# Just Launched: You can now import projects and releases from Google Code onto SourceForge

We are excited to release new functionality to enable a 1-click import from Google Code onto the Allura platform on SourceForge. You can import tickets, wikis, source, releases, and more with a few simple steps.

## pyx-checkins

 [PyX-checkins] SF.net SVN: pyx: [2622] trunk/pyx/design/beziers.tex From: - 2006-04-27 12:53:17 Revision: 2622 Author: m-schindler Date: 2006-04-27 05:53:08 -0700 (Thu, 27 Apr 2006) ViewCVS: http://svn.sourceforge.net/pyx/?rev=2622&view=rev Log Message: ----------- revert to the previous version Modified Paths: -------------- trunk/pyx/design/beziers.tex Modified: trunk/pyx/design/beziers.tex =================================================================== --- trunk/pyx/design/beziers.tex 2006-04-27 11:39:24 UTC (rev 2621) +++ trunk/pyx/design/beziers.tex 2006-04-27 12:53:08 UTC (rev 2622) @@ -320,252 +320,6 @@ not be valid at all. % >>> -\section{Points with extremal curvature in B\'ezier curves} - -% <<< -Extremal curvatures are found at parameter values~$t$ that are determined by -% -\begin{gather} - 0 = \dot\kappa(t) = [\dot x^2(t) + \dot y^2(t)]^{-\frac{5}{2}}\: p(t) \quad\text{with}\\ - p(t) := (\dot x\dddot y - \dddot x\dot y)(\dot x^2 + \dot y^2) - - 3(\dot x\ddot y - \ddot x\dot y)(\dot x\ddot x + \ddot y\dot y) -\end{gather} -% -which is equivalent to setting the polynom $p(t)$ to zero. -It is more convenient to write this equation as a polynom of $t$ and $(1{-}t)$, -where the corresponding powers always add up to 6, -% -- \begin{aligned} - 0 = p(t) = {} - &A_0 t^0 (1{-}t)^6 - + A_1 t^1 (1{-}t)^5 - + A_2 t^2 (1{-}t)^4 - + A_3 t^3 (1{-}t)^3 \\ - {}+{}&A_4 t^4 (1{-}t)^2 - + A_5 t^5 (1{-}t)^1 - + A_6 t^6 (1{-}t)^0. - \end{aligned} - -% -Using Pascal's triangle we can convert this into an ordinary polynom, -% -$$- \setlength\arraycolsep{2pt} - \begin{array}{rrrrrrrrrrrrrrr} - 0 = p(t) = [& A_0 & & & & & & & & & & & & ] & t^0 \\ - {} + [& 6 A_0 &+& A_1 & & & & & & & & & & ] & t^1 \\ - {} + [& 15 A_0 &+& 5 A_1 &+& A_2 & & & & & & & & ] & t^2 \\ - {} + [& 20 A_0 &+& 10 A_1 &+& 4 A_2 &+& A_3 & & & & & & ] & t^3 \\ - {} + [& 15 A_0 &+& 10 A_1 &+& 6 A_2 &+& 3 A_3 &+& A_4 & & & & ] & t^4 \\ - {} + [& 6 A_0 &+& 5 A_1 &+& 4 A_2 &+& 3 A_3 &+& 2 A_4 &+& A_5 & & ] & t^5 \\ - {} + [& A_0 &+& A_1 &+& A_2 &+& A_3 &+& A_4 &+& A_5 &+& A_6] & t^6 - \end{array} -$$ -% -% Pascal's triangle: <<< -% ^0 ^1 ^2 ^3 ^4 ^5 ^6 -% | 1| 6|15|20|15| 6| 1| = (1-t)^6 -% | 1| 5|10|10| 5| 1| | = (1-t)^5 -% | 1| 4| 6| 4| 1| | | = (1-t)^4 -% | 1| 3| 3| 1| | | | = (1-t)^3 -% | 1| 2| 1| | | | | = (1-t)^2 -% | 1| 1| | | | | | = (1-t)^1 -% | 1| | | | | | | = (1-t)^0 -% -% | 1| 6|15|20|15| 6| 1| = t^0 (1-t)^6 -% | | 1| 5|10|10| 5| 1| = t^1 (1-t)^5 -% | | | 1| 4| 6| 4| 1| = t^2 (1-t)^4 -% | | | | 1| 3| 3| 1| = t^3 (1-t)^3 -% | | | | | 1| 2| 1| = t^4 (1-t)^2 -% | | | | | | 1| 1| = t^5 (1-t)^1 -% | | | | | | | 1| = t^6 (1-t)^0 -% -% ein Polynom, das so geschrieben werden kann, -% p = A0 * t^0 (1-t)^6 -% + A1 * t^1 (1-t)^5 -% + A2 * t^2 (1-t)^4 -% + A3 * t^3 (1-t)^3 -% + A4 * t^4 (1-t)^2 -% + A5 * t^5 (1-t)^1 -% + A6 * t^6 (1-t)^0 -% -% ist als reines Polynom in t -% p = [ 1*A0 ] * t^0 -% [ 6*A0 + 1*A1 ] * t^1 -% [15*A0 + 5*A1 + 1*A2 ] * t^2 -% [20*A0 + 10*A1 + 4*A2 + 1*A3 ] * t^3 -% [15*A0 + 10*A1 + 6*A2 + 3*A3 + 1*A4 ] * t^4 -% [ 6*A0 + 5*A1 + 4*A2 + 3*A3 + 2*A4 + 1*A5 ] * t^5 -% [ 1*A0 + 1*A1 + 1*A2 + 1*A3 + 1*A4 + 1*A5 + 1*A6] * t^6 -% -% >>> - -%% Rechnung fuer die Polynom-Koeffizienten <<< -% -% dx = 3(x1-x0)(1-t)^2 + 3(x3-x2)t^2 + 6 (x2-x1)t(1-t) -% dy = 3(y1-y0)(1-t)^2 + 3(y3-y2)t^2 + 6 (y2-y1)t(1-t) -% -% ddx = 6(x0-2*x1+x2)(1-t) + 6(x1-2*x2+x3)t -% ddy = 6(y0-2*y1+y2)(1-t) + 6(y1-2*y2+y3)t -% -% ----------------------------- -% jetzt multipliziere die einzelnen Terme: -% -% dddy*dx = dddy * ( 3 (x1-x0)(1-t)^2 + 3 (x3-x2)t^2 + 6 (x2-x1)t(1-t) ) -% dddx*dy = dddx * ( 3 (y1-y0)(1-t)^2 + 3 (y3-y2)t^2 + 6 (y2-y1)t(1-t) ) -% -% (dx)^2 = 9 (x1-x0)^2 (1-t)^4 -% + 9 (x3-x2)^2 t^4 -% + 36 (x2-x1)^2 t^2(1-t)^2 -% + 18 (x1-x0)(x3-x2) t^2(1-t)^2 -% + 36 (x1-x0)(x2-x1) t (1-t)^3 -% + 36 (x3-x2)(x2-x1) t^3(1-t) -% -% (dy)^2 = 9 (y1-y0)^2 (1-t)^4 -% + 9 (y3-y2)^2 t^4 -% + 36 (y2-y1)^2 t^2(1-t)^2 -% + 18 (y1-y0)(y3-y2) t^2(1-t)^2 -% + 36 (y1-y0)(y2-y1) t (1-t)^3 -% + 36 (y3-y2)(y2-y1) t^3(1-t) -% -% dx*ddx = 18 (x1-x0)(x0-2*x1+x2) (1-t)^3 -% + 18 (x3-x2)(x1-2*x2+x3) t^3 -% + 18 (x1-x0)(x1-2*x2+x3) t(1-t)^2 -% + 36 (x2-x1)(x0-2*x1+x2) t(1-t)^2 -% + 18 (x3-x2)(x0-2*x1+x2) t^2(1-t) -% + 36 (x2-x1)(x1-2*x2+x3) t^2(1-t) -% -% dy*ddy = 18 (y1-y0)(y0-2*y1+y2) (1-t)^3 -% + 18 (y3-y2)(y1-2*y2+y3) t^3 -% + 18 (y1-y0)(y1-2*y2+y3) t(1-t)^2 -% + 36 (y2-y1)(y0-2*y1+y2) t(1-t)^2 -% + 18 (y3-y2)(y0-2*y1+y2) t^2(1-t) -% + 36 (y2-y1)(y1-2*y2+y3) t^2(1-t) -% -% dx*ddy = 18 (x1-x0)(y0-2*y1+y2) (1-t)^3 -% + 18 (x3-x2)(y1-2*y2+y3) t^3 -% + 18 (x1-x0)(y1-2*y2+y3) t(1-t)^2 -% + 36 (x2-x1)(y0-2*y1+y2) t(1-t)^2 -% + 18 (x3-x2)(y0-2*y1+y2) t^2(1-t) -% + 36 (x2-x1)(y1-2*y2+y3) t^2(1-t) -% -% dy*ddx = 18 (y1-y0)(x0-2*x1+x2) (1-t)^3 -% + 18 (y3-y2)(x1-2*x2+x3) t^3 -% + 18 (y1-y0)(x1-2*x2+x3) t(1-t)^2 -% + 36 (y2-y1)(x0-2*x1+x2) t(1-t)^2 -% + 18 (y3-y2)(x0-2*x1+x2) t^2(1-t) -% + 36 (y2-y1)(x1-2*x2+x3) t^2(1-t) -% -% ----------------------------- -% subtrahiere/addiere: -% -% dddy*dx - dddx*dy = -% + 3 [dddy*(x1-x0) - dddx*(y1-y0)] (1-t)^2 -% + 3 [dddy*(x3-x2) - dddx*(y3-y2)] t^2 -% + 6 [dddy*(x2-x1) - dddx*(y2-y1)] t(1-t) -% -% (dx)^2 + (dy)^2 = -% + 9 [(x1-x0)^2 + (y1-y0)^2 ] (1-t)^4 -% + 9 [(x3-x2)^2 + (y3-y2)^2 ] t^4 -% + 36 [(x2-x1)^2 + (y2-y1)^2 ] t^2(1-t)^2 -% + 18 [(x1-x0)(x3-x2) + (y1-y0)(y3-y2)] t^2(1-t)^2 -% + 36 [(x1-x0)(x2-x1) + (y1-y0)(y2-y1)] t (1-t)^3 -% + 36 [(x3-x2)(x2-x1) + (y3-y2)(y2-y1)] t^3(1-t) -% -% -% dx*ddy - dy*ddx = -% + 18 [(x1-x0)(y0-2*y1+y2) - (y1-y0)(x0-2*x1+x2)] (1-t)^3 -% + 18 [(x3-x2)(y1-2*y2+y3) - (y3-y2)(x1-2*x2+x3)] t^3 -% + 18 [(x1-x0)(y1-2*y2+y3) - (y1-y0)(x1-2*x2+x3)] t (1-t)^2 -% + 36 [(x2-x1)(y0-2*y1+y2) - (y2-y1)(x0-2*x1+x2)] t (1-t)^2 -% + 18 [(x3-x2)(y0-2*y1+y2) - (y3-y2)(x0-2*x1+x2)] t^2(1-t) -% + 