Dev:Linking to Blines - Equations

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Revision as of 18:00, 28 March 2008 by Zelgadis (Talk | contribs) (→‎General model: algorythm fixes)

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WARNING: This text is under HEAVY development. Please be patient.

Main equations

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x_1,y_1), (x_2,y_2)} - vertices, defining bline segment
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x,y) } - current bline point
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x_{t1},y_{t1}), (x_{t2},y_{t2})} - tangent points defining bline segment
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x_{t},y_{t})} - tangent coordinates for current bline point
  • u - Amount of current segment, [0,1]
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x,y) = (1-u)^3 (x_1,y_1) + 3 u(1-u)^2 (x_{t1},y_{t1}) + 3 u^2 (1-u) (x_{t2},y_{t2}) + u^3 (x_2,y_2)} - bline point

Relative tangents

O'kay, I know, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x_{t1},y_{t1}), (x_{t2},y_{t2})} defining absolute position of tangents, but in synfig we have their coordinates relative to vertex. Moreover, coordinates of yellow tangent are inverted.

Let's say:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (\Delta x_{t1},\Delta y_{t1})} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (\Delta x_{t2},\Delta y_{t2})} - relative coordinates of tangents

Then:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x_{t1},y_{t1}) = (x,y) + (\Delta x^{t1},\Delta y^{t1})}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x_{t2},y_{t2}) = (x,y) - (\Delta x^{t2},\Delta y^{t2})}

Make substitution:

  • Bline point: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x,y) = (1-(3u^2-2u^3)) (x_1,y_1) + 3u(1-u)^2 (\Delta x_{t1},\Delta y_{t1}) - 3u^2(1-u) (\Delta x_{t2}, \Delta y_{t2}) + (3u^2-2u^3) (x_2,y_2)}

Tangent coordinates

<dooglus> look at etl/_calculus.h
<dooglus> class derivative<hermite<T> >
<dooglus>   T a = func[0], b = func[1], c = func[2], d = func[3];
<dooglus>   typename hermite<T>::argument_type y(1-x);
<dooglus>   return ((b-a)*y*y + (c-b)*x*y*2 + (d-c)*x*x) * 3;
Can't understand, why *3? --Zelgadis 05:31, 8 March 2008 (EST)

For x:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x_{t} = (1-u)^2 \Delta x_{t1} + 2u(1-u) ((x_2 - \Delta x_{t2}) - (x_1 + \Delta x_{t1})) + u^2 \Delta x_{t2} = } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = -2u(1-u)x_1 + (1-u)(1-3u)\Delta x_{t1} + (3u^2 - 2u)\Delta x_{t2} + 2u(1-u)x_2}

Checking formulas

Well, I really wasn't sure if this formulas correct, that's why I did a little research and made some graphics and calculations. They are available here.

As result I got some final corrections to formulas (for x only):

  • Bline point: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x = (1-(3u^2-2u^3)) x_1 + u(1-u)^2 \Delta x_{t1} - u^2(1-u) \Delta x_{t2} + (3u^2-2u^3) x_2}
  • Bline tangent: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x_{t} = -6u(1-u)x_1 + (1-u)(1-3u)\Delta x_{t1} + (3u^2 - 2u)\Delta x_{t2} + 6u(1-u)x_2}

Zelgadis, you wrong!

O'kay I understand what equations above could be wrong. But no one will argue what equations for bline point and tangents generally have form:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x,y) = c_1(u) (x_1,y_1) + c_2(u) (x_{t1},y_{t1}) + c_3(u) (x_{t2},y_{t2}) + c_4(u) (x_2,y_2)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (\Delta x_{t},\Delta y_{t}) = c_{t1}(u) (x_1,y_1) + c_{t2}(u) (\Delta x_{t1}, \Delta y_{t1}) + c_{t3} (\Delta x_{t2}, \Delta y_{t2}) + c_{t4}(u) (x_2,y_2)}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_i(u)} - some function from 'u' argument

So I will use this notation further.

