What is the equation of epicycloid?
The parametric equations are: x=(r+R)cosθ−rcos[(r+R)θr], y=(r+R)sinθ−rsin[(r+R)θr], where r is the radius of the rolling and R that of the fixed circle, and θ is the angle between the radius vector of the point of contact of the circles (see Fig.
What is the use of epicycloid?
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
What is the difference between hypocycloid and epicycloid?
An epicycloid is a plane curve created by tracing a chosen point on the edge of a circle of radius r rolling on the outside of a circle of radius R. A hypocycloid is obtained similarly except that the circle of radius r rolls on the inside of the circle of radius R.
How do you find the area of an epicycloid?
If a = ( m − 1 ) b a = (m – 1)b a=(m−1)b where m is an integer, then the length of the epicycloid is 8 m b 8mb 8mb and its area is \pib 2 ( m 2 + m ) \pib^{2}(m^{2} + m) \pib2(m2+m). The pedal curve, when the pedal point is the centre, is a rhodonea curve.
What is cycloid define Epicycloids and hypocycloid?
Hypocycloid: variant of a cycloid in which a circle rolls on the inside of another circle instead of a line. Epicycloid: variant of a cycloid in which a circle rolls on the outside of another circle instead of a line.
What is epicycloid in engineering drawing?
An epicycloid is defined as the locus of a point on the circumference of a circle which rolls without slip around the outside of another circle.
What is an epicycloid?
The path traced out by a point on the edge of a circle of radius rolling on the outside of a circle of radius . An epicycloid is therefore an epitrochoid with . Epicycloids are given by the parametric equations
What is the form of the epicycloid with modulus $m=1 $?
Depending on the value of the modulus $m=R/r$, the resulting epicycloid has different forms. For $m=1$ it is a cardioid, and if $m$ is an integer, the curve consists of $m$ distinct branches. The cusps $A_1,\\ldots,A_m$ have the polar coordinates $ho=R$, $\\phi=\\pi2k/m$, $k=0,\\ldots,m-1$.
What is the difference between a pedal curve and an epicycloid?
The pedal curve, when the pedal point is the centre, is a rhodonea curve. The evolute of an epicycloid is a similar epicycloid – look at the evolute of the epicycloid above to see it is a similar epicycloid but smaller in size.
What is the difference between an epicycloid and an irrational number?
If k is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius R + 2 r. The epicycloid is a special kind of epitrochoid .