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# Colorful roses with parametric equations

A curve in the plane is said to be parameterized if the set of
coordinates on the curve, $(x,y)$, are represented as functions of a
variable $t$. Namely, \begin{eqnarray*} x = f(t), \quad y = g(t) \quad
\text{with}\quad t\in D \end{eqnarray*} where $D$ is a set of real
numbers. The variable $t$ is called a parameter and the relations
between $x$, $y$ and $t$ are called
**parametric equations**. The set $D$ is called the
domain of $f$ and $g$ and it is the set of values $t$ takes. For
example, while the equation of a circle in Cartesian coordinates can
be given by $r^2=x^2+y^2$, one set of parametric equations for the
circle are given by \begin{eqnarray*} x = r \cos t, \quad y = r \sin t
\quad \text{with}\quad t\in[0, 2\pi). \end{eqnarray*}

The flower in this animation is defined with the following set of parametric equations: \begin{eqnarray*} x & = \big[2.5 v_j + \cos^p\left(n (t - v_j)\right)\big] \cos(t+\phi) \\ y & = \big[2.5 v_j + \sin^q\left(n (t - v_j)\right)\big] \sin (t+\phi) \end{eqnarray*} where $v_j=j\frac{27\pi}{3584}$, for $j=0,1,2,\ldots, 63$ and $\phi\gt 0$ defines an anticlockwise rotation.

Use the controls, at the top-right corner, to change the parameters $n$, $p$ and $q$. Observe how the shape of the flower changes.

Have fun! If you liked this applet, please let me know: [email protected]