Hypotrochoid
Definition
Hypotrochoids are plane curves traced by a fixed point attached to a circle of radius \(r\) that rolls without slipping inside another fixed circle of radius \(R\). The shape of the curve depends on the distance \(d\) from the point to the center of the rolling circle. The general parametric equations for an hypotrochoid 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 becomes a hypocycloid (point lies on the edge).
- If \(d < r\), the curve is smooth and does not loop.
- If \(d > r\), the curve has internal loops.
- If \( \frac{R}{r} \) is rational, the curve is closed and periodic.
Conversion to polar coordinates
To express the curve as \(r(\theta)\), we use:
$$r(\theta) = \sqrt{x\left(\theta\right)^{2} + y\left(\theta\right)^{2}}$$
Substituting \(x(\theta)\) and \(y(\theta)\):
$$\begin{split} r(\theta) &= \left( \left[ \left\lbrace R - r \right\rbrace \cos\theta - d\cos \left\lbrace \frac{R - r}{r} \theta \right\rbrace \right]^{2} \right. \\ &+ \left. \left[ \left\lbrace R - r \right\rbrace \sin\theta - d \sin \left\lbrace \frac{R - r}{r} \theta \right\rbrace \right]^{2}\right)^{\frac{1}{2}} \end{split}$$
After expanding and simplifying, we obtain:
$$r(\theta) = (R - r) + d \cos \left( \frac{R - r}{r} \theta \right)$$
This equation describes the radial distance from the origin to the tracing point in terms of the angle \(θ\), showing the oscillating nature of the hypotrochoid.
Animation
Move sliders \(d\) (point from a certain internal circle center distance), \(R\) (external circle radius) or \(r\) (internal circle radius) to change the hypotrochoid. 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.