searching for Curves and Surfaces in the space

[primo]

suppose someone tell you to plot this equation
Pow(x,4) + Pow(y,4) + Pow(z,4) - Pow(x*x + y*y + z*z,2) + 3 * (x*x + y*y + z*z) = 0
and you don't have any clue how to solve it first such as saying:
z = f(x,y) , so how to plot it ?
there is a plan: is to search the points in space one by one in which this equation is approximately true such as:
If k <= 0.1 And k >= -0.1 Then ...
k is the value of the equation.
change it to:
If k <= 0.5 And k >= -0.5 Then
and you get a thicker surface curve with more points.

now if we tried the above equation but replacing Pow to '^' (uncomment line 178 ) then the time is twice than if use Pow. this is not important issue since we can use the speedier version.
try equation in line 175:
k = Sin(x*y)+Sin(y*z)+Sin(z*x)
it is ineresting
in line 217 : TBGL_PointSize 2 : make it 1 or 3 as necessary
change the range in line 162 as the plotting needed
this is an experimental mathematics
i have used the Entity System to position and rotate and look at using Entity system examples by Petr ,2009
also using Oxygen module to speed the calculations
you may wait from 6 to 30 seconds for every graphics since we check every point in the space within the range.
surface.JPGsurface1.JPG

'Experimental Mathematics                             =
'===============================================================================

Uses "TBGL" , "Math" , "Oxygen"
#Include Once "%APP_INCLUDEPATH%\thinbasic_gl.inc"
#Include Once "%APP_INCLUDEPATH%\thinbasic_glu.inc" 

BEGIN CONST
  ' -- Scene IDs
  %sScene  = 1
 
  ' -- Entity IDs 
  %eCamera = 1
  
  %eContour       
  %eLight
  %texOne
  %lMyPoints       
End Const

Type Point3D
  x As Single
  y As Single
  z As Single
  red As Single
  green As Single
  blue As Single
  tu As Single
  tv As Single
End Type
   
Global Nb As DWord = 2000000 '401880 '8120601
Global x, y, z,zz As Single
Global FrameRate As Long
Global Vertex(Nb) As Point3D ' single
Global total As Long 
'Global ColorA(Nb)  As TBGL_TRGB      ' Byte
  

FUNCTION TBMAIN()
  LOCAL hWnd As DWORD
  Global FrameRate As Double
  Global ang, earthX, earthZ, OrbitRadius As Single
  Global flag, mask As Long
  flag = 1 : mask = 1
  OrbitRadius = 2.5
  
  ' -- Create and show window
  hWnd = TBGL_CreateWindowEx("press Space to toggle the camera position , press 'M' to set a mask , 'R' reset mask to default ", 800, 600, 32, %TBGL_WS_WINDOWED Or %TBGL_WS_CLOSEBOX) 
  TBGL_ShowWindow 
  
  ' -- Create scene
  TBGL_SceneCreate(%sScene)

  ' -- Create basic entities
  ' -- Create camera to look from 0, 8, 20 to 0, 0, 0 
  TBGL_EntityCreateCamera(%sScene, %eCamera)
    TBGL_EntitySetPos(%sScene, %eCamera, 0, 8, 20)
    TBGL_EntitySetTargetPos(%sScene, %eCamera, 0, 0, 0)  
  ' -- Create point light  
  TBGL_EntityCreateLight(%sScene, %eLight)
  TBGL_EntitySetPos(%sScene, %eLight, 10, 20, 50)
  
  ' -- Create something to look at 
  '---Start time
  Dim T0  As Double = Timer
  CreatePointsList()
  Dim T1  As Double = Timer
  MsgBox(0, "Total time to create points List "  & Format$(T1 - T0, "#0.00") & " secs" & " ...   number of points = " & Str$(total))
  
  TBGL_EntityCreateDLSlot(%sScene, %eContour, 0, %lMyPoints) 
    
    TBGL_EntitySetColor(%sScene, %eContour, 255, 255, 255)
        
  ' -- Resets status of all keys 
  TBGL_ResetKeyState() 
  TBGL_EntitySetTexture(%sScene, %eContour, %texOne )
  'TBGL_LoadTexture APP_Path+"SampleScripts\TBGL\GBuffers\Textures\Bricks.bmp", %texOne, %TBGL_TEX_MIPMAP
  TBGL_LoadTexture APP_Path+"SampleScripts\UI\Dialogs\Images\frogduck.bmp", %texOne, %TBGL_TEX_MIPMAP        
  
