Rotations - Demo 07¶

Objective¶

Attempt to rotate the paddles around their center. Learn about rotations. This demo does not work correctly, because of a misunderstanding of how to interpret a sequence of transformations.

How to Execute¶

Load src/modelviewprojection/demo07.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

For another person’s explanation of the trigonometry of rotating in 2D, see

Rotate the Paddles About their Center¶

Besides translate and scale, the third main operation in computer graphics is to rotate an object.

Rotation Around Origin (0,0)¶

We can rotate an object around (0,0) by rotating all of the object’s vertices around (0,0).

In high school math, you will have learned about sin, cos, and tangent. Typically the angles are described on the unit circle, where a rotation starts from the positive x axis.

We can expand on this knowledge, allowing us to rotate all vertices, wherever they are, around the origin (0,0), by some angle \(\theta\).

Let’s take any arbitrary point

\[\vec{a}\]

\[\vec{r}(\vec{a}; \theta)\]

From high school geometry, remember that we can describe a Cartesian (x,y) point by its length r, and its cosine and sine of its angle \(\theta\).

Also remember that when calculating sine and cosine, because right triangles are proportional, that sine and cosine are preserved for a right triangle, even if the length sides are scaled up or down.

So we’ll make a right triangle on the unit circle, but we will remember the length of (x,y), which we’ll call “r”, and we’ll call the angle of (x,y) to be “\(\beta\)”. As a reminder, we want to rotate by a different angle, called “\(\theta\)”.

Before we can rotate by “\(\theta\)”, first we need to be able to rotate by 90 degrees, or \(\pi\)/2. So to rotate (cos(\(\beta\)), sin(\(\beta\))) by \(\pi\)/2, we get (cos(\(\beta\) + \(\pi\)/2), sin(\(\beta\) + \(\pi\)/2)).

Now let’s give each of those vertices new names, x’ and y’, for the purpose of ignoring details for now that we’ll return to later, just let we did for length “r” above.

Now forget about “\(\beta\)”, and remember that our goal is to rotate by angle “\(\theta\)”. Look at the picture below, while turning your head slightly to the left. x’ and y’ look just like our normal Cartesian plane and unit circle, combined with the “\(\theta\)”; it looks like what we already know from high school geometry.

So with this new frame of reference, we can rotate x’ by “\(\theta\)”, and draw a right triangle on the unit circle using this new frame of reference.

So the rotated point can be constructed by the following

\[\vec{r}(\vec{a}; theta) = cos(\theta)*\vec{x'} + sin(\theta)*\vec{y'}\]

Now that we’ve found the direction on the unit circle, we remember to make it length “r”.

Ok, we are now going to stop thinking about geometry, and we will only be thinking about algebra. Please don’t try to look at the formula and try to draw any diagrams.

Ok, now it is time to remember what the values that x’ and y’ are defined as.

\[\begin{split}\vec{r}(\vec{a}; \theta) & = r*(cos(\theta)*\vec{x'} + sin(\theta)*\vec{y'}) \\ & = r*(cos(\theta)*\begin{bmatrix} cos(\beta) \\ sin(\beta) \\ \end{bmatrix} + sin(\theta)*\begin{bmatrix} cos(\beta + \pi/2) \\ sin(\beta + \pi/2) \\ \end{bmatrix})\end{split}\]

A problem we have now is how to calculate cosine and sine of \(\beta\) + \(\pi\)/2, because we haven’t actually calculated \(\beta\); we’ve calculated sine and cosine of \(\beta\) by dividing the x value by the magnitude of the a, and the sine of \(\beta\) by dividing the y value by the magnitude of a.

\[ \begin{align}\begin{aligned}cos(\theta) = \vec{a}_{x} / r\\sine(\theta) = \vec{a}_{y} / r\end{aligned}\end{align} \]

We could try to take the inverse sine or inverse cosine of theta, but there is no need given properties of trigonometry.

\[ \begin{align}\begin{aligned}cos(\theta + \pi/2) = -sin(\theta)\\sin(\theta + \pi/2) = cos(\theta)\end{aligned}\end{align} \]

Therefore

\[ \begin{align}\begin{aligned}cos(\theta) = \vec{a}_{x} / r\\sin(\theta) = \vec{a}_{y} / r\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split}\vec{r}(\vec{a}; \theta) & = r*(cos(\theta)*\vec{x'} + sin(\theta)*\vec{y'}) \\ & = r*(cos(\theta)*\begin{bmatrix} cos(\beta) \\ sin(\beta) \\ \end{bmatrix} + sin(\theta)*\begin{bmatrix} cos(\beta + \pi/2) \\ sin(\beta + \pi/2) \\ \end{bmatrix}) \\ & = r*(cos(\theta)*\begin{bmatrix} \vec{a}_{x} / r \\ \vec{a}_{y} / r \\ \end{bmatrix} + sin(\theta)*\begin{bmatrix} -\vec{a}_{y} / r \\ \vec{a}_{x} / r\\ \end{bmatrix}) \\ & = (cos(\theta)*\begin{bmatrix} \vec{a}_{x} \\ \vec{a}_{y} \\ \end{bmatrix} + sin(\theta)*\begin{bmatrix} -\vec{a}_{y} \\ \vec{a}_{x} \\ \end{bmatrix}) \\ & = (cos(\theta)*\vec{a} + sin(\theta)*\begin{bmatrix} -\vec{a}_{y} \\ \vec{a}_{x} \\ \end{bmatrix})\end{split}\\\begin{split}\vec{r}(\vec{a}; \theta) & = \begin{bmatrix} -\vec{a}_{y} \\ \vec{a}_{x} \\ \end{bmatrix} \text{ if } (\theta = \pi/2) \\ & = (cos(\theta)*\vec{a} + sin(\theta)*\vec{r}(\vec{a}; \pi/2 ) \text{ if } (\theta \ne \pi/2)\end{split}\end{aligned}\end{align} \]

