Dev:Linking to Blines - Equations - Revision history

Zelgadis: moved Linking to Blines - Equations to Dev:Linking to Blines - Equations

2010-02-20T11:03:55Z

moved Linking to Blines - Equations to Dev:Linking to Blines - Equations

Zelgadis: Text replace - ']]' to '}}'

2009-09-08T06:52:08Z

Text replace - ']]' to '}}'

Zelgadis: Text replace - '[[' to '{{l|'

2009-09-08T05:11:51Z

Text replace - '[[' to '{{l|'

Zelgadis: /* General model */ algorythm fixes

2008-03-28T16:00:58Z

‎General model: algorythm fixes

Zelgadis: /* Two splines */ Case 1

2008-03-28T15:46:45Z

‎Two splines: Case 1

Zelgadis: /* One spline */ no inner headers

2008-03-28T15:19:24Z

‎One spline: no inner headers

Zelgadis: /* One spline */ restructured

2008-03-28T15:09:57Z

‎One spline: restructured

Zelgadis: /* One spline */

2008-03-28T15:05:41Z

‎One spline

Zelgadis: /* General model */

2008-03-28T14:43:26Z

‎General model

Zelgadis: /* General model */ coefficient fixes

2008-03-28T14:41:19Z

‎General model: coefficient fixes

@@ Line 37: / Line 37: @@
 </pre>
-: Can't understand, why *3? --{{l|User:Zelgadis|Zelgadis]] 05:31, 8 March 2008 (EST)
+: Can't understand, why *3? --{{l|User:Zelgadis|Zelgadis}} 05:31, 8 March 2008 (EST)
 For x:
@@ Line 73: / Line 73: @@
 == General model ==
-The problem: see {{l|Linking_to_Blines#Creating_loops]]
+The problem: see {{l|Linking_to_Blines#Creating_loops}}
 So we need to determine position of vertex engaged in loop if she attachet ot some position of other bline.

@@ Line 37: / Line 37: @@
 </pre>
-: Can't understand, why *3? --[[User:Zelgadis|Zelgadis]] 05:31, 8 March 2008 (EST)
+: Can't understand, why *3? --{{l|User:Zelgadis|Zelgadis]] 05:31, 8 March 2008 (EST)
 For x:
@@ Line 73: / Line 73: @@
 == General model ==
-The problem: see [[Linking_to_Blines#Creating_loops]]
+The problem: see {{l|Linking_to_Blines#Creating_loops]]
 So we need to determine position of vertex engaged in loop if she attachet ot some position of other bline.

