How to turn a circle inside out (Visual Calculus and the Tractrix)

let's say you go on a walk with your really stubborn dog your dog is so stubborn that he's always one leashes distance away from you the leash is always tight you start facing 90 degrees from the leaf and then you walk in a straight line what shape will your dog's path be now when I ask what shape I don't just mean Trace out with your finger what you think it's going to look like although that is a good thing to do you could probably start with that what we want to do is we want to come up with a way to describe this shape mathematically because if we can do that then maybe we'll find somewhere else that this shape shows up and that's the fun part right that's why we do math so in the last 30 seconds I think you've come up with an idea of what it vaguely looks like in your head so I'm just going to give you the answer right now this is the path that your dog will walk in [Music] you'll notice that I didn't even use a dog I just dragged this Yoda across the lawn and that's the first thing to realize about this problem the shape doesn't depend on the strength of the force or what kind of force or any of the physics at all you can make this shape by pulling Yoda across the grass by walking your dog by pulling a wakeboarder on a boat even in the vacuum of space this would work if you just took a mass in a string and pulled it this is the shape that you would get the only thing it depends on is the length of the leash if it doesn't depend on the physics then what rules does our curve follow we know that the leash is always tight so at any point in time the dog is going to be the same distance away from the owner that's rule number one we also know that the dog is always facing the owner because the owner is pulling the dog forward since the dog is always facing the owner any line tangent to our curve is going to pass through the owner's position let's draw our shape following these two rules and see if we get the right curve instead of worrying about the continuous case where the owner is walking at a constant speed let's start by thinking about the discrete case where the owner takes one step at a time after one step the dog is breaking the second rule it's no longer facing the odor so we turn it to face the owner's new position and then we draw a line between the two because this is the hypotenuse of a triangle we're now breaking the first rule the leak is too long so starting at the owner we follow this line up the distance of the leaf and this point right here is going to be the dog's new position after the second step we repeat the same process draw a line between the dog's old position and the owner's new follow up the length of the leash and that's the dog's new position if you keep on doing this then all of your points are going to draw out the dog's path if you want to do this problem in the continuous case then it turns out it's easier to make the owner walk up the y-axis according to the second rule the dog is always facing the owner so if we draw a line tangent to the curve then it has to pass through the owner's position and we know that by the first rule that this length is a this length down here is X and then we can use the Pythagorean theorem to find this height now we can set the slope dydx equal to rise over run and that gives us this differential equation once we solve it we have a formula for the shape that our dog walks in we call this shape the track tricks the tractix has a few interesting properties but one that I want to look at in particular is the area now that we have a formula for it I guess we could find the area by taking an integral but it's not going to be pretty so instead we're going to use a trick called visual calculus astrophysicist mommycon manatsukanya developed this idea of visual calculus when he was trying to find the area between two concentric circles the area of a ring say that you're given only the length of a chord a line that goes between two points on the Outer Circle and just grazes the Inner Circle it turns out this length is all we need you could do this problem with some algebra or you could notice that all of the lines tangent to the Inner Circle sweep through a total of 360 degrees and at the same time they're always staying the same length so you could cut along all these tangent lines to make a bunch of triangles and then you can reform those triangles into a circle our ring has the same area as a circle where the radius of that circle is the same as half of this chord length and we can use this trick on all kinds of shapes all we need to do is we need to find a curve that tangents are coming off of slice along all those tangents to make a bunch of triangles and then move the corners of those triangles into one point and if we did everything right it should form a circle let's try it out on the track tricks the tangent lines are coming from the curved part so if we take these and then we slice along basically all the spots where the leak has been then we'll be cutting our track tricks into all these triangles now let's take the triangles and move them so that the corners are all in the same spot and if we do that then we're forming them into a circle in fact it's a quarter Circle which makes sense because the dog turns through a total of 90 degrees during its strip it starts facing down on the y-axis and after the owner has walked to Infinity it's facing to the right so now without doing any calculus or even any algebra we know that the area under the tractrix is one quarter pi r squared where R is the length of the leaf if you think about it the tractrix is kind of like an opposite Circle it's like a circle that you unrolled or you turned inside out if you take a quarter Circle and cut it up into all these triangles and then you just let it fall apart then attract tricks is what you get this idea of the tractrix being an inverse Circle shows up again when we look at the 3D version of attractrix a pseudosphere is a shape made by rotating attract tricks around the line that the owner walks on you might recognize this shape because in my last video I used it as a mold for my two-headed crane the two-headed crane is made out of a hyperbolic piece of paper and in order to make that I needed a surface of constant curvature which a pseudosphere is a sphere has constant positive curvature and the pseudosphere has constant negative curvature so just like the tractix and the circle they're kind of opposites just like the tractrix has the same area of a circle the pseudosphere has the same volume of a sphere there's another interesting place that the track tricks shows up remember one more time the rules of walking a dog the leash is always the same length and the dog is always facing the owner another system that follows these two rules is a bicycle because of a bike's design the front and the back wheels are always the same distance apart and the back wheel is always facing towards the front wheel so if our front wheel moves in the same path as a dog owner then the back wheel is going to draw a track tricks of course that's not the only shape that your bike wheels will move in if you ride in a circle then your back tire is going to draw a smaller Circle than the front tire and this should look familiar to you we used visual calculus to find the area between two concentric circles because every bicycle follows these two rules mamacom's visual calculus gives us the area between the front and the back tires no matter what path you write in the only thing that the area depends on is the angle that your back tire turns through so if your back tire turns a full 360 degrees and ends up facing the same direction that it started then the area between the path of the front tire and the path of the back tire is going to be the same as a circle where the radius of the circle is the length of your bike or the distance between the front and the back tires if your back tire turns through 180 degrees then the area is going to be one half pi r squared I just can't get over how cool that is I mean think about all the pads that you could make on a bike even the craziest shape that you could ride in one that doesn't look anything like a circle is still going to have the same area I mean what if you entered a race where you write in laps even if the race goes around an entire city with a complicated path still when you reach the Finish Line the total area between the two tracks is going to be pi r squared when you write on segments of the path that are straight you're going to be contributing nearly zero and when you make a concave turn it's going to subtract a little bit from the area still the total area between the path of the front tire and the path of the back tire is pi r squared when I started thinking about the pseudosphere I just couldn't put it together in my head how this shape that you get when you walk your dog is the same shape that gives you a surface of constant negative curvature when you rotate it it just seems like there should be a deeper reason for that and I still haven't given you a proof of that it would probably be boring if we went through and calculated the curvature but I think that looking at the track tricks through the lens of visual calculus really makes it easy to see how the track tricks is undoing the circle and we learned a pretty neat trick in the process mamacom's visual calculus can come in handy if you need to find the area or calculate an integral with some shape involving tangents so next time you're riding your bike or walking your dog you can tell your friend all about the track tricks

Channel: Physics for the Birds

Views: 88,477

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Id: nQ2PeqGkQfk

Length: 8min 51sec (531 seconds)

Published: Mon Oct 24 2022