osr on some interesting curves... aditya kiran grad 1 st yr applied math
TRANSCRIPT
![Page 1: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/1.jpg)
OSR
On some interesting Curves...
Aditya KiranGrad 1st yrApplied Math
![Page 2: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/2.jpg)
Curvature
• The fact of being curved or the degree to which something is curved..
• Tendency to deviate from moving in a straight line..
![Page 3: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/3.jpg)
How to find curvature?
y =f(x) form
Parametric form y =f(t) , x=f(t)
Polar form (r,theta)
![Page 4: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/4.jpg)
4 ‘interesting’ curves
• We discuss about 4 curves today:
• 1)Catenary• 2)Tractrix• 3)Archimedean Spiral• 4) Tautochrone
![Page 5: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/5.jpg)
#1
![Page 6: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/6.jpg)
Catenary
• Àlso called Alysoid or chain equation..• The word catenary is derived from the Latin
word ‘catena’ for chain.• Solved by Huygens (wave theory of light- guy)
![Page 7: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/7.jpg)
equation
• Y = Coshx(x/a)
• Parametric:• X(t)=t,• Y(t)=aCosh(t/a)
• Curvature:• k(t)=
![Page 8: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/8.jpg)
Special properties..
• A catenary is also the locus of the focus of a parabola rolling on a straight line.
• The catenary is the evolute of the tractrix.
![Page 9: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/9.jpg)
#2
![Page 10: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/10.jpg)
Tractrix
• From the Latin ‘trahere’ meaning drag/pull..
• Also called drag curve or donkey curve.
• The tractrix was first studied by Huygens in 1692
![Page 11: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/11.jpg)
So what’s the problem about?
“What is the path of an object when it is dragged along by a string of constant length being pulled along a straight horizontal line”
![Page 12: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/12.jpg)
The Equation :
• Y=a*Sech(t)• X=a*(t-Tanh(t))
Curvature: k(t)=Cosech(t)/a
![Page 13: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/13.jpg)
Special properties..• Tractrix is also obtained when a hyperbolic
spiral is rolled on a straight line..Then locus of the center of the hyperbola forms a tractrix.
• Evolute of a tractrix is a catenary
![Page 14: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/14.jpg)
• Example in daily life: ..If Car front wheels travel along straight line,
the back wheels follow a tractrix.
![Page 15: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/15.jpg)
#3
![Page 16: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/16.jpg)
Archimedean spiral
• It is the locus of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity.
Rotate around a point with constant angular velocity
Move away from the point with constant
speed
Archimedean spiral=
![Page 17: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/17.jpg)
• Equation in polar coordinates:
• a turns the spiral
• B controls the distance between successive turnings.
![Page 18: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/18.jpg)
equation
• r= bϴ• Parametric: x= b*ϴ*Cos(ϴ) y= b*ϴ*Sin(ϴ)
• Curvature:
• Any ray from origin meets successive turnings at a constant separation.
![Page 19: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/19.jpg)
• Used to convert circular motion to linear motion..
• Used in Archimedes screw
![Page 20: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/20.jpg)
General formula of spiral
• X=1, Archimedean spiral• X=2 ,Fermat spiral• X=-1 ,Hyperbolic spiral
• Logarithmic spiral
![Page 21: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/21.jpg)
Logarithmic spiral
![Page 22: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/22.jpg)
#4
![Page 23: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/23.jpg)
Tautochrone
• A tautochrone or isochrone curve • Tauto=same, and chrono =time
• Solved by Christian Huygens in 1659.
• “the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point”
![Page 24: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/24.jpg)
• Its nothing but a cycloid.
• (
• Curvature: k=Cosec(ϴ/2)/4a
![Page 25: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/25.jpg)
• T=
![Page 26: OSR On some interesting Curves... Aditya Kiran Grad 1 st yr Applied Math](https://reader035.vdocuments.site/reader035/viewer/2022070406/56649e0d5503460f94af6bf9/html5/thumbnails/26.jpg)
Thank you.!!!