Identifying Names/Groups of Curves
Introduction
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.
Hypocyloid
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
Hypertrochoid shapes usually are recognised as having smoother curves than their hypocyloid cousins.
Fig 2: Hypertrochoid
Epicycloid
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
Epitrochoid
Similar to Epicycloid except the cusps are smooth curves.
Fig 4: Epitrochoid
Rose
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 Quatrefoil. The third shows a shape called a Dürer Folium. The fourth shape contains numerous petals that overlap to create a further structure around the centre.
Fig 5: Rose Curves
Combining Shapes
Some interesting patterns can be created by combining shapes - figure 6 shows an Astroid inside an Epitrochoid.
Fig 6
Other Curves
There are too many to mention but useful websites are listed under the paragraph "Resources" at the end of the page.
Graphic Examples
Following links illustrate some examples where parametric equations are run in RISC OS BASIC.
Parametric Equations - Creating the curves 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
Example Program for Creating a Deltoid (figure 1)
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
Example Illustrations
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"
Resources
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 parametric equation for the purposes of getting an equation pair to run in BASIC.