Difference between revisions of "Dev:Linking to Blines - Equations"

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(One spline)
m (Main equations: t->u (typo))
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* <math>(x,y) = (1-u)^3 (x_1,y_1) + 3 u(1-u)^2  (x^t_1,y^t_1) + 3 u^2 (1-u)  (x^t_2,y^t_2) + u^3 (x_2,y_2)</math> - bline point
 
* <math>(x,y) = (1-u)^3 (x_1,y_1) + 3 u(1-u)^2  (x^t_1,y^t_1) + 3 u^2 (1-u)  (x^t_2,y^t_2) + u^3 (x_2,y_2)</math> - bline point
* <math>(x^t_L,y^t_L) = (1-t)^2 (x_1,y_1) + 2t(1-t)(x^t_1,y^t_1) + t^2(x^t_2,y^t_2)</math> - yellow tangent of bline point
+
* <math>(x^t_L,y^t_L) = (1-u)^2 (x_1,y_1) + 2u(1-u)(x^t_1,y^t_1) + u^2(x^t_2,y^t_2)</math> - yellow tangent of bline point
* <math>(x^t_N,y^t_N) = (1-t)^2 (x^t_1,y^t_1) + 2t(1-t)(x^t_2,y^t_2) + t^2(x_2,y_2)</math> - red tangent of bline point
+
* <math>(x^t_N,y^t_N) = (1-u)^2 (x^t_1,y^t_1) + 2u(1-u)(x^t_2,y^t_2) + u^2(x_2,y_2)</math> - red tangent of bline point
  
 
== One spline ==
 
== One spline ==

Revision as of 19:40, 7 March 2008

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)} - points of bline vertex
  • 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_1,y^t_1), (x^t_2,y^t_2)} - tangent points
  • 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^t_L,y^t_L), (x^t_N,y^t_N)} - tangents of current 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^t_1,y^t_1) + 3 u^2 (1-u) (x^t_2,y^t_2) + u^3 (x_2,y_2)} - 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^t_L,y^t_L) = (1-u)^2 (x_1,y_1) + 2u(1-u)(x^t_1,y^t_1) + u^2(x^t_2,y^t_2)} - yellow tangent of 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^t_N,y^t_N) = (1-u)^2 (x^t_1,y^t_1) + 2u(1-u)(x^t_2,y^t_2) + u^2(x_2,y_2)} - red tangent of bline point

One spline

Case: Bline A. A2 linked to A (with tangent).

A2 with its tangent belongs to A, so:

  • 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_A2 = (1-u)^3 x_A1 + 3u(1-u)^2 x^t_A1 + 3u^2(1-u) x^t_A2 + u^3 x_A2}
  • 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_A2 = (1-u)^2 x_A1 + 2u(1-u) x^t_A1 + t^2 x^t_A2}

Two splines

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