What is hypocycloid and epicycloid?
Epicycloid and Hypocycloid. Main Concept. 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.
Is epicycloid and hypocycloid same?
Epicycloid and hypocycloid both describe a family of curves. Epicycloid is a special case of epitrochoid, and hypocycloid is a special case of hypotrochoid. Specifically, epi/hypocycloid is the trace of a point on a circle rolling upon another circle without slipping.
How the angle θ is obtained of epicycloid and hypocycloid?
5. How the angle ᴓ θ is obtained of epicycloid and hypocycloid? Explanation: When r and R are the radii of the rolling and the generating circle, respectively, the hypocycloid is a plane curve generated by a point on the circumference of a circle, when it rolls without slipping on another circle and inside it.
What is epicycloid in math?
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 equation of hypocycloid?
y=(a−b)sinθ−bsin(ϕ−θ) The arc of C1 between P and B is the same as the arc of C2 between A and B. Thus by Arc Length of Sector: aθ=bϕ
How do you find the angle of an epicycloid?
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. a, Fig. b) and the x-axis.
What is the equation of epicycloid?
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.
How do you calculate an epicycloid angle?
How do you find the parametric equation of a hypocycloid?
Thus, the parametric equations of hypocycloid become x = 2bcos t and y = 0. This means point P traces along the x-axis with the interval [–a, a]. If a = b, then a – b = 0. Thus, the parametric equations of the hypocycloid be- come x = a and y = 0, so the graph of g is a point P(a, 0).
What is the polar angle from the center of an epicycloid?
The polar angle from the center is To get cusps in the epicycloid, , because then rotations of bring the point on the edge back to its starting position. An epicycloid with one cusp is called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid .
How does an epicycloid curve work?
Such a curve is called an epicycloid. In this case, the moving circle now gains one revolution each time around the fixed circle and so turns at a rate of $((a/b)+1)t=(a+b)t/b$. I think this is pretty much the same story, however, likewise, I don’t understand the part where it gains a revolution, and how that rate of turning represents the angle.
What are the conditions for an epicycloid to be periodic?
Epicycloid is periodic if and only if $R/r$ is rational 4 Center of wheel travels the length of circumference in one revolution Hot Network Questions What was the longest time it took to assemble a coalitional government?
How do you find the parametric equation for a hypocycloid?
If the initial configuration is such that $P$ is at $(a,0)$, find parametric equations for the curve traced by $P$, using angle $t$ from the positive $x$-axis to the center $B$ of the moving circle. The resulting curve is called a hypocycloid.
