Adding Depth - Z axis Demo 14¶

Purpose¶

Do the same stuff as the previous demo, but use 3D coordinates, where the negative z axis goes into the screen (because of the right hand rule). Positive z comes out of the monitor towards your face.

Things that this demo doesn’t end up doing correctly:

The blue square is always drawn, even when its z-coordinate in world space is less than the paddle’s. The solution will be z-buffering https://en.wikipedia.org/wiki/Z-buffering, and it is implemented in the next demo.

How to Execute¶

On Linux or on MacOS, in a shell, type “python src/demo14/demo.py”. On Windows, in a command prompt, type “python src\demo14\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¶

Vertex data will now have an X, Y, and Z component.
Rotations around an angle in 3D space follow the right hand rule. Here’s a link to them in matrix form, which we have not yet covered.

With open palm, fingers on the x axis, rotating the fingers to y axis, means that the positive z axis is in the direction of the thumb. Positive Theta moves in the direction that your fingers did.
starting on the y axis, rotating to z axis, thumb is on the positive x axis.
starting on the z axis, rotating to x axis, thumb is on the positive y axis.

src/demo14/demo.py¶

@dataclass
class Vertex2D:
    x: float
    y: float

    def __add__(self, rhs: Vertex2D) -> Vertex2D:
        return Vertex2D(x=self.x + rhs.x, y=self.y + rhs.y)

    def translate(self: Vertex2D, translate_amount: Vertex2D) -> Vertex2D:
        return self + translate_amount

    def __mul__(self, scalar: float) -> Vertex2D:
        return Vertex2D(x=self.x * scalar, y=self.y * scalar)

    def __rmul__(self, scalar: float) -> Vertex2D:
        return self * scalar

    def uniform_scale(self: Vertex2D, scalar: float) -> Vertex2D:
        return self * scalar

    def scale(self: Vertex2D, scale_x: float, scale_y: float) -> Vertex2D:
        return Vertex2D(x=self.x * scale_x, y=self.y * scale_y)

    def __neg__(self):
        return -1.0 * self

    def rotate_90_degrees(self: Vertex2D):
        return Vertex2D(x=-self.y, y=self.x)

    def rotate(self: Vertex2D, angle_in_radians: float) -> Vertex2D:
        return math.cos(angle_in_radians) * self + math.sin(angle_in_radians) * self.rotate_90_degrees()


@dataclass
class Vertex:
    x: float
    y: float
    z: float

    def __add__(self, rhs: Vertex) -> Vertex:
        return Vertex(x=self.x + rhs.x, y=self.y + rhs.y, z=self.z + rhs.z)

    def translate(self: Vertex, translate_amount: Vertex) -> Vertex:
        return self + translate_amount

Rotate Z¶

Rotate Z is the same rotate that we’ve used so far, but doesn’t affect the z component at all.

src/demo14/demo.py¶

    def rotate_z(self: Vertex, angle_in_radians: float) -> Vertex:
        xy_on_xy: Vertex2D = Vertex2D(x=self.x, y=self.y).rotate(angle_in_radians)
        return Vertex(x=xy_on_xy.x, y=xy_on_xy.y, z=self.z)

Rotate X¶

src/demo14/demo.py¶

    def rotate_x(self: Vertex, angle_in_radians: float) -> Vertex:
        yz_on_xy: Vertex2D = Vertex2D(x=self.y, y=self.z).rotate(angle_in_radians)
        return Vertex(x=self.x, y=yz_on_xy.x, z=yz_on_xy.y)

Rotate Y¶

src/demo14/demo.py¶

    def rotate_y(self: Vertex, angle_in_radians: float) -> Vertex:
        zx_on_xy: Vertex2D = Vertex2D(x=self.z, y=self.x).rotate(angle_in_radians)
        return Vertex(x=zx_on_xy.y, y=self.y, z=zx_on_xy.y)

Scale¶

src/demo14/demo.py¶

    def __mul__(self, scalar: float) -> Vertex:
        return Vertex(x=self.x * scalar, y=self.y * scalar, z=self.z * scalar)

    def __rmul__(self, scalar: float) -> Vertex:
        return self * scalar

    def uniform_scale(self: Vertex, scalar: float) -> Vertex:
        return self * scalar

    def scale(self: Vertex, scale_x: float, scale_y: float, scale_z: float) -> Vertex:
        return Vertex(x=self.x * scale_x, y=self.y * scale_y, z=self.z * scale_z)

    def __neg__(self):
        return -1.0 * self

Code¶

The only new aspect of the code below is that the paddles have a z-coordinate of 0 in their modelspace.