For now we assume what:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_1(u)=1-(3u^2-2u^3)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_2(u)=u(1-u)^2}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_3(u)=- u^2(1-u)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_4(u)=3u^2-2u^3}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_{t1}(u)=-6u(1-u)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_{t2}(u)=(1-u)(1-3u)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_{t3}(u)=3u^2 - 2u}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c_{t4}(u)=6u(1-u)}

General model

The problem: see 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.

NOTICE: We need such recalculation only for cases when in loop included only 1 segment from each bline. Like this:
cr02.jpg cr04.jpg
We don't need to do recalculation for cases like those:
attach-loop-simple.png

So, we have N blines (N bline segments) engaged in a loop.

To determine the x coordinate of vertex what finalizes the loop we need to solve matrix equation having form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (A - B)*\alpha = \beta}

NOTE: Solution formulas for y coordinate will be the same.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle A, B} - matrices 4N x 4N
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \alpha, \beta} - 4N column vectors
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \alpha} - is unknown quantity

Each line in matrix or column vector corresponding to particular vertex or it's tangent.

After solving system we will find column vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \alpha} which have form: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (x^{[1]}_1,\Delta x^{[1]}_{t1},\Delta x^{[1]}_{t2},x^{[1]}_2, x^{[2]}_1,\Delta x^{[2]}_{t1},\Delta x^{[2]}_{t2},x^{[2]}_2, ... ,x^{[N]}_1,\Delta x^{[N]}_{t1},\Delta x^{[N]}_{t2},x^{[N]}_2)} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[i]}_j} - point j of bline [i] and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[i]}_tj} tangent of point j of bline i. From this vector we could retrieve x coordinate of desired vertex or tangent.

So:

  • the first line of each matrix or column vector corresponding to first vertex of first bline,
  • second line - corresponding to tangent of first vertex of first bline,
  • third line - to tangent of second vertex of first bline,
  • fourth - second vertex of first bline,
  • fifth - first vertex of second bline,
  • sixth - tangent of first vertex of second bline,
  • ...and so on...
NOTE: Talking about bline we are talking about single (!) segment of bline which engaged in loop. That's why I not specify which tangent (yellow or red) we using - it's always tangent for current segment.

Column vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \beta} have following structure. If position of vertex/tangent corresponding to vector element is "static" (i.e. it's not linked any other bline segment of the loop) then vector element is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \beta_i} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \beta_i} is current x coordinate of this vector/tangent. If vertex/tangent is not "static" then corresponding vector element is zero.

NOTE: When we calling vertex position "static" it's not means what vertex not linked to anything. The vertex could be linked to bline and still considered as "static" if that bline is not engaged into loop what we processing.

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 , (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 , (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 , (i-1)*4+3 ) = 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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle E = \begin{pmatrix} 1 & 0 & 0& ... & 0 \\ 0 & 1 & 0& ... & 0 \\ 0 & 0 & 1 & ... & 0 \\...&...&...&...&... \\ 0 & 0 & 0& ... & 1 \\ \end{pmatrix}}

It is modified in such way:

  • 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c^{[j]}_1(u^{[i]}_1), c^{[j]}_2(u^{[i]}_1), c^{[j]}_3(u^{[i]}_1), c^{[j]}_4(u^{[i]}_1)}
  • 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c^{[j]}_{t1}(u^{[i]}_1), c^{[j]}_{t2}(u^{[i]}_1), c^{[j]}_{t3}(u^{[i]}_1), c^{[j]}_{t4}(u^{[i]}_1)}
  • 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c^{[j]}_{t1}(u^{[i]}_2), c^{[j]}_{t2}(u^{[i]}_2), c^{[j]}_{t3}(u^{[i]}_2), c^{[j]}_{t4}(u^{[i]}_2)}
  • 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c^{[j]}_1(u^{[i]}_2), c^{[j]}_2(u^{[i]}_2), c^{[j]}_3(u^{[i]}_2), c^{[j]}_4(u^{[i]}_2)}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle u^{[i]}_k} is a position of vertex/tangent k of bline [i] on bline which it's linked to.

