algebraic formations

These explorations mainly focus on 2 main areas, formations and transformations. Formation is considered as a whole created through aggregation of small entities – in this case, I am using a sine wave pattern as the basic description of all the surfaces. Patterns can be layered, and their formation becomes complex as layers build up. In the first part of this series, I will mainly focus on various pattern making techniques and layering techniques.

Transformations, as we look beyond the buttons in our 3D software, is described mathematically as an equation, such as x’=Sx*x, y’=Sy*y for scaling or x’=x*cos(a)+y*sin(a), y’=y*cos(a)-x*sin(a) for rotation. In the later part of the series of, transformations are applied algebraically at various levels of complexities, from turning a surface into a sinuous form, to embedding multiple ripples onto a spherical form.

The study itself is ongoing, and I will keep updating it as much as I can.

*special thanks to Jess Märtterer for all his help on scripting.

download script01

SR-01

base

SR-02

cosine

parametric form :
	n/a
transformation form :
	x = xt
	y = yt
	z = zt
parametric form :
	a*cos(t)
transformation form :
	x = xt
	y = yt*cos(xt/8)
	z = zt

SR-03

exponential

SR-04

square roots

parametric form :
	a*x^b
transformation form :
	x = xt
	y = (xt^2+yt^2)/32
	z = zt
parametric form :
	a*sqr(x)
transformation form :
	x = xt
	y = yt+xt
	z = sqr(zt^2+xt^2)

SR-05

multiplication

SR-06

arctangent

parametric form :
	a*x*y
transformation form :
	x = xt
	y = yt*xt/16
	z = zt
parametric form :
	a*atn(t)
transformation form :
	x = xt
	y = 4*atn(yt^4/xt^4)
	z = zt

SR-07

helix no.1

SR-08

helix no.2

parametric form :
	x=a*cos(t),y=a*sin(t),z=b*t
transformation form :
	x = (2*xt+100)*cos(yt*Pi/24)
	y = (2*xt+100)*sin(yt*Pi/24)
	z = zt
parametric form :
	x=a*cos(t),y=a*sin(t),z=b*t
transformation form :
	x=(4*xt+100)*cos(zt*Pi/24)
	y=(4*xt+100)*sin(yt*Pi/24)
	z=zt

SR-09

helix no.3

SR-10

ripple

parametric form :
	x=a*cos(t),y=a*sin(t),z=b*t
transformation form :
	x=(2*xt+100)*cos(yt*Pi/24)
	y=(2*xt+100)*sin(sin(sin(yt*
    	Pi/24)))
	z=zt
parametric form :
	a*cos(sqr((x-u)^2+(y-v)^2))
    	/sqr((x-u)^2+(y-v)^2)
transformation form :
	x=xt
	y=yt
	z=10*cos(Sqr(xt^2+(zt-24)^2))
    	/Sqr(xt^2+(zt-24)^2+2)

SR-11

ripple x 2

SR-12

ripple + sine

parametric form :
   a*cos(sqr((x-u)^2+(y-v)^2))
   /sqr((x-u)^2+(y-v)^2)
transformation form :
   a=10*cos(Sqr((xt)^2+(zt-24)^2))
     /Sqr((xt-0)^2+(zt-24)^2+2)
   b=10*cos(Sqr((xt-16)^2+(zt-12)^2))
     /Sqr((xt-16)^2+(zt-12)^2+2)
   x=xt
   y=yt
   z=a+b
parametric form :
   a*cos(sqr((x-u)^2+(y-v)^2))
transformation form :
   x=xt + Sin(Sqr(xt^2+(yt-24)^2)+Pi)
   y=yt
   z=0

One Response to “algebraic formations”

  1. ememery says:

    great job!