How do I calculate the center of a square given its position and orientation?

[sterlingm]

Hi,

I want to find the world coordinates of the center of a square given:

-The top left corner is the reference point
-Its position relative to the world frame
-Its orientation relative to the world frame

The example that I have been using is that the square is at (0,2), has 0 orientation, and the distance to the center from the corner is 0.1524m in the x and y directions (Picture given).

View attachment 1868

This means that the center should be -0.1524, 1.8476. My closest way of doing it was:

x = x_world + (h*cos(-PI/4))
y = y_world + (h*sin(-PI/4))

h = 0.2155m for this example

However, this gives me the location of (0.1524, 1.8476). The x coordinate should be negative. I have tried numerous equations such as:

x = (x_world+h)*(cos(orientation - PI/4))
x = (x_world+h)*(cos(orientation + PI/4))
x = x_world*cos(orientation+PI/4)
(The y formulas being the same but replacing x with y and cos with sin)

However, none of these work. If anyone could help me out, that would be great. I feel like it should be so simple, but I can't seem to figure it out. The oddly specific example is not from homework, this is something I am working on for a hobbyist 2D project.

 

[MarkFL]

The angle you want to use is \(\displaystyle -\frac{3\pi}{4}\)

The center of the given square would then have the coordinates:

\(\displaystyle x_C=0+0.1524\cos\left(-\frac{3\pi}{4} \right)\approx-0.107763\)

\(\displaystyle y_C=2+0.1524\sin\left(-\frac{3\pi}{4} \right)\approx1.892237\)

It looks like your diagram gives the upper right corner as the reference point, and where you have $h$, this is where you should have the measure $0.1524$.

 

[sterlingm]

The angle you want to use is \(\displaystyle -\frac{3\pi}{4}\)

The center of the given square would then have the coordinates:

\(\displaystyle x_C=0+0.1524\cos\left(-\frac{3\pi}{4} \right)\approx-0.107763\)

\(\displaystyle y_C=2+0.1524\sin\left(-\frac{3\pi}{4} \right)\approx1.892237\)

It looks like your diagram gives the upper right corner as the reference point, and where you have $h$, this is where you should have the measure $0.1524$.

Thanks! I think the main problem was using -PI/3 rather than -3PI/4. I probably should have added an arrow to my diagram to show where the "front" of the square was. I added it here:

View attachment 1871

However, I forgot to mention one important detail. The square can rotate. So how does the angle change from -3PI/4 if the square can rotate?

 

[MarkFL]

If the square rotates about the point (0,2) in a counter-clockwise direction, then add the angle of rotation to \(\displaystyle -\frac{3\pi}{4}\). If the rotation is in a clockwise direction then subtract the angle of rotation.

 

[CellCoree]

Hi there,

Calculating the center of a square can be done using basic geometry principles. Here's how you can do it:

1. Determine the length of one side of the square. In this case, the distance from the corner to the center is given as 0.1524m in both the x and y directions. This means that the length of one side of the square is h = 0.1524 * 2 = 0.3048m.

2. Find the midpoint of one side. Since the square is symmetric, the midpoint of one side will also be the center of the square. To find this, divide the length of one side by 2. The midpoint for this example would be 0.3048/2 = 0.1524m.

3. Use the position and orientation information to determine the coordinates of the center. Since the top left corner is the reference point, the coordinates of the center will be relative to the top left corner. This means that the x coordinate of the center will be the x coordinate of the corner plus the x coordinate of the midpoint of one side. Similarly, the y coordinate of the center will be the y coordinate of the corner plus the y coordinate of the midpoint of one side.

In this example, the top left corner is at (0,2). So the x coordinate of the center will be 0 + 0.1524 = 0.1524m. The y coordinate of the center will be 2 + 0.1524 = 2.1524m.

4. Take into account the orientation of the square. Since the square has 0 orientation, the x and y coordinates of the center will remain the same. However, if the square had a different orientation, you would need to use basic trigonometry to find the new x and y coordinates of the center.

In summary, to calculate the center of a square given its position and orientation, determine the length of one side, find the midpoint of one side, use the position and orientation information to determine the coordinates of the center, and take into account the orientation if necessary. I hope this helps!