Hermite spline

< 참고 URL >
*참고한 공식 Site !
http://www.siggraph.org/education/materials/HyperGraph/modeling/splines/hermite.htm
http://blog.naver.com/millinya?Redirect=Log&logNo=40053907534
http://en.wikipedia.org/wiki/Catmull-Rom_spline#Interpolation_on_a_single_interval
*곡선의 데모를 볼수있는곳( Hermite, Bezier, B-Spline )
http://www.siggraph.org/education/materials/HyperGraph/modeling/splines/demoprog/curve.html
* API 로된 소스 & 자료
http://cstudy.co.kr/bbs/view.php?id=na_report&page=3&sn1=&divpage=1&sn=off&ss=on&sc=on&select_arrange=headnum&desc=asc&no=31


참고 URL 을 따라가면...상세한 공식 설명이 나옵니다.
그리고 중요한 공식하나...만!

x(t) = TM_HG_Hx = P_1x(2t^3 - 3t^2 + 1) + P_4x(-2t^3 + 3t^2) + R_1x(t^3 - 2t^2 + t) + R_4X(t^3 - t^2) 

이게 무슨말이냐!? 후훗..
첫 start point, end point 와 탄젠트 값 2개만 있으면 구할수 있습니다.
탄젠트의 값은 그냥 일반적으로 지정해주고 하네요...
다른예제들 보니 특별히 따로 계산하지 않은것 같더라구요. ㅎㅎㅎ

먼저 제가 site 를 보고 계산한 방법 부터 하겠습니다.


My Hermite Source 다운! ㅎ ->?

각 함수 4개를 만듭니다.
double HermiteF1( double t )
{
 double result;
 result = 1 -3*pow(t,2) + 2*pow(t,3); 
 return result;
}
double HermiteF2( double t )
{
 double result;
 result = 3*pow(t,2) - 2*pow(t,3);
 return result;
}
double HermiteF3( double t )
{
 double result;
 result = t-2*pow(t,2) + pow(t,3);
 return result;
}
double HermiteF4( double t )
{
 double result;
 result = -1 * pow(t,2) + pow(t,3);
 return result; 
}

// 각 점의 좌표를 가지고 아래의 공식으로 점의 좌표를 구합니다. ( 계속 그림...반복문을 돌며...)

/* x */ v[0] = CP[i][0]*HermiteF1(t) + CP[i+1][0]*HermiteF2(t) + 10*HermiteF3(t) + 10*HermiteF4(t);
/* y */ v[1] = CP[i][1]*HermiteF1(t) + CP[i+1][1]*HermiteF2(t) + 300*HermiteF3(t) + 300*HermiteF4(t);
   v[1] = wh - v[1]; // 좌표 뒤집기
   glVertex3dv(v);
   t += 0.01;   
   
   if( CP[i+1][0] == 0 && CP[i+1][1] == 0 )
    break;