Examples

Ok, I sure you guys are wondering how is this work. (I personally wondering IF this works or not :D). Let's view some examples.

One spline

Case 1: Given Bline [1]. Vertex 2 of bline [1] linked to bline [1].

cr05.jpg

For x coordinates we have following system of equations:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_1 = X^{[1]}_1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t1} = \Delta X^{[1]}_{t1}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t2} = \Delta X^{[1]}_{t2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 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} - because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_2} belongs to bline [1]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle X^{[i]}_j} is a constant value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[i]}_j} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta X^{[i]}_{j}} is a constant value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[i]}_{j}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle u^{[1]}_2} is a position of vertex 2 of bline [1] on the bline which it's linked to (i.e. bline [1]).

Transform:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_1 = X^{[1]}_1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t1} = \Delta X^{[1]}_{t1}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t2} = \Delta X^{[1]}_{t2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 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}

So, we must solve the 4x4 matrix equation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle ( \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}}


dooglus, can you check how this formula works in code? Something like:

  • Create bline with 2 points - 1 and 2
  • Select point 2, right click on bline -> "Link to bline"
  • 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.


Case 2: Given Bline [1]. Vertex 2 of bline [1] linked to bline [1] with it's tangent. cr05.jpg

For x coordinates we have following system of equations:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_1 = X^{[1]}_1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t1} = \Delta X^{[1]}_{t1}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t2} = c^{[1]}_{t1}(u^{[1]}_2) x^{[1]}_1 + c^{[1]}_{t2}(u^{[1]}_2) \Delta x^{[1]}_{t1} + c^{[1]}_{t3}(u^{[1]}_2) \Delta x^{[1]}_{t2} + c^{[1]}_{t4}(u^{[1]}_2) x^{[1]}_2} - because tangent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t2}} linked to bline [1]
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 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} - because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_2} belongs to bline [1]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle X^{[i]}_j} is a constant value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[i]}_j} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta X^{[i]}_{j}} is a constant value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[i]}_{j}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle u^{[1]}_2} is a position of vertex 2 of bline [1] on the bline which it's linked to (i.e. bline [1]).

Transform:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_1 = X^{[1]}_1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t1} = \Delta X^{[1]}_{t1}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t2} = c^{[1]}_{t1}(u^{[1]}_2) x^{[1]}_1 + c^{[1]}_{t2}(u^{[1]}_2) \Delta x^{[1]}_{t1} + c^{[1]}_{t3}(u^{[1]}_2) \Delta x^{[1]}_{t2} + (c^{[1]}_{t4}(u^{[1]}_2) - 1) x^{[1]}_2}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 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) -1) \Delta x^{[1]}_{t2} + c^{[1]}_4(u^{[1]}_2) x^{[1]}_2 = 0}

So, we must solve the 4x4 matrix equation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle ( \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 &0 \\ c^{[1]}_{t1}(u^{[1]}_2) & c^{[1]}_{t2}(u^{[1]}_2) & c^{[1]}_{t3}(u^{[1]}_2) & c^{[1]}_{t4}(u^{[1]}_2) \\ 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&1&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} \\ 0 \\ 0 \end{pmatrix}}

Two splines

Case 1: Given Blines [1] and [2].

Vertex 1 of bline [1] linked to bline [2].

