Moving Camera in 3D Space - Demo 17¶

Objective¶

Make a moving camera in 3D space. Use Ortho to transform a rectangular prism, defined relative to camera space, into NDC.

Problem purposefully put in¶

When running this demo and moving the viewer, parts of the geometry will disappear. This is because it gets “clipped out”, as the geometry will be outside of NDC, (-1 to 1 on all three axis). We could fix this by making a bigger ortho rectangular prism, but that won’t solve the fundamental problem.

This doesn’t look like a 3D application should, where objects further away from the viewer would appear smaller. This will be fixed in demo17.

How to Execute¶

Load src/modelviewprojection/demo17.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¶

Before starting this demo, run mvpVisualization/modelvieworthoprojection/modelvieworthoprojection.py, as it will show graphically all of the steps in this demo. In the GUI, take a look at the camera options buttons, and once the camera is placed and oriented in world space, use the buttons to change the camera’s position and orientation. This will demonstrate what we have to do for moving the camera in a 3D scene.

There are new keyboard inputs to control the moving camera. As you would expect to see in a first person game, up moves the camera forward (-z), down moves the camera backwards (z), left rotates the camera as would happen if you rotated your body to the left, and likewise for right. Page UP and Page DOWN rotate the camera to look up or to look down.

To enable this, the camera is modeled with a data structure, having a position in x,y,z relative to world space, and two rotations (one around the camera’s x axis, and one around the camera’s y axis).

To position the camera you would

translate to the camera’s position, using the actual position values of camera position in world space coordinates.
rotate around the local y axis
rotate around the local x axis

To visualize this, run “python mvpVisualization/modelvieworthoprojection/modelvieworthoprojection.py”

The ordering of 1) before 2) and 3) should be clear, as we are imagining a coordinate system that moves, just like we do for the modelspace to world space transformations. The ordering of 2) before 3) is very important, as two rotations around different axes are not commutative, meaning that you can’t change the order and still expect the same results https://en.wikipedia.org/wiki/Commutative_property.

Try this. Rotate your head to the right a little more that 45 degrees. Now rotate your head back a little more than 45 degrees.

Now, reset your head (glPopMatrix, which we have not yet covered). Try rotating your head back 45 degrees. Once it is there, rotate your head (not your neck), 45 degrees. It is different, and quite uncomfortable!

We rotate the camera by the y axis first, then by the relative x axis, for the same reason.

(Remember, read bottom up, just like the previous demos for modelspace to world-space data)

Back to the point, we are envisioning the camera relative to the world space by making a moving coordinate system (composed of an origin, 1 unit in the “x” axis, 1 unit in the “y” axis, and 1 unit in the “z” axis), where each subsequent transformation is relative to the previous coordinate system. (This system is beneficial btw because it allows us to think of only one coordinate system at a time, and allows us to forget how we got there, (similar to a Markov process, https://en.wikipedia.org/wiki/Markov_chain))

But this system of thinking works only when we are placing the camera into its position/orientation relative to world space, which is not what we need to actually do. We don’t need to place the camera. We need to move every already-plotted object in world space towards the origin and orientation of NDC. Looking at the following graph,

We want to take the modelspace geometry from, say Paddle1 space, to world space, and then to camera space (which is going in the opposite direction of the arrow, therefore requires an inverse operation, because to plot data we go from modelspace to screen space on the graph.

Given that the inverse of a sequence of transformations is the sequence backwards, with each transformations inverted, we must do that to get from world space to camera space.

Other things added Added rotations around the x axis, y axis, and z axis. https://en.wikipedia.org/wiki/Rotation_matrix

Code¶

The camera now has two angles as instance variables.

src/modelviewprojection/demo17.py¶

@dataclasses.dataclass
class Camera:
    position_ws: mu3d.Vector3D = dataclasses.field(
        default_factory=lambda: mu3d.Vector3D(x=0.0, y=0.0, z=15.0)
    )
    rot_y: float = 0.0
    rot_x: float = 0.0

Since we want the user to be able to control the camera, we need to read the input.

src/modelviewprojection/demo17.py¶

def handle_inputs() -> None:

...

Left and right rotate the viewer’s horizontal angle, page up and page down the vertical angle.

src/modelviewprojection/demo17.py¶

    if glfw.get_key(window, glfw.KEY_RIGHT) == glfw.PRESS:
        camera.rot_y -= 0.03
    if glfw.get_key(window, glfw.KEY_LEFT) == glfw.PRESS:
        camera.rot_y += 0.03
    if glfw.get_key(window, glfw.KEY_PAGE_UP) == glfw.PRESS:
        camera.rot_x += 0.03
    if glfw.get_key(window, glfw.KEY_PAGE_DOWN) == glfw.PRESS:
        camera.rot_x -= 0.03

The up arrow and down arrow make the user move forwards and backwards. Unlike the camera space to world space transformation, here for movement code, we don’t do the rotate around the x axis. This is because users expect to simulate walking on the ground, not flying through the sky. I.e, we want forward/backwards movement to happen relative to the XZ plane at the camera’s position, not forward/backwards movement relative to camera space.

