Simple Beautiful Curves

[primo]

i have found these shapes while experimenting with a variations of the Mystery Curve i have posted before here.
triangleX.PNGtoonX.PNGhexagonX.PNG
note that they all have smooth vertices, look at the triangle carefully it has a slight concave edges so i consider it beautiful. it is suitable as a road sign, and all the people in the city will wonder why the Municipal make the sign concave a little !!.
some suggestions for other shapes:
1-to make it a convex triangle change s2 = 1/3 to s2 = 1/7
2- s = 1/19: s2 = 1/6 ' toon triangle
k = 3
suitable for Fred Flintstone mathematics school and also as a funny road sign
3- s = 0: s2 = 1/19 ' hexagon
k = 6 ' k=4 'square
4- uncomment lines 63 and 64 and you get full circle
there are many shapes by just changing the parameters.
5- note that they all can be considered a deformation of a circle made by a sewing thread manually
6- the usefulness of plotting these shapes in 3D engine instead of 2D canvas is to be able to rotate the shapes in any direction, and to be able to look at it from any perspective, also we can surround the curves by tubes (as demonstrated somewhere here) for 3D printing

Uses "TBGL"
Uses "math"
Begin Const
  %listPoints = 1
End Const

Function TBMain()
  Local hWnd      As DWord
  Local FrameRate As Double
  ' -- Create and show window
  hWnd = TBGL_CreateWindowEx("Triangle proposed as a Road Sign - press ESC to quit", 600, 600, 32, %TBGL_WS_WINDOWED)' Or %TBGL_WS_CLOSEBOX) 
  TBGL_ShowWindow 
  TBGL_BackColor 255,255,255 
  
  FillArrays ' call the sub to create the List of vertices and color
    
  ' -- Resets status of all keys 
  TBGL_ResetKeyState()
  TBGL_PointSize 2 'if you want thick points uncomment this line
    ' -- Main loop
  While TBGL_IsWindow(hWnd)
  'init 
    FrameRate = TBGL_GetFrameRate
    TBGL_ClearFrame
      TBGL_Camera(0, 0, 5, 0, 0, 0)
      ' -- Turn triangle around Z axis
      TBGL_Rotate GetTickCount/50, 0, 0, 1
      
      ' -- Render it                              
      TBGL_CallList %listPoints
      TBGL_DrawFrame
     ' -- ESCAPE key to exit application
    If TBGL_GetWindowKeyState(hWnd, %VK_ESCAPE) Then Exit While
  Wend       
  ' -- Destroying the buffer is not necessary,
  ' -- the garbage collector will take care of it
  
  ' -- Destroy window
  TBGL_DestroyWindow
End Function 
 
Sub FillArrays()
Dim x, y, z, t As Single
Dim N As DWord = 1
local s, s2, k as single

s =  0:  s2 = 1/3 ' triangle
k = 3

's =  1/19: s2 = 1/6 ' toon triangle
'k = 3

's = 0: s2 = 1/19 ' hexagon
'k = 6

TBGL_NewList %listPoints
TBGL_BeginPoly(%GL_POINTS)   
    
While t <= 2*Pi
          
         x = s*Sin(t)* Sin(k* t) - s* Cos(k* t)* Cos(t) + Cos(t) + s2* Sin(k* t)* Cos(t) - s2* Sin(t)* Cos(k* t)
         y = Sin(t) + s2* Sin(t)* Sin(k* t) + s2* Cos(t)* Cos(k* t) - s* Sin(t)* Cos(k* t) - s* Sin(k* t)* Cos(t)
         'x = Cos(t) + s* Sin(k* t)* Cos(t) - s* Sin(t)* Cos(k* t)
         'y = Sin(t) + s* Sin(t)* Sin(k* t) + s* Cos(t)* Cos(k* t)
          
          TBGL_Color 255, 0, 0
          TBGL_Vertex x, y , z
          t + 2*PI/1000               
    
Wend
TBGL_EndPoly
  TBGL_EndList
  
End Sub