Geometry – Calculate Position of Rectangle Lower Left Corner on Center Rotation

geometry

Since I forgot all the basics of math, I'm asking you to help me out with simple task.

enter image description here

I need to rotate text box in PDF. Rotation point is lower left corner, but I need to rotate it as if the rotation point would be on the center.

Since I am using a programming language, I can convert an angle in degrees to its radian equivalent (although I'm not sure if this will help)

Can you help me out to find lower left corner coordinates if rotation point is center?

Best Answer

First of all, we need to convert the coordinates so the center is (0, 0). Doing this, the coordinates of the right lower corner are (131.5, 49.5). Rotation formula is

$$(x_1, y_1) = (x, y) * \left(\begin{matrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{matrix}\right) = (x \cos(\theta)-y \sin(\theta), x \sin(\theta) + y \cos(\theta))$$

By putting $\theta=\frac{5\pi}{180}$, you get the new coordinates

$$ x_1 \approx 126.69, y_1 \approx 60.77$$

To know the distance we need to move the initial rectangle, need to subtract the initial and resulting coordinates

$$x_d = x_0-x_1 \approx 4.81$$ $$y_d = y_1-y_0 \approx 11.27$$

You need to add $x_d$ to the initial $x$ coordinates and $y_d$ to initial $y$ coordinates.

Related Solutions

Geometry – Find a Point Position on a Rotated Rectangle

The idea is to first translate your coordinates to a new coordinate system such that the origin is at $(0,0)$ (subtract $(0,50)$ from all your original coordinates), perform the rotation, and then translate back.

So, in this new coordinate system, the bottom of your rectangle has endpoints at $(0,0)$ and $(100,0)$. You can then use a(n anticlockwise) rotation matrix to transform the point $(100,0)$ to its rotated image:

$$\begin{align*}100\cos45^\circ-0\sin45^\circ&=50\sqrt{2}\\100\sin45^\circ+0\cos45^\circ&=50\sqrt{2}\end{align*}$$

and then undo the translation you made before rotating, which yields the point $(50\sqrt{2},50+50\sqrt{2})$ as the coordinates for the upper left corner in its new position.

[Math] How to rotate a matrix transformation with a centered origin

Let xc, yc be the coordinates of the center of the rectangle.

Translate your points such that the center is the new origin:
```
xt = x1 - xc;
yt = y1 - yc;
```

Rotate around the origin by the angle a:

c = cos(a); // compute trig. functions only once
s = sin(a);
xr = xt * c - yt * s;
yr = xt * s + yt * c;

Translate points back:
```
x2 = xr + xc;
y2 = yr + yc;
```

If you do this for all 4 corners of the rectangle and then draw lines between the transformed corners, you get the rotated rectangle. The rotation angle $a$ is measured from the x-axis ($0$) to to the y-axis ($\pi/2$). In 2D graphic libraries the x-axis goes to the right and the y-axis goes down, such that the angle $+35^\circ$ corresponds to your figure.

Best Answer

Related Solutions

Geometry – Find a Point Position on a Rotated Rectangle

[Math] How to rotate a matrix transformation with a centered origin

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