Square cardioid

[ValeryOchkov]

Can you create this "square cardioid" in Mathcad.

With animation!

 

[Werner_E]

I wouldn't call it a square cardiod (even though it would look similar).

The path of the point is relatively unspectacular and consists of three different curves:
1) four quarter circles
2) two parts of an involute of a circle, such as those used in the construction of a particular type of gear
3) two parts of a parallel curve to an involute of a circle

You can also do it the other way around and unroll the circle on the square. The point path then consists of alternating circular and cycloid pieces. That doesn't look particularly exciting either.

Apropos Cardioid - you sure remember the bicycle with the square wheels

https://thumbs.dreamstime.com/z/driewieler-voor-jonge-geitjes-met-vierkante-wielen-110677055.jpg

https://bike-blog.info/wp-content/uploads/2020/04/fahrrad-mit-eckigen-raedern-800.jpg

But did you know that there also are vehicles with wheels in the shape of cardioids?

See https://mathematikalpha.de/zykloide where you also can download software to play around with some kind of animations 🙂

[ValeryOchkov]

Thanks, Werner!

But it will be good to see animtion!

[Werner_E]

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@ValeryOchkov wrote:

Thanks, Werner!

But it will be good to see animtion!

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Sure! Feel free to create and show your animation 🙂

EDIT: @ValeryOchkov  I'm still waiting to see your animation!

[ValeryOchkov]

I am waiting too! It is one task for my students!

[Werner_E · Accepted solution]

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@ValeryOchkov wrote:

I am waiting too! It is one task for my students!

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Lazy guy!! 🙂

But that sure shouldn't take that long, should it?

Waiting for students is easy, but you can only do a thing if you do it yourself.

Couldn't wait any longer 😉

Now its your turn! How about a triangle "rolling" on a square or a circle, or, again, the other way round.

[ValeryOchkov]

Nice! Colors on the closet curves - parts of circle, involute and cycloida!

I cannot ask to send Mathcad-sheet! Can I ask to send the origin avi-file (without grids)!

PS

Sometimes coming up with a problem is harder than solving it! 😉

[Werner_E]

As it was done in rather short time the Mathcad Sheet is quite a chaotic mess with a lot of case distinctions and probably unnecessarily blown up and inefficient. The sheet sure would need a lot of cleaning up before posting it.

The original avi would have shown the grid, too, and I didn't keep it. But I sure can create a new one without the grid if you have use for it.

EDIT: Video (MP4, H264, for massive size reduction) is attached

For test purpose its also embedded:

[ValeryOchkov]

Thank You, Werner!

But can you send the avi-file! Please!

[Werner_E]

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@ValeryOchkov wrote:

Thank You, Werner!

But can you send the avi-file! Please!

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Why? The avi created by MC is already deleted but you can always use an online converter. Find attached the file converted to avi.

MC uses a very old and inefficient codec and produces way too large video files. Converting it results in much smaller files of the same quality.

[ValeryOchkov]

Thanks!

Will be better with changed words: cycloid, ivolute, arc of cyrcle. 

And what about a triangle with cyrcle?