The curves created by parametric equations are many and varied. They are used to describe geometric figures or curves some of which can be complex. A mechanical device, the Spirograph, often sold as a children's toy can be used to create many of the patterns. The following show the more common graphic geometric curves.

One of the most commonly known is the **Deltoid** recognised by its 3 sharp cusps and shown in figure 1 on the left whilst an **Astroid** is on the right.

Fig 1: Deltoid and Astroid

Hypertrochoid shapes usually are recognised as having smoother curves than their hypocyloid cousins.

Fig 2: Hypertrochoid

Curves which have sharp cusps on the inside indicate Epicycloid geometry. Figure 3 illustrates the progression from the basic Cardioid (top-left-hand corner).

Fig 3: Epicycloids

Similar to Epicycloid except the cusps are smooth curves.

Fig 4: Epitrochoid

Characterised by petals coming from a central point (figure 5). Some fascinating shapes can be created with this graphic. The simplest are the 3-petalled, ** Regular Trifolium** and the 4-petalled

Fig 5: Rose Curves

Some interesting patterns can be created by combining shapes - figure 6 shows an Astroid inside an Epitrochoid.

Fig 6

There are too many to mention but useful websites are listed under the paragraph "**Resources**" at the end of the page.

Following links illustrate some examples where parametric equations are run in RISC OS BASIC.

As well as the slightly more complex shapes described above the standard geometric graphics ie. straight line, polygon, ellipse, spiral can also be created with these equations.

Consider one of the simplest shapes - the circle. Any point around its circumference can be calculated using only 2 equations. The equations are used to set the X and Y co-ordinates with some simple trigonometry.

X=SIN(T)

Y=COS(T)

where T is the angle of rotation in PI radians. The X and Y co-ordinates are too small to be useful. Simply multiply each X and Y value by a multiplication factor (eg *1000) to give...

X=SIN(T)*1000

Y=COS(T)*1000

Typical program to draw a circle becomes...

MODE 1920,1080,32

ORIGIN 1920,1080

FOR T = 0 TO 2*PI STEP PI/100

** X=SIN(T)*1000**

** Y=COS(T)*1000**

IF T=0 THEN MOVE X,Y ELSE DRAW X,Y

NEXT T

MODE 1920,1080,32

ORIGIN 1920,1080

FOR T=0 TO 2*PI STEP PI/100

**X=(2*COS(T)+COS(2*T))*400**

** Y=(2*SIN(T)-SIN(2*T))*400**

IF T=0 THEN MOVE X,Y ELSE DRAW X,Y

NEXT T

Figure 7, 8, 9 and 10 show some of the many graphic shapes that can be created with parametric equations.

*Some of the illustrations have been drawn in ArtWorks to improve rendering quality. Others are sprite screensaves which when resized may not give optimum quality. When programs are run in BASIC render quality is good.*

Fig 7 Programmed using "Parametric Equations"

Fig 8 Programmed using "Parametric Equations"

Fig 9 Programmed using "Parametric Equations"

Fig 10 Programmed using "Parametric Equations"

Wikipedia is the first port of call:

https://en.wikipedia.org/wiki/Parametric_equation

**Mathcurve.com** is a French site that provides an excellent resource for anything related to curves. David Michel has translated the **2D Curves** (courbes 2D) section into English.

Some of the examples contain equations in the form of ** cartesian parametrization**. This can be considered as a close equivalent to a