What shape is a hypocycloid?

What shape is a hypocycloid?

In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created by rolling a circle on a line.

What is Epicycloid and hypocycloid?

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.

What are hypocycloids used for?

Hypocycloids are relatives of the curves you make when you play with a Spirograph. On a Spirograph, you roll a circle around inside of a larger circle, with your pen on a point inside the circle, tracing out a curve called a hypotrochoid.

How do you find the area of a hypocycloid?

Area Enclosed by the Hypocycloid x = a cos3 t, y = a sin3 t, 0 ≤ t ≤ 2π. (a sin3 t)(3a cos2 t · − sin t) dt = 3 32 πa2. Multiplying the result by 4 for the full area gives Area of a Hypocycloid = 3 8 πa2.

What is a 3-cusped hypocycloid planetary gear system?

A hypocycloid mechanism with simple planetary gear train comprises a fixed ring gear, a, a planetary gear, b, and a planetary carrier,c. In a 3-cusped hypocycloid planetary gear system, Figure 5, the number of teeth in ring gear a is Ta, which is thrice that of gear b, i.e., Ta =3Tb.

What is a hypocycloid curve with 3 cusps called?

A hypocycloid with three cusps is known as a deltoid. A hypocycloid curve with four cusps is known as an astroid. The hypocycloid with two cusps is a degenerate but still very interesting case, known as the Tusi couple.

What is a N-cusped hypocycloid?

When the carrier makes a full rotation, the arbitrary points P on the pitch circle of the planetary gears Bor B′ describes a n-cusped hypocycloid curve. The results of the designed 3-cusped and 4-cusped hypocycloid mechanisms are shown as Figure 14 and 15, respectively.

How do the cusps of Each hypocycloid maintain continuous contact with each other?

The cusps of each of the smaller curves maintain continuous contact with the next-larger hypocycloid. Any hypocycloid with an integral value of k, and thus k cusps, can move snugly inside another hypocycloid with k +1 cusps, such that the points of the smaller hypocycloid will always be in contact with the larger.