Vertex 2 of bline [2] linked to bline [1].

cr07.jpg

For x coordinates we have following system of equations:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_1 = c^{[2]}_1(u^{[1]}_1) x^{[2]}_1 + c^{[2]}_2(u^{[1]}_1) \Delta x^{[2]}_{t1} + c^{[2]}_3(u^{[1]}_1) \Delta x^{[2]}_{t2} + c^{[2]}_4(u^{[1]}_1) x^{[2]}_2} - because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_1} belongs to bline [2]
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t1} = \Delta X^{[1]}_{t1}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t2} = \Delta X^{[1]}_{t2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_2 = X^{[1]}_2}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[2]}_1 = X^{[2]}_1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[2]}_{t1} = \Delta X^{[2]}_{t1}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[2]}_{t2} = \Delta X^{[2]}_{t2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[2]}_{2} = c^{[1]}_1(u^{[2]}_2) x^{[1]}_1 + c^{[1]}_2(u^{[2]}_2) \Delta x^{[1]}_{t1} + c^{[1]}_3(u^{[2]}_2) \Delta x^{[1]}_{t2} + c^{[1]}_4(u^{[2]}_2) x^{[1]}_2} - because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[2]}_2} belongs to bline [1]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle X^{[i]}_j} is a constant value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[i]}_j} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta X^{[i]}_{j}} is a constant value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[i]}_{j}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle u^{[2]}_2} is a position of vertex 2 of bline [2] on the bline which it's linked to (i.e. bline [1]). Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle u^{[1]}_1} is a position of vertex 1 of bline [1] on the bline which it's linked to (i.e. bline [2]).

Transform:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle -x^{[1]}_1 + c^{[2]}_1(u^{[1]}_1) x^{[2]}_1 + c^{[2]}_2(u^{[1]}_1) \Delta x^{[2]}_{t1} + c^{[2]}_3(u^{[1]}_1) \Delta x^{[2]}_{t2} + c^{[2]}_4(u^{[1]}_1) x^{[2]}_2 = 0} - because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_1} belongs to bline [2]
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t1} = \Delta X^{[1]}_{t1}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[1]}_{t2} = \Delta X^{[1]}_{t2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[1]}_2 = X^{[1]}_2}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[2]}_1 = X^{[2]}_1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[2]}_{t1} = \Delta X^{[2]}_{t1}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Delta x^{[2]}_{t2} = \Delta X^{[2]}_{t2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle c^{[1]}_1(u^{[2]}_2) x^{[1]}_1 + c^{[1]}_2(u^{[2]}_2) \Delta x^{[1]}_{t1} + c^{[1]}_3(u^{[2]}_2) \Delta x^{[1]}_{t2} + c^{[1]}_4(u^{[2]}_2) x^{[1]}_2 - x^{[2]}_{2} = 0} - because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x^{[2]}_2} belongs to bline [1]


So, we must solve the 8x8 matrix equation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle ( \begin{pmatrix} 0 & 0 & 0 & 0 & c^{[2]}_1(u^{[1]}_1) & c^{[2]}_2(u^{[1]}_1) & c^{[2]}_3(u^{[1]}_1) & c^{[2]}_4(u^{[1]}_1) \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ c^{[1]}_1(u^{[2]}_2) & c^{[1]}_2(u^{[2]}_2) & c^{[1]}_3(u^{[2]}_2) & c^{[1]}_4(u^{[2]}_2) & 0 & 0 & 0 & 0 \end{pmatrix} - \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 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} \\ x^{[2]}_1 \\ \Delta x^{[2]}_{t1} \\ \Delta x^{[2]}_{t2} \\ x^{[2]}_{2} \end{pmatrix} = \begin{pmatrix} 0 \\ \Delta X^{[1]}_{t1} \\ \Delta X^{[1]}_{t2} \\ X^{[1]}_{2} \\ X^{[2]}_1 \\ \Delta X^{[2]}_{t1} \\ \Delta X^{[2]}_{t2} \\ 0 \end{pmatrix}}

Conclusion

Maybe formulas are not correct, but I think you've got the idea: For N Bline segments in loop we have 4N equations. If vertex is linked then we got appropriate equation, depending on what is linked. If vertex is not linked - just assigning constants.

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