Dogs on leashes dragging their owners. Surely, this has nothing to do with maths! Well, if they are running along a line perpendicular to the initial direction of the leash, the owners will be dragged along a curve called tractrix.🐶 [Wiki bit.ly/2stpcUl] #50FamousCurves
Look at them doggos go. Aren't they cute? Unfortunately, I couldn't find the source of these cute drawings, hence the lack of image credit. If anyone finds the OP let me know!
Fun fact about the tractrix: The envelope of lines perpendicular to the tractrix is a hyperbolic cosine, which is also called a catenary, because it's the curve describing the shape of a hanging chain. #50FamousCurves
Fun fact about the tractrix: The area between a tractrix of parameter 'a' and its asymptote equals a²π/2. #50FamousCurves
I've found the source code for this animation here. demonstrations.wolfram.com/AreaUnderTheTr…
Fun fact about the tractrix: It is used in making horn loudspeakers. In fact, a horn with tractrix shaped contour minimizes distortion caused by internal reflection of sound within the horn. #50FamousCurves [Source: bit.ly/2RPFNjl] 
Fun fact: If you rotate a tractrix about its asymptote you'll get a surface with constant negative Gaussian curvature¹. This is the pseudosphere². It serves as a model for hyperbolic geometry. #50FamousCurves
_
¹Except for the equator.
²The sphere has constant positive curvature.
_
¹Except for the equator.
²The sphere has constant positive curvature.
How to draw a tractrix? You'll need the following:
1) Marked points 1,2,3,... on a base-line L at equal intervals.
2) Lines at right angles to L through 1,3,...
3) Quarter circles of a fixed radius with centres at points 2,4,...
4) A point C₁ on line 1.
(1/2) #50FamousCurves
1) Marked points 1,2,3,... on a base-line L at equal intervals.
2) Lines at right angles to L through 1,3,...
3) Quarter circles of a fixed radius with centres at points 2,4,...
4) A point C₁ on line 1.
(1/2) #50FamousCurves
5) Draw a tangent from C₁ to circle 2 to get point T₂.
6) Draw an arc centred at C₁ through T₂ to circle 4 to get T₄.
7) The line C₁T₂ meets line 3 at C₃.
Repeat 6&7 with C₃,C₅,... & T₄,T₆,...
Arcs joining T₂,T₄,T₆,... approximate a tractrix. (2/2) #50FamousCurves
6) Draw an arc centred at C₁ through T₂ to circle 4 to get T₄.
7) The line C₁T₂ meets line 3 at C₃.
Repeat 6&7 with C₃,C₅,... & T₄,T₆,...
Arcs joining T₂,T₄,T₆,... approximate a tractrix. (2/2) #50FamousCurves
Fun fact: The timing belts in many cars/motorbikes have small teeth with tractrix shaped profiles, because this shape minimizes the friction of the belt teeth and the pulley. #50FamousCurves
Is that a giant tractrix horn in @MarkRober's video? 🤩 #50FamousCurves
• • •
Missing some Tweet in this thread? You can try to
force a refresh
