Suggest parametric equations for a given curve

Question

Can anyone suggest a pair of parametric equations involving trigonometric functions that could give a curve with the following shape?

Answer 1

The "right-angle-like" curve reminds of the astroid, which is a particular case of the hypocycloid, a rolling curve (with four cusps).

To obtain the small loop, you keep the rolling but take the current point a little outside the rolling circle, and this yields an hypotrochoid.

http://mathworld.wolfram.com/Astroid.html

http://mathworld.wolfram.com/Hypocycloid.html

http://mathworld.wolfram.com/Hypotrochoid.html

Remains to rotate it by 45°, which is elementary.

Answer 2

You can do without trigonometric functions by taking a cubic Bezier curve like this one (see fig. below):

$$\binom{x}{y}=s^3\binom{x_A}{y_A}+3s^2t\binom{x_B}{y_B}+3st^2\binom{x_C}{y_C}+t^3\binom{x_D}{y_D}$$

$$ \text{with} \ s:=1-t \ \text{and ''guiding'' points :} \ A(0,0), \ B(9,-1), \ C(5,4), \ D(4,-3), \ $$

giving:

$$\begin{cases}x=3*9*(t-2t^2+t^3)+3*5*(t^2-t^3)+4*t^3\\ y=3*(-1)*(t-2t^2+t^3)+3*4*(t^2-t^3)+(-3)*t^3\end{cases}$$

(I don't expand and reduce these expressions because small adjustments are easier to do under this form).

Answer 3

R = 5; r = 4; d = 2;
ParametricPlot[{(R - r) Cos[t] + d Cos[(R/r - 1) t], (R - r) Sin[t] - 
   d Sin[(R/r - 1) t]}, {t, 0, 25}, PlotStyle -> {Red, Thick}, 
 GridLines -> Automatic]

You can play with constants ( radii of big circle $R$, small circle $r$, crank length $d$) producing a variety of Hypo-Cycloids. Loop size can be varied and start angle can be also added and set inside the sin/cos trig arguments.