Epitrochoid
Definition
Epitrochoids are plane curves traced by a fixed point on a circle of radius \(r\) that rolls without slipping around the outside of a fixed circle of radius \(R\). The tracing point is located at a distance \(d\) from the center of the rolling circle. The general parametric equations for an epitrochoid are:
$$ x \left(\theta\right) = \left(R + r\right)\cos \theta - d \cos \left(\frac{R + r}{r} \theta \right) $$
$$ y \left(\theta\right) = \left(R + r\right)\sin \theta - d \sin \left(\frac{R + r}{r} \theta \right) $$
where:
- \(R\) is the radius of the fixed circle.
- \(r\) is the radius of the rolling circle.
- \(d\) is the distance from the traced point to the center of the rolling circle.
- \(\theta\) is the rolling angle.
Characteristics
- If \(d = r\), the curve is called an epicycloid.
- If \(d < r\), the curve is smooth and without loops.
- If \(d > r\), the curve forms loops or cusps.
- If \( \frac{R}{r} \) is a rational number, the curve is closed and periodic.
Conversion to polar coordinates
To write the curve as \(r(\theta)\), we use:
$$r(\theta) = \sqrt{x\left(\theta\right)^{2} + y\left(\theta\right)^{2}}$$
Substituting the expressions:
$$\begin{split} r(\theta) &= \left( \left[ \left\lbrace R + r \right\rbrace \cos\theta - d\cos \left\lbrace \frac{R + r}{r} \theta \right\rbrace \right]^{2} \right. \\ &+ \left. \left[ \left\lbrace R + r \right\rbrace \sin\theta - d \sin \left\lbrace \frac{R + r}{r} \theta \right\rbrace \right]^{2}\right)^{\frac{1}{2}} \end{split}$$
After simplifying:
$$r(\theta) = \sqrt{(R + r)^{2} + d^{2} - 2d (R + r) \cos \left(\frac{R + r}{r}\theta\right)}$$
This equation represents the radial distance from the origin (center of the fixed circle) to the tracing point as a function of \(\theta\).
Animation
Move sliders \(d\) (point from a certain external circle center distance), \(R\) (internal circle radius) or \(r\) (external circle radius) to change the epitrochoid. Press the "Start" button to watch the animation, this will move the \(t\) (time) slider. Press the "Stop" button to interrupt the animation and press the "Reset" button to reboot the starting values.
Cardioid
The cardioid is a type of curve classified as an epicycloid with one cusp. It is generated by tracing a point on a circumference as it rolls around a fixed circle of the same radius.
Mathematically, the cardioid can be represented in polar coordinates as:
$$r = 2a \left( 1 + \cos \theta \right)$$
where \(a\) is a constant that determines the size of the cardioid.
In Cartesian coordinates, its equation can be expressed as:
$$\left( x^{2} + y^{2} - 2ax \right)$$
To obtain a cardioid, we set \(R = r\) and \(d = r\), meaning the rolling circle has the same radius as the fixed one, and the traced point lies on the circumference of the rolling circle. Substituting these values:
$$x\left( \theta \right) = 2r \cos \theta - r \cos \left( 2 \theta \right)$$
$$y\left( \theta \right) = 2r \sin \theta - r \sin \left( 2 \theta \right)$$
Rewriting in polar coordinates \((x = r\cos \theta, y = r\sin \theta)\), we obtain:
$$r = 2a \left( 1 + \cos \theta \right)$$
which is the standard polar equation of a cardioid.