Rotation + Translation = Rotation. Animated proof | #SoME3

Every orientation-preserving rigid motion is either a translation or a rotation. This video shows an animated, geometric proof of this fact.

Submitted to Summer of Math Exposition #SoME3

Illustrations by Aleks Pankratova.

[Qwerasd]

And of course a translation is just a rotation about a point at infinity.

[CalculusPhysics]

i LOVE the demon analogy in place of "proof by contradiction"!! It makes it way more fun and intuitive than the way it's formally phrased

[RyanLynch]

great storytelling and animation. i like how you used the demon to prove some version of being unable to create a counter example

[borstenpinsel]

I think it was a numberphile video where they said "translation is really just rotation with an infinitely distant center of rotation"

[hvok]

This is wonderful, I would LOVE for this to win the SOME3 competition this year, the visual style and explanatory power of your video was on point! It really felt like I was engaged in proof making throughout the whole video and took something I had some intuition about and really went deep. Great work

[mathemelo]

Absolutely great! The illustrations in particular give this video so much personality! If you keep posting content with the same vibe, I can really see this unique artstyle and approach become your core identity, instantly recognizable from the rest.

[isingelan]

Thats some nice and refreshing animations you got there

[janopa]

Just wanted to share in case it's not obvious at 9:09:
If B' *does* lie on the line A' C, then during the transformation it must have moved closer to C (inside the circle), further away from C (outside the circle), or fall on the same point as A'. In the first two cases, the distance between C and B isn't preserved, and in the third case the distance between A and B isn't preserved (they smoosh together).

[bixenteartola]

Cool stuff! Nice little proof and the animation style is on point

[hauntedsunsets]

even though this was intuitively true to me at first glance (just like, the spatial reasoning that since you can choose any point of rotation I can visualize how it'd work out) I quickly realized I'd have no idea how to go about proving it short of "it makes sense to me" so it was really cool seeing the actual math that goes into proving this kind of thing! goes to show how understanding something on one level doesn't necessarily mean you understand all the optics of it

[Q_]

rotation = reflection + reflection
reflection = -1 scale
rotation + rotation = rotation + translation
but also reflection + reflection could be translation

[Rudxain]

This is exactly the reason why rotation matrices are so important

[denistusca]

Congrats on HM! This video was really fun, you deserved it :)

[DisisSid]

I used to edit videos sometime back, I noticed this phenomenon too while rotation and positioning images. Now i understand why its like that. Thanks!

[kasugaryuichi]

Oh interesting, keep it up! This was fun!

[alienmoonstalker]

In kinematics, we call this the pole point or "finding the pole of rotation"

[ItsSkylah]

This video isn't actually useless at what it first seems, it unironically helped me as I was animating something in after effects.

I was debugging my animation, as the center point was offset, and I was trying to fix it. I had both rotation and position animated, and it was causing issues with the other animations in the scene (reason why is very complicated, expressions and stuff). Suddenly I remembered this video, and I realized I could just simplify the movement to a single rotate keyframe.

[ayte]

I may be biased because I like all stuff related to linear algebra, but I really liked this video!!

[rklehm]

Very cool video! The side quests with the deamon was awesome!!!

[romancontreras]

Your proof falls apart if A and A' are antipodes, I think this case needs special considerations. Anyways very cool video!