@@ Line 112: / Line 112: @@
 B matrix is zero-filled 4Nx4N matrix which we modifying in folowing way (rows and columns in matrix are numbered from 1):
-* if first vertex of bline [i] is linked to bline [j] then element at ( (i-1)*4+1 , (j-1)*4+1 ) = 1
+* if first vertex of bline [i] is linked to bline [j] then element at ( (i-1)*4+1 , (i-1)*4+1 ) = 1
-* if tangent of first vertex of bline [i] is linked to bline [j] then element at ( (i-1)*4+2 , (j-1)*4+2 ) = 1
+* if tangent of first vertex of bline [i] is linked to bline [j] then element at ( (i-1)*4+2 , (i-1)*4+2 ) = 1
-* if tangent of second vertex of bline [i] is linked to bline [j] then element at ( (i-1)*4+3 , (j-1)*4+3 ) = 1
+* if tangent of second vertex of bline [i] is linked to bline [j] then element at ( (i-1)*4+3 , (i-1)*4+3 ) = 1
-* if second vertex of bline [i] is linked to bline [j] then element at ( (i-1)*4+4 , (j-1)*4+4 ) = 1
+* if second vertex of bline [i] is linked to bline [j] then element at ( (i-1)*4+4 , (i-1)*4+4 ) = 1
 A matrix is modified E matrix. <math>E = \begin{pmatrix} 1 & 0 & 0& ... & 0 \\ 0 & 1 & 0& ... & 0 \\ 0 & 0 & 1 & ... & 0 \\...&...&...&...&... \\ 0 & 0 & 0& ... & 1 \\ \end{pmatrix}</math>
 It is modified in such way:
-* if first vertex k of bline [i] is linked to bline [j] then elements in line (j-1)*4+1 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_1(u^{[i]}_k), c^{[j]}_2(u^{[i]}_k), c^{[j]}_3(u^{[i]}_k), c^{[j]}_4(u^{[i]}_k)</math>
+* if first vertex of bline [i] is linked to bline [j] then element at position ( (i-1)*4+1, (i-1)*4+1 ) replaced with zero and elements in line (i-1)*4+1 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_1(u^{[i]}_1), c^{[j]}_2(u^{[i]}_1), c^{[j]}_3(u^{[i]}_1), c^{[j]}_4(u^{[i]}_1)</math>
-* if tangent of first vertex of bline [i] is linked to bline [j] then elements in line (j-1)*4+2 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_{t1}(u^{[i]}_k), c^{[j]}_{t2}(u^{[i]}_k), c^{[j]}_{t3}(u^{[i]}_k), c^{[j]}_{t4}(u^{[i]}_k)</math>
+* if tangent of first vertex of bline [i] is linked to bline [j] then element at position ( (i-1)*4+2, (i-1)*4+2 ) replaced with zero and elements in line (i-1)*4+2 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_{t1}(u^{[i]}_1), c^{[j]}_{t2}(u^{[i]}_1), c^{[j]}_{t3}(u^{[i]}_1), c^{[j]}_{t4}(u^{[i]}_1)</math>
-* if tangent of second vertex of bline [i] is linked to bline [j] then elements in line (j-1)*4+3 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_{t1}(u^{[i]}_k), c^{[j]}_{t2}(u^{[i]}_k), c^{[j]}_{t3}(u^{[i]}_k), c^{[j]}_{t4}(u^{[i]}_k)</math>
+* if tangent of second vertex of bline [i] is linked to bline [j] then element at position ( (i-1)*4+3, (i-1)*4+3 ) replaced with zero and elements in line (i-1)*4+3 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_{t1}(u^{[i]}_2), c^{[j]}_{t2}(u^{[i]}_2), c^{[j]}_{t3}(u^{[i]}_2), c^{[j]}_{t4}(u^{[i]}_2)</math>
-* if second vertex of bline [i] is linked to bline [j] then elements in line (j-1)*4+4 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_1(u^{[i]}_k), c^{[j]}_2(u^{[i]}_k), c^{[j]}_3(u^{[i]}_k), c^{[j]}_4(u^{[i]}_k)</math>
+* if second vertex of bline [i] is linked to bline [j] then element at position ( (i-1)*4+4, (i-1)*4+4 ) replaced with zero and elements in line (i-1)*4+4 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_1(u^{[i]}_2), c^{[j]}_2(u^{[i]}_2), c^{[j]}_3(u^{[i]}_2), c^{[j]}_4(u^{[i]}_2)</math>
 where <math>u^{[i]}_k</math> is a position of vertex/tangent k of bline [i] on bline which it's linked to.

@@ Line 187: / Line 187: @@
 === Two splines ===
-If we got circular reference with two blines, then we must solve equation with matrix 8x8.
+'''Case 1: Given Blines [1] and [2]. '''
 == Conclusion ==

@@ Line 133: / Line 133: @@
 === One spline ===
-====Case 1: Given Bline [1]. Vertex 2 of bline [1] linked to bline [1].====
+'''Case 1: Given Bline [1]. Vertex 2 of bline [1] linked to bline [1].'''
 http://zelgadis.profusehost.net/files/synfig/circular_references/cr05.jpg
@@ Line 163: / Line 163: @@
-==== Case 2: Given Bline [1]. Vertex 2 of bline [1] linked to bline [1] with it's tangent. ====
+''' Case 2: Given Bline [1]. Vertex 2 of bline [1] linked to bline [1] with it's tangent. '''
-−
 http://zelgadis.profusehost.net/files/synfig/circular_references/cr05.jpg

← Older revision	Revision as of 11:03, 20 February 2010
(No difference)