참고 URL 을 따라가심 나온..소스! 아래 부붙 참조바랍니다.
API 소스   - 다운받기 ->

Hermite_Spline.rar

1. // Hermite Spline.cpp : Defines the entry point for the application.
2. //
3.  
4. #include "stdafx.h"
5. #include "resource.h"
6.  
7. // 좌표를 저장할 구조체 변수
8. // POINT형을 사용하면 되나 소수점 좌표도 처리하기 위해서 사용한다.
9. typedef struct Apt {
10.         double x;
11.         double y;
12. } Apt;
13.  
14. #define MAX_LOADSTRING 100
15.  
16. // 변수 선언 & 초기화 및 함수 선언
17. Apt HermiteSpline(Apt G[4], double t);
18. void DrawHermite(HDC hdc, Apt G[4]);
19. Apt G[4] = {{190, 230}, {570, 230}, {380, 115}, {380, 345}};
20.  
21. // Global Variables:
22. HINSTANCE hInst;                                                                // current instance
23. TCHAR szTitle[MAX_LOADSTRING];                                                                // The title bar text
24. TCHAR szWindowClass[MAX_LOADSTRING];                                                                // The title bar text
25.  
26. // Foward declarations of functions included in this code module:
27. ATOM                                MyRegisterClass(HINSTANCE hInstance);
28. BOOL                                InitInstance(HINSTANCE, int);
29. LRESULT CALLBACK        WndProc(HWND, UINT, WPARAM, LPARAM);
30. LRESULT CALLBACK        About(HWND, UINT, WPARAM, LPARAM);
31.  
32. int APIENTRY WinMain(HINSTANCE hInstance,
33.                      HINSTANCE hPrevInstance,
34.                      LPSTR     lpCmdLine,
35.                      int       nCmdShow)
36. {
37.          // TODO: Place code here.
38.         MSG msg;
39.         HACCEL hAccelTable;
40.  
41.         // Initialize global strings
42.         LoadString(hInstance, IDS_APP_TITLE, szTitle, MAX_LOADSTRING);
43.         LoadString(hInstance, IDC_HERMITESPLINE, szWindowClass, MAX_LOADSTRING);
44.         MyRegisterClass(hInstance);
45.  
46.         // Perform application initialization:
47.         if (!InitInstance (hInstance, nCmdShow))
48.         {
49.                 return FALSE;
50.         }
51.  
52.         hAccelTable = LoadAccelerators(hInstance, (LPCTSTR)IDC_HERMITESPLINE);
53.  
54.         // Main message loop:
55.         while (GetMessage(&msg, NULL, 0, 0))
56.         {
57.                 if (!TranslateAccelerator(msg.hwnd, hAccelTable, &msg))
58.                 {
59.                         TranslateMessage(&msg);
60.                         DispatchMessage(&msg);
61.                 }
62.         }
63.  
64.         return msg.wParam;
65. }
66.  
67.  
68.  
69. //
70. //  FUNCTION: MyRegisterClass()
71. //
72. //  PURPOSE: Registers the window class.
73. //
74. //  COMMENTS:
75. //
76. //    This function and its usage is only necessary if you want this code
77. //    to be compatible with Win32 systems prior to the 'RegisterClassEx'
78. //    function that was added to Windows 95. It is important to call this function
79. //    so that the application will get 'well formed' small icons associated
80. //    with it.
81. //
82. ATOM MyRegisterClass(HINSTANCE hInstance)
83. {
84.         WNDCLASSEX wcex;
85.  
86.         wcex.cbSize = sizeof(WNDCLASSEX);
87.  
88.         wcex.style                        = CS_HREDRAW | CS_VREDRAW;
89.         wcex.lpfnWndProc        = (WNDPROC)WndProc;
90.         wcex.cbClsExtra                = 0;
91.         wcex.cbWndExtra                = 0;
92.         wcex.hInstance                = hInstance;
93.         wcex.hIcon                        = LoadIcon(hInstance, (LPCTSTR)IDI_HERMITESPLINE);
94.         wcex.hCursor                = LoadCursor(NULL, IDC_ARROW);
95.         wcex.hbrBackground        = (HBRUSH)(COLOR_WINDOW+1);
96.         wcex.lpszMenuName        = (LPCSTR)IDC_HERMITESPLINE;
97.         wcex.lpszClassName        = szWindowClass;
98.         wcex.hIconSm                = LoadIcon(wcex.hInstance, (LPCTSTR)IDI_SMALL);
99.  
100.         return RegisterClassEx(&wcex);
101. }
102.  
103. //
104. //   FUNCTION: InitInstance(HANDLE, int)
105. //
106. //   PURPOSE: Saves instance handle and creates main window
107. //
108. //   COMMENTS:
109. //
110. //        In this function, we save the instance handle in a global variable and
111. //        create and display the main program window.
112. //
113. BOOL InitInstance(HINSTANCE hInstance, int nCmdShow)
114. {
115.    HWND hWnd;
116.  
117.    hInst = hInstance; // Store instance handle in our global variable
118.  
119.    hWnd = CreateWindow(szWindowClass, szTitle, WS_OVERLAPPEDWINDOW,
120.       CW_USEDEFAULT, 0, 760, 460, NULL, NULL, hInstance, NULL);
121.  
122.    if (!hWnd)
123.    {
124.       return FALSE;
125.    }
126.  
127.    ShowWindow(hWnd, nCmdShow);
128.    UpdateWindow(hWnd);
129.  
130.    return TRUE;
131. }
132.  
133. //
134. //  FUNCTION: WndProc(HWND, unsigned, WORD, LONG)
135. //
136. //  PURPOSE:  Processes messages for the main window.
137. //
138. //  WM_COMMAND        - process the application menu
139. //  WM_PAINT        - Paint the main window
140. //  WM_DESTROY        - post a quit message and return
141. //
142. //
143. LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam)
144. {
145.         int wmId, wmEvent;
146.         PAINTSTRUCT ps;
147.         HDC hdc;       
148.  
149.         switch (message)
150.         {
151.                 case WM_COMMAND:
152.                         wmId    = LOWORD(wParam);
153.                         wmEvent = HIWORD(wParam);
154.                         // Parse the menu selections:
155.                         switch (wmId)
156.                         {
157.                                 case IDM_ABOUT:
158.                                    DialogBox(hInst, (LPCTSTR)IDD_ABOUTBOX, hWnd, (DLGPROC)About);
159.                                    break;
160.                                 case IDM_EXIT:
161.                                    DestroyWindow(hWnd);
162.                                    break;
163.                                 default:
164.                                    return DefWindowProc(hWnd, message, wParam, lParam);
165.                         }
166.                         break;