  ' -- Main loop
  While TBGL_IsWindow(hWnd) 
    FrameRate = TBGL_GetFrameRate
    TBGL_SetWindowTitle( hWnd, "FPS = "+Str$(FrameRate)+"  press Space to toggle the camera position , press 'M' to set a mask , 'R' reset mask to default ")
    TBGL_ClearFrame 
    If TBGL_GetWindowKeyOnce( hWnd, %VK_SPACE) Then
    If flag = 1 Then
     TBGL_EntitySetPos(%sScene, %eCamera, 0, 25, 0.1)
     TBGL_EntitySetTargetPos(%sScene, %eCamera, 0, 0, 0)
     
     flag * -1  
     Else 
     TBGL_EntitySetPos(%sScene, %eCamera, 0, 12, 20)
     TBGL_EntitySetTargetPos(%sScene, %eCamera, 0, 0, 0) 
     flag * -1  
    End If
    End If
    
    If TBGL_GetWindowKeyOnce( hWnd, %VK_M) Then 
      If mask = 1 Then
        TBGL_EntitySetColorMask(%sScene, %eCamera, %TBGL_RED  )
        mask * -1
        Else
        TBGL_EntitySetColorMask(%sScene, %eCamera, %TBGL_GREEN  )
        mask * -1
      End If
    End If
     
     If TBGL_GetWindowKeyOnce( hWnd, %VK_R) Then
       TBGL_EntitySetColorMask(%sScene, %eCamera, %TBGL_DEFAULT)
     End If
     
     TBGL_EntitySetRot(%sScene, %eContour, 0, GetTickCount/40, 0)
      
    TBGL_SceneRender(%sScene)
   
    TBGL_DrawFrame 
 
    ' -- ESCAPE key to exit application
    If TBGL_GetWindowKeyState(hWnd, %VK_ESCAPE) Then Exit While 
  
  Wend 

  TBGL_DestroyWindow
END FUNCTION
        

'==================================
Sub CreatePointsList() 

  String src = " 
  ' -- Re-declare types
    
  Type Point3D
  x As Single
  y As Single
  z As Single
  red As Single
  green As Single
  blue As Single
  tu As Single
  tv As Single
  End Type
  
  dim oVertex(2000000) At #Vertex As Point3D
  dim oTexA(2000000)At #TexA As Point3D
  
  dim N At #total As long
  'dim oColorA(2000000) At #ColorA As TBGL_TRGB
  Long a, b, c, Nb
  Single x, z, k, xMin, yMin, zMin, xMax, yMax, zMax, range, step1
    
  Long NbX
  Long NbZ
  Long NbY
  nbx = 300
  nby = 300
  nbz = 300
  
  'xMin = -2 : yMin = -2: zMin = -2 : xMax = 2: yMax = 2 : zMax = 2
  'xMin = -3 : yMin = -3: zMin = -3 : xMax = 3: yMax = 3 : zMax = 3
  xMin = -4 : yMin = -4: zMin = -4 : xMax = 4: yMax = 4 : zMax = 4
  'xMin = -5 : yMin = -5: zMin = -5 : xMax = 5: yMax = 5 : zMax = 5

  range = xMax - xMin
  step1 = range / NbX
  x = xMin: z = zMin : y = yMin
  N = 0  
  For c = 0 To NbY
     For b=0 To NbZ
    
    For a=0 To NbX
      
      'k = 320*(((x*x + (9*y*y)/4 +z*z -1)^3) - x*x*z*z*z - (9*y*y*z*z*z)/80)
      'k = Sin(x*y)+Sin(y*z)+Sin(z*x)
      'k = x^3+y^2+z^2
       k = Pow(x,4) + Pow(y,4) + Pow(z,4) - Pow(x*x + y*y + z*z,2) + 3 * (x*x + y*y + z*z)
      'k = x^4 + y^4 + z^4 - (x^2 + y^2 + z^2)^2 + 3 * (x^2 + y^2 + z^2)
      
             
      If k <= 0.1 And k >= -0.1 Then 
      'If k <= 0.5 And k >= -0.5 Then 
        N = N+1
        oVertex(N).x = x
        oVertex(N).y = y 
        oVertex(N).z = z
        oVertex(N).tu = a/NbX
        oVertex(N).tv = b/Nbz
             
      End If
      x  = x + step1
      
    Next a
    x = xMin
    z = z + step1
    
  Next b
  x = xMin
  z = zMin
  y = y + step1
Next c
'print str N 
 terminate    
  "        
   
  ' -- Pass the source
  'O2_Asmo src
  o2_basic src
  If Len(o2_error) Then
    MsgBox 0, o2_error : Stop
  End If              
  ' -- Execute
  o2_exec
  
  Long N       
  TBGL_NewList %lMyPoints
    TBGL_PointSize 2
    TBGL_UseLighting %FALSE
    
    TBGL_BeginPoly %GL_POINTS
     For N=1 To Nb
       glTexCoord2f(Vertex(N).tu, Vertex(N).tv)
       glVertex3f(Vertex(N).x,Vertex(N).y,Vertex(N).z)
     Next   
   
    TBGL_EndPoly
    TBGL_EndList
    TBGL_UseLighting %TRUE
End Sub