これはエクセルＶＢＡのプログラムです。
これを走らせると、表に、座標データが出力されますので、それを、散布図でグラフ化すれば曲線が見られます。
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'****************************
'クロソイド曲線を描く
'              2005.5.11 崩彦
'****************************
Option Explicit                                 '変数宣言を強制

Const Pi = 3.14159265358979                     '円周率πを定数宣言

Sub clothoid()
 '----------変数宣言------------------------------------------------------------------
    Dim x As Double, y As Double                'グラフの座標
    Dim a As Double, b As Double, h As Double   'a, b は積分範囲, h は分割区間幅
    Dim div As Integer                          '積分範囲の分割数
    Dim i As Integer                            'ループ変数
    Dim v As Double                             'パラメーター(曲線の長さに対応)
    Dim n As Integer                            'ｖの分割数
    Dim delta As Double                         'ｖの増し分
    Dim row As Integer                          'シートに座標データを書き込むための行番号
    '----------初期値--------------------------------------------------------------------
    v = 5           'パラメーターｖは５まで
    n = 1000        '曲線を１０００に分割して伸ばしていく
    div = 1000      '積分区間の分割数は１０００
   a = 0           '積分区間の初期値
    b = 0           '同上
    row = 0         '行番号の初期値
    '------------------------------------------------------------------------------------
    delta = v / n
    For i = 1 To n
        b = b + delta
        '----------シンプソン則による数値積分--------------------------------------------
        x = simpson(a, b, div, 0)
        y = simpson(a, b, div, 1)
        '--------------------------------------------------------------------------------
        row = row + 1
        Cells(row, 1).Value = x
        Cells(row, 2).Value = y
    Next i
End Sub

'-----------------------------------------------------------------------------------
'引数：a, b は積分区間。m は積分区間の分割数。flag は被積分関数の切り替え用フラグ
'-----------------------------------------------------------------------------------
Function simpson(a As Double, b As Double, m As Integer, flag As Integer) As Double
    Dim d As Double, sum As Double
    Dim x As Double
    Dim i As Integer

    d = (b - a) / (2 * m)
    If (flag = 0) Then
        sum = f(a)
        For i = 1 To 2 * m - 1
            x = a + i * d
            If (i Mod 2) = 0 Then
                sum = sum + 2 * f(x)
            Else
                sum = sum + 4 * f(x)
            End If
        Next i
        sum = sum + f(b)
    ElseIf (flag = 1) Then
        sum = g(a)
        For i = 1 To 2 * m - 1
            x = a + i * d
            If (i Mod 2) = 0 Then
                sum = sum + 2 * g(x)
            Else
                sum = sum + 4 * g(x)
            End If
        Next i
        sum = sum + g(b)
    End If
    simpson = sum * d / 3
End Function



================================================================================
これは、上のプログラムを十進ＢＡＳＩＣ用に書き換えたものです。
「!」はＲＥＭですので、それ以降は実行時、無視されます。

!----------変数宣言------------------------------------------------------------------
!    Dim x As Double, y As Double                'グラフの座標
!    Dim a As Double, b As Double, h As Double   'a, b は積分範囲, h は分割区間幅
!    Dim div As Integer                          '積分範囲の分割数
!    Dim i As Integer                            'ループ変数
!    Dim v As Double                             'パラメーター(曲線の長さに対応)
!    Dim n As Integer                            'ｖの分割数
!    Dim delta As Double                         'ｖの増し分
!    Dim row As Integer                          'シートに座標データを書き込むための行番号
!-------------------------------------------------------------------------------
DECLARE EXTERNAL FUNCTION simpson
DECLARE EXTERNAL FUNCTION f
DECLARE EXTERNAL FUNCTION g

SET WINDOW -0.1, 1,-0.1, 1
SET POINT STYLE 1
DRAW AXES
!'------------------------------------------------------------------------------------
LET v = 6           !'パラメーターｖは６まで
LET n = 1000        !'曲線を１０００に分割して伸ばしていく
LET div = 1000      !'積分区間の分割数は１０００
LET a = 0           !'積分区間の初期値
LET b = 0           !'同上
!'------------------------------------------------------------------------------------

LET delta = v / n
FOR i = 1 TO n
   LET b = b + delta
   !'----------シンプソン則による数値積分--------------------------------------------
   LET x = simpson(a, b, div, 0)
   LET y = simpson(a, b, div, 1)
   !'--------------------------------------------------------------------------------
   !PLOT POINTS:x, y
   PLOT LINES:x, y;
NEXT i
END

!'-----------------------------------------------------------------------------------
!'引数：a, b は積分区間。m は積分区間の分割数。flag は被積分関数の切り替え用フラグ
!'-----------------------------------------------------------------------------------
external Function simpson(a, b, m, flag)
LET d = (b - a) / (2 * m)
IF (flag = 0) THEN
   LET sum = f(a)
   FOR i = 1 TO 2 * m - 1
      LET x = a + i * d
      IF (MOD(i , 2) ) = 0 THEN
         LET sum = sum + 2 * f(x)
      ELSE
         LET sum = sum + 4 * f(x)
      END IF
   NEXT i
   LET sum = sum + f(b)
ELSEIF (flag = 1) THEN
   LET sum = g(a)
   FOR i = 1 TO 2 * m - 1
      LET x = a + i * d
      IF MOD(i, 2) = 0 THEN
         LET sum = sum + 2 * g(x)
      ELSE
         LET sum = sum + 4 * g(x)
      END IF
   NEXT i
   LET sum = sum + g(b)
END IF 
LET simpson = sum * d / 3
END FUNCTION

!'----------被積分関数--------------------------------------------------------------------
EXTERNAL FUNCTION f(x)
LET f = Cos((Pi * x * x) / 2)
END FUNCTION

EXTERNAL FUNCTION g(x)
LET g = Sin((Pi * x * x) / 2)
END FUNCTION
================================================================================