
Purpose¶
Implement a perspective projection so that objects further away are smaller than the would be if they were close by
Demo 17¶
Frustum 1¶
Frustum 2¶
Frustum 3¶
Frustum 4¶
Frustum 5¶
Frustum 6¶
How to Execute¶
On Linux or on MacOS, in a shell, type “python src/demo17/demo.py”. On Windows, in a command prompt, type “python src\demo17\demo.py”.
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¶
------------------------------- far z
\ | /
\ | /
\ (x,z) *----|(0,z) /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\---*-------- near z
| | /
|\ | /
| \ | /
| \ | /
| \|/
-----------------*----*-(0,0)-------------------
(x,0)
If we draw a straight line between (x,z) and (0,0), we will have a right triangle with vertices (0,0), (0,z), and (x,z).
There also will be a similar right triangle with vertices (0,0), (0,nearZ), and whatever point the line above intersects the line at z=nearZ. Let’s call that point (projX, nearZ)
because right angle and tan(theta) = tan(theta)
x / z = projX / nearZ
projX = x / z * nearZ
So use projX as the transformed x value, and keep the distance z.
----------- far z
| |
| |
(x / z * nearZ,z) * | non-linear -- the transformation of x depends on its z value
| |
| |
| |
| |
| |
| |
| |
| |
------------ near z
\ | /
\ | /
\ | /
\ | /
\|/
----------------------*-(0,0)-------------------
Top calculation based off of vertical field of view
/* top
/ |
/ |
/ |
/ |
/ |
/ |
/ |
/ |
/ |
/ |
/ |
/ |
origin/ |
/ fov/2 |
z----*---------------*
|\ |-nearZ
| \ |
| \ |
x \ |
\ |
\ |
\ |
\ |
\ |
\ |
\ |
\ |
\ |
\ |
\|
Right calculation based off of Top and aspect ration
top
-------------------------------------------------------
| |
| y |
| | |
| | |
| *----x | right =
| origin | top * aspectRatio
| |
| | aspect ratio should be the viewport's
| | width/height, not necessarily the
------------------------------------------------------- window's
149@dataclass
150class Vertex:
...
215 def perspective(self: Vertex,
216 fov: float,
217 aspectRatio: float,
218 nearZ: float,
219 farZ: float) -> Vertex:
220 # fov, field of view, is angle of y
221 # aspectRatio is xwidth / ywidth
222
223 top: float = -nearZ * math.tan(math.radians(fov) / 2.0)
224 right: float = top * aspectRatio
225
226 scaled_x: float = self.x / self.z * nearZ
227 scaled_y: float = self.y / self.z * nearZ
228 retangular_prism: Vertex = Vertex(scaled_x,
229 scaled_y,
230 self.z)
231
232 return retangular_prism.ortho(left=-right,
233 right=right,
234 bottom=-top,
235 top=top,
236 near=nearZ,
237 far=farZ)
238
239 def camera_space_to_ndc_space_fn(self: Vertex) -> Vertex:
240 return self.perspective(fov=45.0,
241 aspectRatio=1.0,
242 nearZ=-.1,
243 farZ=-1000.0)
368while not glfw.window_should_close(window):
...
400 # draw paddle 1
401 glColor3f(paddle1.r, paddle1.g, paddle1.b)
402
403 glBegin(GL_QUADS)
404 for paddle1_vertex_in_model_space in paddle1.vertices:
405 paddle1_vertex_in_world_space: Vertex = paddle1_vertex_in_model_space.rotate_z(paddle1.rotation) \
406 .translate(paddle1.position)
407 # paddle1_vertex_in_world_space: Vertex = paddle1_vertex_in_camera_space.rotate_x(camera.rot_x) \
408 # .rotate_y(camera.rot_y) \
409 # .translate(camera.position_worldspace)
410 paddle1_vertex_in_camera_space: Vertex = paddle1_vertex_in_world_space.translate(-camera.position_worldspace) \
411 .rotate_y(-camera.rot_y) \
412 .rotate_x(-camera.rot_x)
413 paddle1_vertex_in_ndc_space: Vertex = paddle1_vertex_in_camera_space.camera_space_to_ndc_space_fn()
414 glVertex3f(paddle1_vertex_in_ndc_space.x, paddle1_vertex_in_ndc_space.y, paddle1_vertex_in_ndc_space.z)
415 glEnd()