# Maths Flowers

4°E Siciliani CZ

4°E Siciliani CZ

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. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.

Various sizes of circles generates different hypocycloids. Let the radius of the fixed circle be a, the radius of the rolling circle be b.

The parametric equation of the hypocycloid is:

{(a + b)*Cos[t] + b*Cos[(a + b)/b*t], (a + b)*Sin[t] + b*Sin[(a + b)/b*t]}

If we use this equation we can obtain different types of hypocycloid because the parameters a Continua >

The epicycloid is the locus of a point on the circumference of a circle that rolls without slipping on the circumference of another circle.

The red curve is an epicycloid traced as the small circle (radius = 2) rolls around the outside of the large circle (radius = 4). The epicycloid can have both a parametric equation and a polar equation.

The parametric equation is:

X = r (k + 1) cos (t) – r cos ( (k + 1) t)

Y = r (k + 1) sin (t) – r sin ( (k + 1) t)

If we use these equations we can obtain Continua >

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**Polar Graphing in GeoGebra **

**Theoretical veiw **

It is not possible to plot graphics by Geogebra in polar coordinates, instead it is possible to simulate polar graphs using GeoGebra’s to graph parametric equations. Polar equations are relatively easy to write parametrically using

An expression for r will be substituted into the Parametric Curve Command let us to graph polar equations.

GeoGebra contains the following command that will assist us in creating the polar grid:

**Sequence[Expression, Variable i, Number a, Number b, <Increment>]**

The Parametric Curve Command lets us to create or graph polar functions.

It is

**Curve[Expressione1, Expressione2, Parameter t, Continua >**

In mathematics, a **Rose** or **Rhodonea curve** is a sinusoid plotted. Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728

Up to similarity, these curves can all be expressed by

**a polar equation** of the form:

R= acos (kα).

Since sin (kα) = cos (kα – π/2) = cos(k(α- π/2k))

for all θ, the curves given by the polar equations r = cos(kα) or r = sin (kα), these expressions

are identical except for a rotation of π/2*k* radians.

**Parametric equations **of Rose Curves are:

X(t)=aCos(kα)cos(t)

Y(t)=acos (kα)sin(t)

If *k* is an integer, the curve will be rose shaped with

- 2
*k*petals if*k*is even *k*petals if*k*is odd.

The coefficient a say Continua >

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