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cycloid    
n. 摆线,轮转线
a. 圆形的,循环性格的

摆线,轮转线圆形的,循环性格的

cycloid
adj 1: resembling a circle [synonym: {cycloid}, {cycloidal}]
n 1: a line generated by a point on a circle rolling along a
straight line

Cycloid \Cy"cloid\ (s?"kloid), n. [Cyclo- -oid: cf. F.
cyclo["i]de.] (Geom.)
A curve generated by a point in the plane of a circle when
the circle is rolled along a straight line, keeping always in
the same plane.
[1913 Webster]

Note: The common cycloid is the curve described when the
generating point (p) is on the circumference of the
generating circle; the curtate cycloid, when that point
lies without the circumference; the prolate or
inflected cycloid, when the generating point (p) lies
within that circumference.
[1913 Webster]


Cycloid \Cy"cloid\, a. (Zool.)
Of or pertaining to the Cycloidei.
[1913 Webster]

{Cycloid scale} (Zool.), a fish scale which is thin and shows
concentric lines of growth, without serrations on the
margin.
[1913 Webster]


Cycloid \Cy"cloid\, n. (Zool.)
One of the Cycloidei.
[1913 Webster]


Brachystochrone \Bra*chys"to*chrone\, n. [Incorrect for
brachistochrone, fr. Gr. bra`chistos shortest (superl. of
brachy`s short) ? time : cf. F. brachistochrone. ] (Math.)
A curve, in which a body, starting from a given point, and
descending solely by the force of gravity, will reach another
given point in a shorter time than it could by any other
path. This curve of quickest descent, as it is sometimes
called, is, in a vacuum, the same as the {cycloid}.
[1913 Webster]


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  • geometry - How to find the parametric equation of a cycloid . . .
    26 "A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line " - Wikipedia In many calculus books I have, the cycloid, in parametric form, is used in examples to find arc length of parametric equations This is the parametric equation for the cycloid: x = r(t − sint) y = r(1 − cost)
  • definite integrals - Whats the area of one arch of a cycloid . . .
    So, we get that the area below one arch of a cycloid equals three areas of a circle which forms that cycloid My question is: I don't understand anything about this problem :) How did the teacher integrate this parametric equation, why did he write the integral of y(t)x′(t) y (t) x ′ (t), why did he need a derivative of x (t) and what does it represent Can you please explain this to me
  • Finding the equation for a (inverted) cycloid given two points
    To summarize that link: Draw an arbitrary cycloid along with the line connecting the two endpoints The arc of the cycloid cut off by the line has the correct shape but wrong scale for the brachistochrone, so it just needs to be rescaled to actually connect the two endpoints
  • How can I find the formula for a cycloid with a given speed?
    I know that for a cycloid with radius R R and time t t, it can be defined as x = R(t − sin t) x = R (t − sin t) and y = R(1 − cos t) y = R (1 − cos t) However what if it's not at unit speed and we have a speed z z sec such that after t t seconds the centre is at (tz, R) (t z, R)
  • calculus - Surface area by the revolution of cycloid - Mathematics . . .
    How to find the surface area of the solid generated by the revolution of the cycloid about x x -axis? I know the formula to find out the surface area but I'm getting the point that in the formula why we take the integration limit as 0 to 2π 2 π Please, help me out!
  • ordinary differential equations - The curvature of a Cycloid at its . . .
    My lecturer proposed a question to particular result regarding the curvature of a Cycloid (generated by circle of radius 1) at its cusps Having left it as an open problem, I thought it'd be inter
  • Characterizations of cycloid - Mathematics Stack Exchange
    There are several motions that create a cycloid I have some examples here Are there any others? Trace of a fixed point on a rolling circle Evolute of another cycloid (the locus of all its centers
  • Cycloid (Maths HL IA) - Mathematics Stack Exchange
    I have chosen to investigate the fact that cycloid is a quicker path than the straight line for my HL Maths IA I did my own experiment and was advised to only explain up to 'timing the fall' of the brachistochrone problem by my teacher
  • Proof of the Cycloid Parametric Equation - Mathematics Stack Exchange
    Here we establish that the distance PT is equal to the distance OT, which then (alongside other steps) allows us to derive the parametric equation of the cycloid
  • Proof that the tautochrone is a cycloid - Mathematics Stack Exchange
    1 In the Wikipedia article about the tautochrone curve, there is a proof of the fact that the tautochrone curve must be a cycloid The proof starts with the following statement: One way the curve can be an isochrone is if the Lagrangian is that of a simple harmonic oscillator: the height of the curve must be proportional to the arclength squared





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