Dev:Linking to Blines - Equations

From Synfig Studio :: Documentation
Revision as of 19:47, 7 March 2008 by Zelgadis (Talk | contribs) (One spline: calculations and typos)

Jump to: navigation, search

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}}

Let's find 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}} 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 x^t_{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_{A2} = \frac{(1-u)^3 x_{A1} + 3u(1-u)^2 x^t_{A1} + 3u^2(1-u) x^t_{A2} }{1 - u^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 x^t_{A2} = \frac{(1-u)^2 x_{A1} + 2u(1-u) x^t_{A1}} {1- t^2 }}

Two splines

...