src/modelviewprojection/mathutils2d.py¶

@dataclass
class Vector2D:
    x: float  #: The x-component of the 2D Vector
    y: float  #: The y-component of the 2D Vector

src/modelviewprojection/mathutils2d.py¶

def rotate_90_degrees() -> InvertibleFunction[Vector2D]:
    def f(vector: Vector2D) -> Vector2D:
        return Vector2D(-vector.y, vector.x)

    def f_inv(vector: Vector2D) -> Vector2D:
        return Vector2D(vector.y, -vector.x)

    return InvertibleFunction(f, f_inv)


def rotate(angle_in_radians: float) -> InvertibleFunction:
    r90: InvertibleFunction[Vector2D] = rotate_90_degrees()

    def f(vector: Vector2D) -> Vector2D:
        parallel: Vector2D = math.cos(angle_in_radians) * vector
        perpendicular: Vector2D = math.sin(angle_in_radians) * r90(vector)
        return parallel + perpendicular

    def f_inv(vector: Vector2D) -> Vector2D:
        parallel: Vector2D = math.cos(angle_in_radians) * vector
        perpendicular: Vector2D = math.sin(angle_in_radians) * inverse(r90)(
            vector
        )
        return parallel + perpendicular

    return InvertibleFunction(f, f_inv)

Note the definition of rotate, from the description above. cos and sin are defined in the math module.

src/modelviewprojection/demo07.py¶

@dataclass
class Paddle:
    vertices: list[Vector2D]
    color: Color3
    position: Vector2D
    rotation: float = 0.0

a rotation instance variable is defined, with a default value of 0

src/modelviewprojection/demo07.py¶

def handle_movement_of_paddles() -> None:
    global paddle1, paddle2

    if glfw.get_key(window, glfw.KEY_S) == glfw.PRESS:
        paddle1.position.y -= 1.0
    if glfw.get_key(window, glfw.KEY_W) == glfw.PRESS:
        paddle1.position.y += 1.0
    if glfw.get_key(window, glfw.KEY_K) == glfw.PRESS:
        paddle2.position.y -= 1.0
    if glfw.get_key(window, glfw.KEY_I) == glfw.PRESS:
        paddle2.position.y += 1.0

    if glfw.get_key(window, glfw.KEY_A) == glfw.PRESS:
        paddle1.rotation += 0.1
    if glfw.get_key(window, glfw.KEY_D) == glfw.PRESS:
        paddle1.rotation -= 0.1
    if glfw.get_key(window, glfw.KEY_J) == glfw.PRESS:
        paddle2.rotation += 0.1
    if glfw.get_key(window, glfw.KEY_L) == glfw.PRESS:
        paddle2.rotation -= 0.1

Cayley Graph¶

Code¶

The Event Loop¶

src/modelviewprojection/demo07.py¶

while not glfw.window_should_close(window):

So to rotate paddle 1 about its center, we should translate to its position, and then rotate around the paddle’s center.

src/modelviewprojection/demo07.py¶

    glColor3f(*astuple(paddle1.color))

    glBegin(GL_QUADS)
    for p1_v_ms in paddle1.vertices:
        # doc-region-begin compose transformations on paddle 1
        fn: InvertibleFunction[Vector2D] = compose(
            uniform_scale(1.0 / 10.0),
            rotate(paddle1.rotation),
            translate(paddle1.position),
        )
        paddle1_vector_ndc: Vector2D = fn(p1_v_ms)
        # doc-region-end compose transformations on paddle 1
        glVertex2f(paddle1_vector_ndc.x, paddle1_vector_ndc.y)
    glEnd()

\[\vec{f}_{p1}^{w}\]

\[\vec{f}_{w}^{ndc}\]

...

Likewise, to rotate paddle 2 about its center, we should translate to its position, and then rotate around the paddle’s center.

src/modelviewprojection/demo07.py¶

    glColor3f(*astuple(paddle2.color))

    glBegin(GL_QUADS)
    for p2_v_ms in paddle2.vertices:
        fn: InvertibleFunction[Vector2D] = compose(
            uniform_scale(1.0 / 10.0),
            rotate(paddle2.rotation),
            translate(paddle2.position),
        )
        paddle2_vector_ndc: Vector2D = fn(p2_v_ms)
        glVertex2f(paddle2_vector_ndc.x, paddle2_vector_ndc.y)
    glEnd()

\[\vec{f}_{p2}^{w}\]

\[\vec{f}_{w}^{ndc}\]

Why it is Wrong¶

Turns out, our program doesn’t work as predicted, even though translate, scale, and rotate are all defined correctly. The paddles are not rotating around their center.

Let’s take a look in detail about what our paddle-space to world space transformations are doing.

src/modelviewprojection/demo07.py¶

        fn: InvertibleFunction[Vector2D] = compose(
            uniform_scale(1.0 / 10.0),
            rotate(paddle1.rotation),
            translate(paddle1.position),
        )
        paddle1_vector_ndc: Vector2D = fn(p1_v_ms)