How to find a center of rotation?

hello there, I’m trying to find centers of rotation for movable pieces, I manually scketchs some arcs to get an aproximation.
Is there a way where one can pick start - end points and then a angle range to obtain the center of the arc?
In the image below, I know the initial and final position of the piece which rotates 270 degrees from the center I want to locate.

I’ve been testing to imput the degree value on each arc and circle command but no results yet.

I tried other approach with perpendicular lines from midpoint between 2 pairs of ends
and it works but is a little tricky

Using the arc could be more simple or fast I believe.

Update 2
Using circle from diameter in those two pairs of ends, gave me the same result in the intersection. pretty fast, pretty easy.

To easily find the center of a true arc or circle use the Cen OSnap. Start a command which needs a point location, place the cursor near the arc or circle and the cursor, and the center should be the selected location.

Here’s the proper way you hopefully learnt in 6th grade ; )

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I only have the pieces in each position, the arcs on the first image were a hand made aproximation.

yep. that was the first method I tested, then using the circles works better

Hi Diego,

I couldn’t work out how the circles thing worked so I made another example - and it didn’t. See attached.
CirclesDontWorkHere.3dm (32.4 KB)
The point is the true centre of rotation and the circles don’t intersect on it as they do in your example. Any thoughts?


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Hi Jeremy, thanks for pointing that out, it seems that the circle method only works in some cases, my example was for 90 degrees, in your example. the first method of mid points normal lines works perfectly. maybe I have to search for a macro or script which automatizes it. I will keep trying this thing.
It seems to only work with 90 degrees… I was happy with this damn it! haha

you can also use area centroid, or mass centroid from the analysis menu. My biz partner uses this to find centers of crazy fidget spinner sculptures he makes.

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Area centroid does not provide the center of rotation of a set of entities.

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I agree but is an interesting point of view. now I’m testing with the internal angles between the reference points if they have something to work wth.

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As @Lagom showed there is only one efficient way to find the rotation point you are looking for, for any given set of rotated objects.
A shortcut script would just be something like the orient command with scale-1d of a horizontal line with long perp bisector. Orient with copy to a couple of your reference points and done. Then you can make a circle or arc from that rotation point.

You can also make circles of the same radius on corresponding points of the two objects - the center lies on the line defined by the points of intersection of the circles. If you do this twice - for two pairs of points on the object - the center will be at the intersection of the two lines. Different construction, same idea…

Also, to take it one step farther - if you do not know the plane of rotation, use two pairs of spheres instead of circles, and the axis will be the intersection line of the two planes defined by the intersecting spheres.


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Mathematically, one can prove in 2D and also 3D that there is one solution, and that is shown in the video linked to in the post above. Why try to reinvent geometry when it’s not even possible ; )

You never know when someone might just want to get it done in Rhino.


Alors, monsieur Pascal,

the correct method shown in the video is easy and quick to do in Rhino, with only 4 straight lines. How could it be easier? The goal is always: How can we finish work most efficiently so as to enter the pub at 5pm sharp.

Ok, when we want to find the center of rotation of two tesseracts in Minkowski space… donc, it’s different matter : )

have you always all the answers to everything? I don’t think so and If I want to spend time and efforts searching for other options, is my problem.
Also, as you luckily learnt to read at 1st grade, the "solution"you posted. was the first attempt I posted in the initial thread, before your answer.

Geometric proof is underestimated. Is it math or geometry that is the ultimate reference for truth? (regarding anything geometric).

Does math prove geometry, as if math exist somehow and that geometry is, at best, but not always, a way of representating math, or is it the other way around?

Clifford Algebra (or more modern term, Geometric algebra) indicates that geometry (and geometric methods) is truth and that math is struggling to catch up and at best makes for some useful attempts at representing it, although often being über complex compared to geometric approaches.

An intersting articles by slehar on the subject:

Quoting Clifford:

“geometry is the gate of science, and the gate is so low and small that one can only enter it as a child.”

Geometry rocks!

// Rolf


It actually is. Mathematical proofs are above the four fundamental forces of nature, because they exist beyond their possible reach. Geometry, as a child of Mathematics, should better obey and empty the dishwasher on time : D

Did I say that somewhere? Certainly not.

Next topic.

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And the proof? Or is it a axiom? :slight_smile:

BTW, did you read the article?

I think geometry (in the form of space-time) came first.

// Rolf

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