Tangent to a Curve

[primo]

this example can be used for demonstrations in the calculus lessons in the secondary schools.
this is the tangent and the normal
tangent+normal.png
look for explanation:
https://www.siyavula.com/maths/grade...s-04.cnxmlplus

to find a tangent is to find a derivative at that point (one of the Calculus lessons), the best place to find the derivative is wolfram alpha, such as to find the derivative of y = Sin(x) go to
http://www.wolframalpha.com/input/?i...tive+of+sin(x)
and it will give you Cos(x)
the derivative of x^2 is 2*x
look at the folowing curve and its tangent, at the newTurn of the curve it seems to slow down because there is a lot of points there (condensed).

i have used a box to represent the tangent, because i find tbgl_Cylinder rotate around one of its sides and not from its center, (useful for tree branches generation).
if i use tbgl_cylinder instead of the tbgl_box like this way:

TBGL_Translate x,y,0
          'TBGL_Rotate 0,0,slope
          TBGL_Rotate slope,0,0
          TBGL_Rotate -90,0,0  
          TBGL_Cylinder 0.1,0.1,1

then at the newTurn of the curve it will not stay like the classical tangent.

curve y = x^2 press Space to move the tangent over the curve

Uses "TBGL", "Math" 
Begin Const
  %listPoints = 1
End Const

Dim FrameRate  As Double
Local hWnd As DWord

hWnd = TBGL_CreateWindowEx("curve tangent .... press 'Space' to move the Tangent line over the curve", 800, 600, 32, %TBGL_WS_WINDOWED Or %TBGL_WS_CLOSEBOX) 
TBGL_ShowWindow 


TBGL_BackColor(255,255,255) 

TBGL_ResetKeyState() 

Global  x,y As Single
' catch the data and store it to %listPoints
TBGL_NewList %listPoints
      TBGL_Color(0,0,0)
      TBGL_PointSize 3 
      For x = -2.5 To 2.5 Step 0.01
          y = x^2 - 3  ' just to shift down the curve   
          
          TBGL_BeginPoly(%GL_POINTS) 
               TBGL_Color(150, 150, 255 )
               TBGL_Vertex(x,y)
          TBGL_EndPoly
         
      Next
        
TBGL_EndList

TBGL_ResetKeyState() 
x = -2.5: y = 0
Global slope, slopeAngle As Single

While TBGL_IsWindow(hWnd)
     
    FrameRate = TBGL_GetFrameRate
    
    TBGL_ClearFrame 

      TBGL_RenderMatrix2D( -5, -5, 5, 5 ) 
                                         
      TBGL_CallList %listPoints
      
          If TBGL_GetWindowKeyState( hWnd, %VK_SPACE) Then 
          x + 0.01
          y = x^2 - 3 ' just to shift down the curve 
        
          slope = 2*x ' This is the derivative of f(y)=x^2
          slopeAngle = RadToDeg(Atn(slope)) 'find the angle correspend to slope and convert it to degrees
          'slopeAngle = (Atn(slope))*180/Pi 'find the angle correspend to slope and convert it to degrees
          End If
          TBGL_Translate x,y,0
          TBGL_Rotate slopeAngle,0,0
          TBGL_Color(255, 0, 0 )
          TBGL_Box 1, 0.05, 0.2
          TBGL_Rotate 180,0,0 
          TBGL_Cylinder 0.02,0.02,0.5
          
    TBGL_DrawFrame 

    If TBGL_GetWindowKeyState(hWnd, %VK_ESCAPE) Then Exit While
        
Wend

TBGL_DestroyWindow

Curve y = Sin(x)

Uses "TBGL", "Math" 
Begin Const
  %listPoints = 1
End Const

Dim FrameRate  As Double
' Handle for our window
Local hWnd As DWord

hWnd = TBGL_CreateWindowEx("curve tangent .... press 'Space' to move the Tangent line over the curve", 800, 600, 32, %TBGL_WS_WINDOWED Or %TBGL_WS_CLOSEBOX) 
TBGL_ShowWindow 
  
' Set background  to white
TBGL_BackColor(255,255,255) 

TBGL_ResetKeyState() 

Global  x,y As Single
' catch the data and store it to %listPoints
TBGL_NewList %listPoints
      TBGL_Color(0,0,0)
      TBGL_PointSize 3 
      For x = -5 To 5 Step 0.01
          y = Sin(x)   
          
          TBGL_BeginPoly(%GL_POINTS) 
               TBGL_Color(150, 150, 255 )
               TBGL_Vertex(x,y)
          TBGL_EndPoly
          
      Next
        
TBGL_EndList

TBGL_ResetKeyState() 
x = -5: y = 0
Global slope, slopeAngle As Single

While TBGL_IsWindow(hWnd)
     
    FrameRate = TBGL_GetFrameRate
    
    TBGL_ClearFrame 

      TBGL_RenderMatrix2D( -5, -5, 5, 5 ) 
                                         
      TBGL_CallList %listPoints
       
          If TBGL_GetWindowKeyState( hWnd, %VK_SPACE) Then 
          x + 0.01
          y = Sin(x)
        
          slope = Cos(x) ' This is the derivative of f(y)=Sine(x)
          slopeAngle = RadToDeg(Atn(slope)) 'find the angle correspend to slope and convert it to degrees
          'slopeAngle = (Atn(slope))*180/Pi 'find the angle correspend to slope and convert it to degrees
          End If
          TBGL_Translate x,y,0
          TBGL_Rotate slopeAngle,0,0
          TBGL_Color(255, 0, 0 )
          TBGL_Box 1, 0.05, 0.2
          TBGL_Cylinder 0.02,0.02,0.5
          'TBGL_Box 0.1, 1, 0.1
          
     TBGL_DrawFrame 

    If TBGL_GetWindowKeyState(hWnd, %VK_ESCAPE) Then Exit While
        
Wend

TBGL_DestroyWindow

tangent.JPG