Rotations - Demo 07¶

Purpose¶

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/demo07/demo.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 a given vertex, wherever it is, around the origin (0,0). This is done by separating the x and y value, rotating each component separately, and then adding the results together.

That might not have been fully clear. Let me try again. The vertex (0.5,0.4) can be separated into two vertices, (0.5,0) and (0,0.4).

These vertices can be added together to create the original vertex. But, before we do that, let’s rotate each of the vertices.

(0.5,0) is on the x-axis, so rotating it by “angle” degrees, results in vertex (0.5*cos(angle), 0.5*sin(angle)). This is high school geometry, and won’t be explained here in detail.

But what may not be obvious, is what happens to the y component? Turns out, it is easy. Just rotate your point of view, and it is the same thing, with one minor difference, in that the x value is negated.

(0,0.4) is on the y-axis, so rotating it by “angle” degrees, results in vertex (0.4*-sin(angle), 0.4*cos(angle)).

Wait. Why is negative sin applied to the angle to make the x value, and cos applied to angle to make the y value? It is because positive 1 unit on the x axis goes downwards. The top of the plot is in units of -1.

After the rotations have been applied, sum the results to get your vertex rotated around the origin!

(0.5*cos(angle), 0.5*sin(angle)) + (0.4*-sin(angle), 0.4*cos(angle)) = (0.5*cos(angle) + 0.4*-sin(angle), 0.5*sin(angle) + 0.4*cos(angle))

src/demo07/demo.py¶

@dataclass
class Vertex:
    x: float
    y: float

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

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

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

    def __neg__(self):
        return -1.0 * self

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

    # fmt: off
    def rotate(self: Vertex, angle_in_radians: float) -> Vertex:
        return math.cos(angle_in_radians) * self + math.sin(angle_in_radians) * self.rotate_90_degrees()
    # fmt: on

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

src/demo07/demo.py¶

@dataclass
class Paddle:
    vertices: list[Vertex]
    r: float
    g: float
    b: float
    position: Vertex
    rotation: float = 0.0

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

src/demo07/demo.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/demo07/demo.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/demo07/demo.py¶

    glColor3f(paddle1.r, paddle1.g, paddle1.b)

    glBegin(GL_QUADS)
    for paddle1_vertex_ms in paddle1.vertices:

$\vec{f}_{p1}^{w}$

src/demo07/demo.py¶

        paddle1_vertex_ws: Vertex = paddle1_vertex_ms.translate(
            paddle1.position
        ).rotate(paddle1.rotation)

$\vec{f}_{w}^{ndc}$

src/demo07/demo.py¶

        paddle1_vertex_ndc: Vertex = paddle1_vertex_ws.uniform_scale(1.0 / 10.0)

src/demo07/demo.py¶

        glVertex2f(paddle1_vertex_ndc.x, paddle1_vertex_ndc.y)
    glEnd()

...

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

src/demo07/demo.py¶

    glBegin(GL_QUADS)
    for paddle2_vertex_ms in paddle2.vertices:

$\vec{f}_{p2}^{w}$

src/demo07/demo.py¶

        paddle2_vertex_ws: Vertex = paddle2_vertex_ms.translate(
            paddle2.position
        ).rotate(paddle2.rotation)

$\vec{f}_{w}^{ndc}$

src/demo07/demo.py¶

        paddle2_vertex_ndc: Vertex = paddle2_vertex_ws.uniform_scale(1.0 / 10.0)

src/demo07/demo.py¶

        glVertex2f(paddle2_vertex_ndc.x, paddle2_vertex_ndc.y)
    glEnd()

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/demo07/demo.py¶

        paddle1_vertex_ws: Vertex = paddle1_vertex_ms.translate(
            paddle1.position
        ).rotate(paddle1.rotation)

Modelspace vertices

Translate

Reset the coordinate system

Modelspace¶

Rotate around World Spaces’s origin

Modelspace¶

Reset the coordinate system

Modelspace¶

Final world space coordinates

Modelspace¶

So then what the heck are we supposed to do in order to rotate around an object’s center? The next section provides a solution.

Updated explanation¶

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.

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

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.

$cos(\theta) = \vec{a}_{x} / r sine(\theta) = \vec{a}_{y} / r$

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

$cos(\theta + \pi/2) = -sin(\theta) sin(\theta + \pi/2) = cos(\theta)$

Therefore

$cos(\theta) = \vec{a}_{x} / r sin(\theta) = \vec{a}_{y} / r$

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