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 it’s 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 verticies (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
@dataclass
class Vertex:
...
def perspective(self: Vertex,
fov: float,
aspectRatio: float,
nearZ: float,
farZ: float) -> Vertex:
# fov, field of view, is angle of y
# aspectRatio is xwidth / ywidth
top: float = -nearZ * math.tan(math.radians(fov) / 2.0)
right: float = top * aspectRatio
scaled_x: float = self.x / self.z * nearZ
scaled_y: float = self.y / self.z * nearZ
retangular_prism: Vertex = Vertex(scaled_x,
scaled_y,
self.z)
return retangular_prism.ortho(left=-right,
right=right,
bottom=-top,
top=top,
near=nearZ,
far=farZ)
def camera_space_to_ndc_space_fn(self: Vertex) -> Vertex:
return self.perspective(fov=45.0,
aspectRatio=1.0,
nearZ=-0.1,
farZ=-10000.0)
while not glfw.window_should_close(window):
...
# draw paddle 1
glColor3f(paddle1.r, paddle1.g, paddle1.b)
glBegin(GL_QUADS)
for model_space in paddle1.vertices:
world_space: Vertex = model_space.rotate_z(paddle1.rotation) \
.translate(tx=paddle1.position.x,
ty=paddle1.position.y,
tz=0.0)
# world_space: Vertex = camera_space.rotate_x(camera.rot_x) \
# .rotate_y(camera.rot_y) \
# .translate(tx=camera.position_worldspace.x,
# ty=camera.position_worldspace.y,
# tz=camera.position_worldspace.z)
camera_space: Vertex = world_space.translate(tx=-camera.position_worldspace.x,
ty=-camera.position_worldspace.y,
tz=-camera.position_worldspace.z) \
.rotate_y(-camera.rot_y) \
.rotate_x(-camera.rot_x)
ndc_space: Vertex = camera_space.camera_space_to_ndc_space_fn()
glVertex3f(ndc_space.x, ndc_space.y, ndc_space.z)
glEnd()