How do you find the equation of a hypocycloid?
A hypocycloid is a Hypotrochoid with . To derive the equations of the hypocycloid, call the Angle by which a point on the small Circle rotates about its center , and the Angle from the center of the large Circle to that of the small Circle . Then Call .
When is a hypocycloid an epicycloid?
Theorem 1: The hypocycloid [a, b, t] is an epicycloid for b < 0 and b > a. Figure 1a . a = 3,b = 1, b/a = 1/3 Figure 1b. a = 2, b = 1 .2, b/a = 3/5
How to find the COS( )Cos of hypocycloid and epicycloid?
We denote the hypocycloid by [a, b, t] and the epicycloid by [a, c, w] wheret andware parameters for each curve, respectively. Since b > a > 0, let c = b – a so that b = c + a. We replacecin the parametric equations of [a, b, t] to get cos ( )cos c x c t ac t ac − =− ++ + and sin ( )sin .
What is the proof of the hypocycloid theorem?
Proof of Theorems Theorem 1: For any non-zero rational numbers a> 0, b, andc, the hypocycloid [a, b, t] is an epicycloid if b < 0or b > a. Proof: We will first show that ( )cos cos
Which path is a hypocycloid?
The red path is a hypocycloid traced as the smaller black circle rolls around inside the larger black circle (parameters are R=4.0, r=1.0, and so k=4, giving an astroid ). 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.
What is a hypocycloid in printing?
Girolamo Cardano was the first to describe these hypocycloids and their applications to high-speed printing. If k is a rational number, say k = p / q expressed in simplest terms, then the curve has p cusps. If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R − 2 r .
How do you know if a hypocycloid is epicycloid?
A hypocycloid (or epicycloid) with n cusps can move inside a hypocycloid (or epicycloid) with n + 1 cusps in such a way that the cusps of one of the curves always touches the other curve. Red hypocycloids and blue epicycloids rolling on the inside.