Adding Depth - Z axis Demo 14¶

Objective¶

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¶

Load src/modelviewprojection/demo14.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¶

Vector 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/modelviewprojection/mathutils3d¶

@dataclass
class Vector3D:
    x: float  #: The x-component of the 3D Vector
    y: float  #: The y-component of the 3D Vector
    z: float  #: The z-component of the 3D Vector

    def __add__(self, rhs: Vector3D) -> Vector3D:
        return Vector3D(
            x=(self.x + rhs.x), y=(self.y + rhs.y), z=(self.z + rhs.z)
        )

    def __sub__(self, rhs: Vector3D) -> Vector3D:
        return Vector3D(
            x=(self.x - rhs.x), y=(self.y - rhs.y), z=(self.z - rhs.z)
        )

    def __mul__(vector, scalar: float) -> Vector3D:
        return Vector3D(
            x=(vector.x * scalar), y=(vector.y * scalar), z=(vector.z * scalar)
        )

    def __rmul__(vector, scalar: float) -> Vector3D:
        return vector * scalar

    def __neg__(vector):
        return -1.0 * vector


def translate(translate_amount: Vector3D) -> InvertibleFunction:
    def f(vector: Vector3D) -> Vector3D:
        return vector + translate_amount

    def f_inv(vector: Vector3D) -> Vector3D:
        return vector - translate_amount

    return InvertibleFunction(f, f_inv)

Rotate Z¶

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

src/modelviewprojection/mathutils3d.py¶

def rotate_z(angle_in_radians: float) -> Vector3D:
    fn = rotate2D(angle_in_radians)

    def f(vector: Vector3D) -> Vector3D:
        xy_on_xy: Vector2D = fn(Vector2D(x=vector.x, y=vector.y))
        return Vector3D(x=xy_on_xy.x, y=xy_on_xy.y, z=vector.z)

    def f_inv(vector: Vector3D) -> Vector3D:
        xy_on_xy: Vector2D = inverse(fn)(Vector2D(x=vector.x, y=vector.y))
        return Vector3D(x=xy_on_xy.x, y=xy_on_xy.y, z=vector.z)

    return InvertibleFunction(f, f_inv)

Rotate X¶

src/modelviewprojection/mathutils3d.py¶

def rotate_x(angle_in_radians: float) -> Vector3D:
    fn = rotate2D(angle_in_radians)

    def f(vector: Vector3D) -> Vector3D:
        yz_on_xy: Vector2D = fn(Vector2D(x=vector.y, y=vector.z))
        return Vector3D(x=vector.x, y=yz_on_xy.x, z=yz_on_xy.y)

    def f_inv(vector: Vector3D) -> Vector3D:
        yz_on_xy: Vector2D = inverse(fn)(Vector2D(x=vector.y, y=vector.z))
        return Vector3D(x=vector.x, y=yz_on_xy.x, z=yz_on_xy.y)

    return InvertibleFunction(f, f_inv)

Rotate Y¶

src/modelviewprojection/mathutils3d.py¶

def rotate_y(angle_in_radians: float) -> Vector3D:
    fn = rotate2D(angle_in_radians)

    def f(vector: Vector3D) -> Vector3D:
        zx_on_xy: Vector2D = fn(Vector2D(x=vector.z, y=vector.x))
        return Vector3D(x=zx_on_xy.y, y=vector.y, z=zx_on_xy.x)

    def f_inv(vector: Vector3D) -> Vector3D:
        zx_on_xy: Vector2D = inverse(fn)(Vector2D(x=vector.z, y=vector.x))
        return Vector3D(x=zx_on_xy.y, y=vector.y, z=zx_on_xy.x)

    return InvertibleFunction(f, f_inv)

Scale¶

src/modelviewprojection/mathutils3d.py¶

def uniform_scale(scalar: float) -> InvertibleFunction:
    def f(vector: Vector3D) -> Vector3D:
        return vector * scalar

    def f_inv(vector: Vector3D) -> Vector3D:
        if scalar == 0:
            raise ValueError("Not invertible.  Scaling factor cannot be zero.")

        return vector / scalar

    return InvertibleFunction(f, f_inv)

Code¶

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