OSR On some interesting Curves... Aditya Kiran Grad

OSR On some interesting Curves... Aditya Kiran Grad

OSR On some interesting Curves... Aditya Kiran Grad 1st yr

Applied Math Curvature The fact of being curved or the degree to which something is curved.. Tendency to deviate from moving in a straight

line.. How to find curvature? y =f(x) form Parametric form y =f(t) , x=f(t)

Polar form (r,theta) 4 interesting curves We discuss about 4 curves today:

1)Catenary 2)Tractrix

3)Archimedean Spiral 4) Tautochrone #1 Catenary

lso 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) equation

Y = Coshx(x/a) Parametric: X(t)=t, Y(t)=aCosh(t/a) Curvature: k(t)=

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.

#2 Tractrix From the Latin trahere meaning drag/pull.. Also called drag curve or donkey curve.

The tractrix was first studied by Huygens in 1692 So whats 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 The Equation : Y=a*Sech(t) X=a*(t-Tanh(t))

Curvature: k(t)=Cosech(t)/a 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 Example in daily life: ..If Car front wheels travel along straight line,

the back wheels follow a tractrix. #3 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 Equation in polar coordinates:

a turns the spiral B controls the distance between successive turnings. equation

r= b Parametric: x= b**Cos() y= b**Sin() Curvature:

Any ray from origin meets successive turnings at a constant separation. Used to convert circular motion to linear motion.. Used in Archimedes screw

General formula of spiral X=1, Archimedean spiral X=2 ,Fermat spiral X=-1 ,Hyperbolic spiral

Logarithmic spiral Logarithmic spiral #4

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 Its nothing but a cycloid. ( Curvature: k=Cosec(/2)/4a

T= Thank you.!!!

Recently Viewed Presentations

  • Minerals Study Guide - part 1

    Minerals Study Guide - part 1

    A mineral is a naturally formed, inorganic solid that has a definite crystalline structure. Solid Cannot be a liquid or a gas Naturally Occurring Found in nature, not man-made Inorganic Is not alive and never was, non-living Fixed composition Has...
  • Digital Tools - DMPS Instructional Technology

    Digital Tools - DMPS Instructional Technology

    Create your own or use the many ones that are in the public file. Best of all you can adjust already existing KAHOOTs to specifically meet you classroom needs and then export the data to assist you in grouping of...
  • Heizer/Render 11e

    Heizer/Render 11e

    Heizer and Render . Operations Management, Eleventh Edition ... Continuous improvement, Benchmarking, Just-in-time, Tools of TQM. Yields: How to do what is important and to be. accomplished. Employee Fulfillment. ... An operations manager's objective is to build a total quality...
  • The History of Artificial Intelligence

    The History of Artificial Intelligence

    For a long time after the feudal system had passed away, robota continued to be used to describe work that one wasn't exactly doing voluntarily or for fun, while today's younger Czechs and Slovaks tend to use robota to refer...
  • Figure 10.9B_3

    Figure 10.9B_3

    (2) The big res dpi gat eth ere dra g. (3) The big rep iga tet her edr ag. 2. The authors have noted elsewhere that "A random mutation is like a random shot in the dark. It is not...
  • Title of Presentation (40 pt, Calibri, Bold)

    Title of Presentation (40 pt, Calibri, Bold)

    Lesson phases are defined as distinct shifts in activity structure in a given lesson (e.g., whole class lectures, whole class answer checking, individual seatwork ; Luke, Freebody, Cazden, & Lin, 2004)
  • AllStar Link Presentation - wshiggins.net

    AllStar Link Presentation - wshiggins.net

    AllStar Link Presentation Sullivan Amateur Radio Club September 8th, 2016 Bill Higgins N0NOE The EchoLink Node EchoLink on 146.805 PC running Windows 7 and EchoLink software Motorola Maxtrac radio Kludged up interface with RIGblaster Nomic Located at my home QTH...
  • PDHymns


    Words by John S. B. Monsell / Music by Sir Arthur S. Sullivan Father, Blessing Every Seedtime Here we bless Thy hand that gave us Thought and feeling, life and limb; Bless Thy Son, Who died to save us, In...