167.                 case WM_LBUTTONDOWN:
168.                 case WM_RBUTTONDOWN:
169.                 case WM_MOUSEMOVE:
170.                         // 마우스 왼쪽 버튼이나 오른쪽 버튼이 클릭되어 있는 동시에 control키나 shift키가
171.                         // 눌러졌을 시에
172.                         if (((wParam & MK_LBUTTON) && ((wParam & MK_CONTROL) || (wParam & MK_SHIFT)))
173.                                 || ((wParam & MK_RBUTTON) && ((wParam & MK_CONTROL) || (wParam & MK_SHIFT))))
174.                         {
175.                                 hdc = GetDC (hWnd);
176.  
177.                                 // 기존에 있던 그래프를 하얀펜으로 다시 그려서 지우는 효과를 낸다.
178.                                 SelectObject (hdc, GetStockObject (WHITE_PEN));
179.                                 DrawHermite (hdc, G);
180.  
181.                                 // 마우스 왼쪽 버튼이 클릭되어 있을 시에
182.                                 if (wParam & MK_LBUTTON)
183.                                 {
184.                                         // 마우스 왼쪽 버튼과 CONTROL키가 동시에 눌러졌을 때는
185.                                         // 시작점의 좌표 값을 조정할 수 있도록 한다.
186.                                         if (wParam & MK_CONTROL) {
187.                                                 G[0].x = LOWORD (lParam);
188.                                                 G[0].y = HIWORD (lParam);
189.                                         }
190.                                         // 마우스 왼쪽 버튼과 SHIFT키가 동시에 눌러졌을 때는
191.                                         // 첫번 째 벡터값을 조정할 수 있도록 한다.
192.                                         else if(wParam & MK_SHIFT) {
193.                                                 G[2].x = LOWORD (lParam);
194.                                                 G[2].y = HIWORD (lParam);                                               
195.                                         }                                       
196.                                 }
197.  
198.                                 // 마우스 오른쪽 버튼이 클릭되어 있을 시에
199.                                 if (wParam & MK_RBUTTON)
200.                                 {
201.                                         // 마우스 오른쪽 버튼과 CONTROL키가 동시에 눌러졌을 때는
202.                                         // 끝점의 좌표 값을 조정할 수 있도록 한다.
203.                                         if (wParam & MK_CONTROL) {
204.                                                 G[1].x = LOWORD (lParam);
205.                                                 G[1].y = HIWORD (lParam);                                               
206.                                         }
207.                                         // 마우스 오른쪽 버튼과 SHIFT키가 동시에 눌러졌을 때는
208.                                         // 두번 째 벡터값을 조정할 수 있도록 한다.
209.                                         else if(wParam & MK_SHIFT) {
210.                                                 G[3].x = LOWORD (lParam);
211.                                                 G[3].y = HIWORD (lParam);                                               
212.                                         }
213.                                 }
214.  
215.                                 // 검정펜으로 다시 바꾸고 그려준다.
216.                                 SelectObject (hdc, GetStockObject (BLACK_PEN));
217.                                 DrawHermite (hdc, G);
218.                                 ReleaseDC (hWnd, hdc);
219.                         }
220.                        
221.                         break;
222.                 case WM_PAINT:
223.                         hdc = BeginPaint(hWnd, &ps);
224.                         // TODO: Add any drawing code here...
225.  
226.                         DrawHermite(hdc, G);
227.  
228.                         EndPaint(hWnd, &ps);
229.                         break;
230.                 case WM_DESTROY:
231.                         PostQuitMessage(0);
232.                         break;
233.                 default:
234.                         return DefWindowProc(hWnd, message, wParam, lParam);
235.    }
236.    return 0;
237. }
238.  
239. // Mesage handler for about box.
240. LRESULT CALLBACK About(HWND hDlg, UINT message, WPARAM wParam, LPARAM lParam)
241. {
242.         switch (message)
243.         {
244.                 case WM_INITDIALOG:
245.                                 return TRUE;
246.  
247.                 case WM_COMMAND:
248.                         if (LOWORD(wParam) == IDOK || LOWORD(wParam) == IDCANCEL)
249.                         {
250.                                 EndDialog(hDlg, LOWORD(wParam));
251.                                 return TRUE;
252.                         }
253.                         break;
254.         }
255.     return FALSE;
256. }
257.  
258. Apt HermiteSpline(Apt G[4], double t)
259. {
260.         double t2, t3;
261.         Apt Q;
262.  
263.         // 이해하기 쉽기 위해서 t의 제곱, 세제곱의 값을 변수에 저장한다.
264.         t2 = t * t;
265.         t3 = t * t * t;
266.        
267.         /* 시작점이 (x₁, y₁), 끝점이 (x₂, y₂)이고 벡터값이 (x₃, y₃), (x₄, y₄) 일 때
268.            Hermite Spline을 구하기 위한 혼합함수는
269.            x(t) = (2t³- 3t²+ 1)x₁+ (-2t³+ 3t²)x₂+ (t³- 2t²+ t)x₃+ (t³- t²)x₄
270.        y(t) = (2t³- 3t²+ 1)y₁+ (-2t³+ 3t²)y₂+ (t³- 2t²+ t)y₃+ (t³- t²)y₄이다. */
271.         Q.x = ((2 * t3 - 3 * t2 + 1) * G[0].x) + ((-2 * t3 + 3 * t2) * G[1].x)
272.                 + ((t3 - 2 * t2 + t) * G[2].x) + ((t3 - t2) * G[3].x);
273.         Q.y = ((2 * t3 - 3 * t2 + 1) * G[0].y) + ((-2 * t3 + 3 * t2) * G[1].y)
274.                 + ((t3 - 2 * t2 + t) * G[2].y) + ((t3 - t2) * G[3].y);
275.  
276.         return Q;
277. }
278.  
279. // Hermite Spline을 그리는 함수
280. void DrawHermite(HDC hdc, Apt G[4])
281. {
282.         double t;
283.         Apt Q;
284.  
285.         // t가 0부터 1까지 변하는 값에 대해 곡선이 그려진다.
286.         for(t=0; t<=1.0; t+=0.01) {
287.                 Q = HermiteSpline(G, t);               
288.                 if(t==0) {
289.                         MoveToEx (hdc, (int) Q.x, (int) Q.y, NULL);
290.                 }
291.                 else {
292.                         LineTo (hdc, (int) Q.x, (int) Q.y);
293.                 }
294.         }
295. }
     