src/demo14/demo.py¶

paddle1: Paddle = Paddle(
    vertices=[
        Vertex(x=-1.0, y=-3.0, z=0.0),
        Vertex(x=1.0, y=-3.0, z=0.0),
        Vertex(x=1.0, y=3.0, z=0.0),
        Vertex(x=-1.0, y=3.0, z=0.0),
    ],
    r=0.578123,
    g=0.0,
    b=1.0,
    position=Vertex(x=-9.0, y=0.0, z=0.0),
)

paddle2: Paddle = Paddle(
    vertices=[
        Vertex(x=-1.0, y=-3.0, z=0.0),
        Vertex(x=1.0, y=-3.0, z=0.0),
        Vertex(x=1.0, y=3.0, z=0.0),
        Vertex(x=-1.0, y=3.0, z=0.0),
    ],
    r=1.0,
    g=1.0,
    b=0.0,
    position=Vertex(x=9.0, y=0.0, z=0.0),
)

The only new aspect of the square below is that the paddles have a z-coordinate of 0 in their modelspace. N.B that since we do a sequence transformations to the modelspace data to get to world-space coordinates, the X, Y, and Z coordinates are subject to be different.

src/demo14/demo.py¶

@dataclass
class Camera:
    position_worldspace: Vertex = field(default_factory=lambda: Vertex(x=0.0, y=0.0, z=0.0))


camera: Camera = Camera()

The camera now has a z-coordinate of 0 also.

src/demo14/demo.py¶

square: Paddle = [
    Vertex(x=-0.5, y=-0.5, z=0.0),
    Vertex(x=0.5, y=-0.5, z=0.0),
    Vertex(x=0.5, y=0.5, z=0.0),
    Vertex(x=-0.5, y=0.5, z=0.0),
]

Event Loop¶

src/demo14/demo.py¶

while not glfw.window_should_close(window):

...

Draw Paddle 1

src/demo14/demo.py¶

    glColor3f(paddle1.r, paddle1.g, paddle1.b)
    glBegin(GL_QUADS)
    for paddle1_vertex_in_model_space in paddle1.vertices:
        paddle1_vertex_in_world_space: Vertex = paddle1_vertex_in_model_space.rotate_z(paddle1.rotation) \
                                         .translate(paddle1.position)
        paddle1_vertex_in_camera_space: Vertex = paddle1_vertex_in_world_space.translate(-camera.position_worldspace)
        paddle1_vertex_in_ndc_space: Vertex = paddle1_vertex_in_camera_space.uniform_scale(scalar=1.0 / 10.0)
        glVertex2f(paddle1_vertex_in_ndc_space.x, paddle1_vertex_in_ndc_space.y)
    glEnd()

The square should not be visible when hidden behind the paddle1, as we do a translate by -1. But in running the demo, you see that the square is always drawn over the paddle.

Draw the Square

src/demo14/demo.py¶

    # draw square
    glColor3f(0.0, 0.0, 1.0)
    glBegin(GL_QUADS)
    for model_space in square:
        paddle_1_space: Vertex =  model_space.rotate_z(square_rotation) \
                                             .translate(Vertex(x=2.0,
                                                               y=0.0,
                                                               z=0.0)) \
                                             .rotate_z(rotation_around_paddle1) \
                                             .translate(Vertex(x=0.0,
                                                               y=0.0,
                                                               z=-1.0))
        world_space: Vertex = paddle_1_space.rotate_z(paddle1.rotation) \
                                            .translate(paddle1.position)
        camera_space: Vertex = world_space.translate(-camera.position_worldspace)
        ndc_space: Vertex = camera_space.uniform_scale(scalar=1.0 / 10.0)
        glVertex3f(ndc_space.x, ndc_space.y, ndc_space.z)
    glEnd()

This is because without depth buffering, the object drawn last clobbers the color of any previously drawn object at the pixel. Try moving the square drawing code to the beginning, and you will see that the square can be hidden behind the paddle.

Draw Paddle 2

src/demo14/demo.py¶

    # draw paddle 2
    glColor3f(paddle2.r, paddle2.g, paddle2.b)
    glBegin(GL_QUADS)
    for paddle2_vertex_model_space in paddle2.vertices:
        paddle2_vertex_world_space: Vertex = paddle2_vertex_model_space.rotate_z(paddle2.rotation) \
                                                                       .translate(paddle2.position)
        paddle2_vertex_camera_space: Vertex = paddle2_vertex_world_space.translate(-camera.position_worldspace)
        paddle2_vertex_ndc_space: Vertex = paddle2_vertex_camera_space.uniform_scale(scalar=1.0 / 10.0)
        glVertex3f(paddle2_vertex_ndc_space.x, paddle2_vertex_ndc_space.y, paddle2_vertex_ndc_space.z)
    glEnd()

Added translate in 3D. Added scale in 3D. These are just like the 2D versions, just with the same process applied to the z axis.

They direction of the rotation is defined by the right hand rule.

https://en.wikipedia.org/wiki/Right-hand_rule