@@ Line 133: / Line 133: @@
 === One spline ===
-Case 1: Bline A. A2 linked to A with tangent.
+http://zelgadis.profusehost.net/files/synfig/circular_references/cr05.jpg
 For x coordinates we have following system of equations:
-* <math>x_{A1} = c_1</math>
+* <math>x^{[1]}_1 = X^{[1]}_1</math>
-* <math>x^{t2}_{A1} = c_2</math>
+* <math>\Delta x^{[1]}_{t1} = \Delta X^{[1]}_{t1}</math>
-* <math>\Delta x^{t1}_{A2} = -2u(1-u)x_{A1} + (1-u^2)\Delta x^{t2}_{A1} + u(2-u)\Delta x^{t1}_{A2} + u(1-u)x_{A2}</math>
+* <math>\Delta x^{[1]}_{t2} = \Delta X^{[1]}_{t2}</math>
-* <math>x_{A2} = (1-(3u^2-2u^3)) x_{A1} + 3u(1-u)^2 \Delta x ^{t2}_{A1} - 3u^2(1-u) \Delta x^{t1}_{A2} + (3u^2-2u^3) x_{A2}</math>
+* <math>x^{[1]}_{2} = c^{[1]}_1(u^{[1]}_2) x^{[1]}_1 + c^{[1]}_2(u^{[1]}_2) \Delta x^{[1]}_{t1} + c^{[1]}_3(u^{[1]}_2) \Delta x^{[1]}_{t2} + c^{[1]}_4(u^{[1]}_2) x^{[1]}_2</math> - because <math>x^{[1]}_2</math> belongs to bline [1]
 Transform:
+* <math>x^{[1]}_1 = X^{[1]}_1</math>
-* <math>x_{A1} = c_1</math>
+* <math>\Delta x^{[1]}_{t1} = \Delta X^{[1]}_{t1}</math>
-* <math>x^{t2}_{A1} = c_2 </math>
+* <math>\Delta x^{[1]}_{t2} = \Delta X^{[1]}_{t2}</math>
-* <math>-2u(1-u)x_{A1} + (1-u^2)\Delta x^{t2}_{A1} + (u(2-u)-1)\Delta x^{t1}_{A2} + u(1-u)x_{A2} = 0</math>
+* <math>c^{[1]}_1(u^{[1]}_2) x^{[1]}_1 + c^{[1]}_2(u^{[1]}_2) \Delta x^{[1]}_{t1} + c^{[1]}_3(u^{[1]}_2) \Delta x^{[1]}_{t2} + (c^{[1]}_4(u^{[1]}_2) - 1) x^{[1]}_2 = 0</math>
-* <math>(1-(3u^2-2u^3)) x_{A1} + 3u(1-u)^2 \Delta x ^{t2}_{A1} - 3u^2(1-u) \Delta x^{t1}_{A2} + (3u^2-2u^3-1) x_{A2} = 0</math>
 So, we must solve the 4x4 matrix equation:
-<math>( \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 &0 \\ (-2u(1-u) & (1-u^2) & u(2-u) & u(1-u) \\ (1-(3u^2-2u^3)) & 3u(1-u)^2 & -3u^2(1-u) & (3u^2-2u^3)  \end{pmatrix} -\begin{pmatrix} 0&0&0&0 \\ 0&0&0&0\\ 0&0&1&0 \\ 0&0&0&1\end{pmatrix})  \cdot \begin{pmatrix} x_{A1} \\ x^{t2}_{A1} \\ x^{t1}_{A2} \\ x_{A2} \end{pmatrix} = \begin{pmatrix} c_1 \\ c_2 \\ 0 \\ 0 \end{pmatrix}</math>
+<math>( \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 &0 \\ 0 & 0 & 1 & 0 \\ c^{[1]}_1(u^{[1]}_2) & c^{[1]}_2(u^{[1]}_2) & c^{[1]}_3(u^{[1]}_2) & c^{[1]}_4(u^{[1]}_2)  \end{pmatrix} -\begin{pmatrix} 0&0&0&0 \\ 0&0&0&0\\ 0&0&0&0 \\ 0&0&0&1\end{pmatrix})  \cdot \begin{pmatrix} x^{[1]}_1 \\ \Delta x^{[1]}_{t1} \\ \Delta x^{[1]}_{t2} \\ x^{[1]}_{2} \end{pmatrix} = \begin{pmatrix} X^{[1]}_1 \\ \Delta X^{[1]}_{t1} \\ \Delta X^{[1]}_{t2} \\ 0 \end{pmatrix}</math>
 dooglus, can you check how this formula works in code? Something like:
-* Create bline with 2 points - A1,A2
+* Create bline with 2 points - 1 and 2
-* Select point A2, right click on bline -> "Link to bline"
+* Select point 2, right click on bline -> "Link to bline"
-* Place (not link!) at the position calculated by this formula. If we'll have A2 on bline after that then it's ok for now, if not - something wrong.
+* Place (not link!) point 2 at the position calculated by this formula. If we'll have point 2 on bline after that then it's ok for now, if not - something wrong.
-I know we need a general formula but if this won't work then I definitely not right.
-Case 2: Bline A. A2 linked to A without tangent.
+'''Case 2: Given Bline [1]. Vertex 2 of bline [1] linked to bline [1] with it's tangent.'''
-The 4th line just replaced with:
+For x coordinates we have following system of equations:
-* <math>\Delta x^{t1}_{A2} = c_3</math>
 === Two splines ===

@@ Line 125: / Line 125: @@
 * if second vertex of bline [i] is linked to bline [j] then elements in line (j-1)*4+4 at positions (j-1)*4+1, (j-1)*4+2, (j-1)*4+3, (j-1)*4+4 are replaced with <math>c^{[j]}_1(u^{[i]}_k), c^{[j]}_2(u^{[i]}_k), c^{[j]}_3(u^{[i]}_k), c^{[j]}_4(u^{[i]}_k)</math>
-where <math>u^{[i]}_k</math> is a position of vertex/tangent k of bline [i] on bline [j] to which it's linked.
+where <math>u^{[i]}_k</math> is a position of vertex/tangent k of bline [i] on bline which it's linked to.
 == Examples ==