     
     

그리고 참고했던 비주얼베이직으로 된 소스입니다.

Hermite.zip

Option Explicit

Dim u As Double
Dim x As Double
Dim y As Double
Dim x0 As Double, x1 As Double, y0 As Double, y1 As Double
Dim x0_p As Double, x1_p As Double, y0_p As Double, y1_p As Double
Dim k As Double
Private Sub Command1_Click()
setScaleMode
k = 3
x0 = 1: y0 = 1
x1 = 3: y1 = 1
x0_p = 1: y0_p = k
x1_p = 1: y1_p = k

u = 0
Do While u < 1
   
    'x = x0 * f1(u) + x1 * f2(u) + x0_p * f3(u) + x1_p * f4(u)
    'y = y0 * f1(u) + y1 * f2(u) + y0_p * f3(u) + y1_p * f4(u)
   
    ''x = 138 * f1(u) + 409 * f2(u) + x0_p * f3(u) + x1_p * f4(u)
    ''y = 322 * f1(u) + 248 * f2(u) + y0_p * f3(u) + y1_p * f4(u)
   
    'x = 1 * f1(u) + 4 * f2(u) + x0_p * f3(u) + x1_p * f4(u)
    'y = 3 * f1(u) + 2 * f2(u) + y0_p * f3(u) + y1_p * f4(u)
   
    x = 1 * f1(u) + 3 * f2(u) + x0_p * f3(u) + x1_p * f4(u)
    y = 1 * f1(u) + 1 * f2(u) + y0_p * f3(u) + y1_p * f4(u)
   
    Debug.Print "u:" + CStr(u), "x:" + CStr(x), "y:" + CStr(y)
   
    u = u + 0.01
   
    If x > 5 Then Exit Do
    Picture1.PSet (x, y)
Loop
End Sub
Private Function f1(u) As Double
f1 = 1 - 3 * (u ^ 2) + 2 * (u ^ 3)
End Function
Private Function f2(u) As Double
f2 = 3 * (u ^ 2) - 2 * (u ^ 3)
End Function
Private Function f3(u) As Double
f3 = u - 2 * (u ^ 2) + (u ^ 3)
End Function
Private Function f4(u) As Double
f4 = -1 * (u ^ 2) + (u ^ 3)
End Function
Private Sub Command2_Click()
setScaleMode
Picture1.Line (x0, y0)-(x0 + x0_p, y0 + y0_p)
Picture1.Line -(x1 - x1_p, y1 - y1_p)
Picture1.Line -(x1, y1)

End Sub
Private Sub setScaleMode()
With Picture1
    .AutoRedraw = True
    .ScaleWidth = 6
    .ScaleHeight = 6
    .ScaleMode = 0
    .ScaleLeft = 0
    .ScaleTop = 0
End With
End Sub
Private Sub Form_Load()
End Sub