src/modelviewprojection/demo17.py¶

    if glfw.get_key(window, glfw.KEY_UP) == glfw.PRESS:
        forwards_cs = mu3d.Vector3D(x=0.0, y=0.0, z=-1.0)
        forward_ws = mu3d.compose(
            [mu3d.translate(camera.position_ws), mu3d.rotate_y(camera.rot_y)]
        )(forwards_cs)
        camera.position_ws = forward_ws
    if glfw.get_key(window, glfw.KEY_DOWN) == glfw.PRESS:
        forwards_cs = mu3d.Vector3D(x=0.0, y=0.0, z=1.0)
        forward_ws = mu3d.compose(
            [mu3d.translate(camera.position_ws), mu3d.rotate_y(camera.rot_y)]
        )(forwards_cs)
        camera.position_ws = forward_ws

Ortho is the function call that shrinks the viewable region relative to camera space down to NDC, by moving the center of the rectangular prism to the origin, and scaling by the inverse of the width, height, and depth of the viewable region.

src/modelviewprojection/mathutils3d.py¶

def ortho(
    left: float,
    right: float,
    bottom: float,
    top: float,
    near: float,
    far: float,
) -> mu.InvertibleFunction:
    midpoint = Vector3D(
        x=(left + right) / 2.0, y=(bottom + top) / 2.0, z=(near + far) / 2.0
    )
    length_x: float
    length_y: float
    length_z: float
    length_x, length_y, length_z = right - left, top - bottom, far - near

    fn = mu.compose(
        [
            scale(
                m_x=(2.0 / length_x),
                m_y=(2.0 / length_y),
                m_z=(2.0 / (-length_z)),
            ),
            mu.translate(-midpoint),
        ]
    )

    def f(vector: mu.Vector) -> mu.Vector:
        return fn(vector)

    def f_inv(vector: mu.Vector) -> mu.Vector:
        return mu.inverse(fn)(vector)

    return mu.InvertibleFunction(f, f_inv)

We will make a wrapper function camera_space_to_ndc_space_fn which calls ortho, setting the size of the rectangular prism.

src/modelviewprojection/mathutils3d.py¶

def cs_to_ndc_space_fn(
    vector: Vector3D,
) -> mu.InvertibleFunction:
    return perspective(
        field_of_view=45.0, aspect_ratio=1.0, near_z=-0.1, far_z=-1000.0
    )

Event Loop¶

The amount of repetition in the code below in starting to get brutal, as there’s too much detail to think about and retype out for every object being drawn, and we’re only dealing with 3 objects. The author put this repetition into the book on purpose, so that when we start using matrices later, the reader will fully appreciate what matrices solve for us.

src/modelviewprojection/demo17.py¶

while not glfw.window_should_close(window):

...

the square should not be visible when hidden behind the paddle1, as we did a translate by -10 in the z direction.

src/modelviewprojection/demo17.py¶

    # cameraspace to NDC
    with mu3d.push_transformation(
        mu3d.ortho(
            left=-10.0, right=10.0, bottom=-10.0, top=10.0, near=-0.1, far=-30.0
        )
    ):
        # world space to camera space, which is mu3d.inverse of camera space to world space
        with mu3d.push_transformation(
            mu3d.inverse(
                mu3d.compose(
                    [
                        mu3d.translate(camera.position_ws),
                        mu3d.rotate_y(camera.rot_y),
                        mu3d.rotate_x(camera.rot_x),
                    ]
                )
            )
        ):
            # paddle 1 space to world space
            with mu3d.push_transformation(
                mu3d.compose(
                    [
                        mu3d.translate(paddle1.position),
                        mu3d.rotate_z(paddle1.rotation),
                    ]
                )
            ):
                GL.glColor3f(*iter(paddle1.color))
                GL.glBegin(GL.GL_QUADS)
                for p1_v_ms in paddle1.vertices:
                    paddle1_vector_ndc = mu3d.fn_stack.modelspace_to_ndc_fn()(
                        p1_v_ms
                    )
                    GL.glVertex3f(
                        paddle1_vector_ndc.x,
                        paddle1_vector_ndc.y,
                        paddle1_vector_ndc.z,
                    )
                GL.glEnd()

                # square space to paddle 1 space
                with mu3d.push_transformation(
                    mu3d.compose(
                        [
                            mu3d.translate(mu3d.Vector3D(x=0.0, y=0.0, z=-1.0)),
                            mu3d.rotate_z(rotation_around_paddle1),
                            mu3d.translate(mu3d.Vector3D(x=2.0, y=0.0, z=0.0)),
                            mu3d.rotate_z(square_rotation),
                        ]
                    )
                ):
                    # draw square
                    GL.glColor3f(0.0, 0.0, 1.0)
                    GL.glBegin(GL.GL_QUADS)
                    for ms in square:
                        square_vector_ndc = (
                            mu3d.fn_stack.modelspace_to_ndc_fn()(ms)
                        )
                        GL.glVertex3f(
                            square_vector_ndc.x,
                            square_vector_ndc.y,
                            square_vector_ndc.z,
                        )
                    GL.glEnd()

            # paddle 2 space to world space
            with mu3d.push_transformation(
                mu3d.compose(
                    [
                        mu3d.translate(paddle2.position),
                        mu3d.rotate_z(paddle2.rotation),
                    ]
                )
            ):
                # draw paddle 2
                GL.glColor3f(*iter(paddle2.color))
                GL.glBegin(GL.GL_QUADS)
                for p2_v_ms in paddle2.vertices:
                    paddle2_vector_ndc = mu3d.fn_stack.modelspace_to_ndc_fn()(
                        p2_v_ms
                    )
                    GL.glVertex3f(
                        paddle2_vector_ndc.x,
                        paddle2_vector_ndc.y,
                        paddle2_vector_ndc.z,
                    )
                GL.glEnd()

    glfw.swap_buffers(window)