36 [(x2-x1)(y1-2*y2+y3) - (y2-y1)(x1-2*x2+x3)] t^2(1-t) -% -% dx*ddx + dy*ddy = -% + 18 [(x1-x0)(x0-2*x1+x2) + (y1-y0)(y0-2*y1+y2)] (1-t)^3 -% + 18 [(x3-x2)(x1-2*x2+x3) + (y3-y2)(y1-2*y2+y3)] t^3 -% + 18 [(x1-x0)(x1-2*x2+x3) + (y1-y0)(y1-2*y2+y3)] t (1-t)^2 -% + 36 [(x2-x1)(x0-2*x1+x2) + (y2-y1)(y0-2*y1+y2)] t (1-t)^2 -% + 18 [(x3-x2)(x0-2*x1+x2) + (y3-y2)(y0-2*y1+y2)] t^2(1-t) -% + 36 [(x2-x1)(x1-2*x2+x3) + (y2-y1)(y1-2*y2+y3)] t^2(1-t) -% -% ----------------------------- -% und multipliziere die grossen Terme miteinander -% -% [dddy*dx - dddx*dy] * [(dx)^2 + (dy)^2] = -% + 27 [dddy*(x1-x0) - dddx*(y1-y0)] [(x1-x0)^2 + (y1-y0)^2 ] t^0(1-t)^6 -% + 108 [dddy*(x1-x0) - dddx*(y1-y0)] [(x1-x0)(x2-x1) + (y1-y0)(y2-y1)] t^1(1-t)^5 -% + 54 [dddy*(x2-x1) - dddx*(y2-y1)] [(x1-x0)^2 + (y1-y0)^2 ] t^1(1-t)^5 -% + 108 [dddy*(x1-x0) - dddx*(y1-y0)] [(x2-x1)^2 + (y2-y1)^2 ] t^2(1-t)^4 -% + 54 [dddy*(x1-x0) - dddx*(y1-y0)] [(x1-x0)(x3-x2) + (y1-y0)(y3-y2)] t^2(1-t)^4 -% + 27 [dddy*(x3-x2) - dddx*(y3-y2)] [(x1-x0)^2 + (y1-y0)^2 ] t^2(1-t)^4 -% + 216 [dddy*(x2-x1) - dddx*(y2-y1)] [(x1-x0)(x2-x1) + (y1-y0)(y2-y1)] t^2(1-t)^4 -% + 108 [dddy*(x1-x0) - dddx*(y1-y0)] [(x3-x2)(x2-x1) + (y3-y2)(y2-y1)] t^3(1-t)^3 -% + 108 [dddy*(x3-x2) - dddx*(y3-y2)] [(x1-x0)(x2-x1) + (y1-y0)(y2-y1)] t^3(1-t)^3 -% + 108 [dddy*(x2-x1) - dddx*(y2-y1)] [(x1-x0)(x3-x2) + (y1-y0)(y3-y2)] t^3(1-t)^3 -% + 216 [dddy*(x2-x1) - dddx*(y2-y1)] [(x2-x1)^2 + (y2-y1)^2 ] t^3(1-t)^3 -% + 27 [dddy*(x1-x0) - dddx*(y1-y0)] [(x3-x2)^2 + (y3-y2)^2 ] t^4(1-t)^2 -% + 108 [dddy*(x3-x2) - dddx*(y3-y2)] [(x2-x1)^2 + (y2-y1)^2 ] t^4(1-t)^2 -% + 216 [dddy*(x2-x1) - dddx*(y2-y1)] [(x3-x2)(x2-x1) + (y3-y2)(y2-y1)] t^4(1-t)^2 -% + 54 [dddy*(x3-x2) - dddx*(y3-y2)] [(x1-x0)(x3-x2) + (y1-y0)(y3-y2)] t^4(1-t)^2 -% + 108 [dddy*(x3-x2) - dddx*(y3-y2)] [(x3-x2)(x2-x1) + (y3-y2)(y2-y1)] t^5(1-t)^1 -% + 54 [dddy*(x2-x1) - dddx*(y2-y1)] [(x3-x2)^2 + (y3-y2)^2 ] t^5(1-t)^1 -% + 27 [dddy*(x3-x2) - dddx*(y3-y2)] [(x3-x2)^2 + (y3-y2)^2 ] t^6(1-t)^0 -% -% [dx*ddy - dy*ddx] * [dx*ddx + dy*ddy] = -% + 18 18 [(x1-x0)(y0-2*y1+y2) - (y1-y0)(x0-2*x1+x2)] [(x1-x0)(x0-2*x1+x2) + (y1-y0)(y0-2*y1+y2)] t^0(1-t)^6 -% + 18 18 [(x1-x0)(y0-2*y1+y2) - (y1-y0)(x0-2*x1+x2)] [(x1-x0)(x1-2*x2+x3) + (y1-y0)(y1-2*y2+y3)] t^1(1-t)^5 -% + 18 36 [(x1-x0)(y0-2*y1+y2) - (y1-y0)(x0-2*x1+x2)] [(x2-x1)(x0-2*x1+x2) + (y2-y1)(y0-2*y1+y2)] t^1(1-t)^5 -% + 18 18 [(x1-x0)(y1-2*y2+y3) - (y1-y0)(x1-2*x2+x3)] [(x1-x0)(x0-2*x1+x2) + (y1-y0)(y0-2*y1+y2)] t^1(1-t)^5 -% + 36 18 [(x2-x1)(y0-2*y1+y2) - (y2-y1)(x0-2*x1+x2)] [(x1-x0)(x0-2*x1+x2) + (y1-y0)(y0-2*y1+y2)] t^1(1-t)^5 -% + 18 18 [(x1-x0)(y1-2*y2+y3) - (y1-y0)(x1-2*x2+x3)] [(x1-x0)(x1-2*x2+x3) + (y1-y0)(y1-2*y2+y3)] t^2(1-t)^4 -% + 18 36 [(x1-x0)(y1-2*y2+y3) - (y1-y0)(x1-2*x2+x3)] [(x2-x1)(x0-2*x1+x2) + (y2-y1)(y0-2*y1+y2)] t^2(1-t)^4 -% + 36 18 [(x2-x1)(y0-2*y1+y2) - (y2-y1)(x0-2*x1+x2)] [(x1-x0)(x1-2*x2+x3) + (y1-y0)(y1-2*y2+y3)] t^2(1-t)^4 -% + 36 36 [(x2-x1)(y0-2*y1+y2) - (y2-y1)(x0-2*x1+x2)] [(x2-x1)(x0-2*x1+x2) + (y2-y1)(y0-2*y1+y2)] t^2(1-t)^4 -% + 18 18 [(x1-x0)(y0-2*y1+y2) - (y1-y0)(x0-2*x1+x2)] [(x3-x2)(x0-2*x1+x2) + (y3-y2)(y0-2*y1+y2)] t^2(1-t)^4 -% + 18 36 [(x1-x0)(y0-2*y1+y2) - (y1-y0)(x0-2*x1+x2)] [(x2-x1)(x1-2*x2+x3) + (y2-y1)(y1-2*y2+y3)] t^2(1-t)^4 -% + 18 18 [(x3-x2)(y0-2*y1+y2) - (y3-y2)(x0-2*x1+x2)] [(x1-x0)(x0-2*x1+x2) + (y1-y0)(y0-2*y1+y2)] t^2(1-t)^4 -% + 36 18 [(x2-x1)(y1-2*y2+y3) - (y2-y1)(x1-2*x2+x3)] [(x1-x0)(x0-2*x1+x2) + (y1-y0)(y0-2*y1+y2)] t^2(1-t)^4 -% + 36 18 [(x2-x1)(y0-2*y1+y2) - (y2-y1)(x0-2*x1+x2)] [(x3-x2)(x0-2*x1+x2) + (y3-y2)(y0-2*y1+y2)] t^3(1-t)^3 -% + 36 36 [(x2-x1)(y0-2*y1+y2) - (y2-y1)(x0-2*x1+x2)] [(x2-x1)(x1-2*x2+x3) + (y2-y1)(y1-2*y2+y3)] t^3(1-t)^3 -% + 18 18 [(x3-x2)(y0-2*y1+y2) - (y3-y2)(x0-2*x1+x2)] [(x1-x0)(x1-2*x2+x3) + (y1-y0)(y1-2*y2+y3)] t^3(1-t)^3 -% + 18 36 [(x3-x2)(y0-2*y1+y2) - (y3-y2)(x0-2*x1+x2)] [(x2-x1)(x0-2*x1+x2) + (y2-y1)(y0-2*y1+y2)] t^3(1-t)^3 -% + 36 18 [(x2-x1)(y1-2*y2+y3) - (y2-y1)(x1-2*x2+x3)] [(x1-x0)(x1-2*x2+x3) + (y1-y0)(y1-2*y2+y3)] t^3(1-t)^3 -% + 36 36 [(x2-x1)(y1-2*y2+y3) - (y2-y1)(x1-2*x2+x3)] [(x2-x1)(x0-2*x1+x2) + (y2-y1)(y0-2*y1+y2)] t^3(1-t)^3 -% + 18 18 [(x1-x0)(y1-2*y2+y3) - (y1-y0)(x1-2*x2+x3)] [(x3-x2)(x0-2*x1+x2) + (y3-y2)(y0-2*y1+y2)] t^3(1-t)^3 -% + 18 36 [(x1-x0)(y1-2*y2+y3) - (y1-y0)(x1-2*x2+x3)] [(x2-x1)(x1-2*x2+x3) + (y2-y1)(y1-2*y2+y3)] t^3(1-t)^3 -% + 18 18 [(x1-x0)(y0-2*y1+y2) - (y1-y0)(x0-2*x1+x2)] [(x3-x2)(x1-2*x2+x3) + (y3-y2)(y1-2*y2+y3)] t^3(1-t)^3 -% + 18 18 [(x3-x2)(y1-2*y2+y3) - (y3-y2)(x1-2*x2+x3)] [(x1-x0)(x0-2*x1+x2) + (y1-y0)(y0-2*y1+y2)] t^3(1-t)^3 -% + 18 18 [(x3-x2)(y1-2*y2+y3) - (y3-y2)(x1-2*x2+x3)] [(x1-x0)(x1-2*x2+x3) + (y1-y0)(y1-2*y2+y3)] t^4(1-t)^2 -% + 18 36 [(x3-x2)(y1-2*y2+y3) - (y3-y2)(x1-2*x2+x3)] [(x2-x1)(x0-2*x1+x2) + (y2-y1)(y0-2*y1+y2)] t^4(1-t)^2 -% + 36 18 [(x2-x1)(y1-2*y2+y3) - (y2-y1)(x1-2*x2+x3)] [(x3-x2)(x0-2*x1+x2) + (y3-y2)(y0-2*y1+y2)] t^4(1-t)^2 -% + 18 18 [(x3-x2)(y0-2*y1+y2) - (y3-y2)(x0-2*x1+x2)] [(x3-x2)(x0-2*x1+x2) + (y3-y2)(y0-2*y1+y2)] t^4(1-t)^2 -% + 36 18 [(x2-x1)(y0-2*y1+y2) - (y2-y1)(x0-2*x1+x2)] [(x3-x2)(x1-2*x2+x3) + (y3-y2)(y1-2*y2+y3)] t^4(1-t)^2 -% + 18 36 [(x3-x2)(y0-2*y1+y2) - (y3-y2)(x0-2*x1+x2)] [(x2-x1)(x1-2*x2+x3) + (y2-y1)(y1-2*y2+y3)] t^4(1-t)^2 -% + 18 18 [(x1-x0)(y1-2*y2+y3) - (y1-y0)(x1-2*x2+x3)] [(x3-x2)(x1-2*x2+x3) + (y3-y2)(y1-2*y2+y3)] t^4(1-t)^2 -% + 36 36 [(x2-x1)(y1-2*y2+y3) - (y2-y1)(x1-2*x2+x3)] [(x2-x1)(x1-2*x2+x3) + (y2-y1)(y1-2*y2+y3)] t^4(1-t)^2 -% + 36 18 [(x2-x1)(y1-2*y2+y3) - (y2-y1)(x1-2*x2+x3)] [(x3-x2)(x1-2*x2+x3) + (y3-y2)(y1-2*y2+y3)] t^5(1-t)^1 -% + 18 18 [(x3-x2)(y1-2*y2+y3) - (y3-y2)(x1-2*x2+x3)] [(x3-x2)(x0-2*x1+x2) + (y3-y2)(y0-2*y1+y2)] t^5(1-t)^1 -% + 18 36 [(x3-x2)(y1-2*y2+y3) - (y3-y2)(x1-2*x2+x3)] [(x2-x1)(x1-2*x2+x3) + (y2-y1)(y1-2*y2+y3)] t^5(1-t)^1 -% + 18 18 [(x3-x2)(y0-2*y1+y2) - (y3-y2)(x0-2*x1+x2)] [(x3-x2)(x1-2*x2+x3) + (y3-y2)(y1-2*y2+y3)] t^5(1-t)^1 -% + 18 18 [(x3-x2)(y1-2*y2+y3) - (y3-y2)(x1-2*x2+x3)] [(x3-x2)(x1-2*x2+x3) + (y3-y2)(y1-2*y2+y3)] t^6(1-t)^0 -% -% Koeffizienten: 27 ist GGT -% 27 = 3^3 = 27 * 1 -% 54 = 3^3 * 2 = 27 * 2 -% 108 = 3^3 * 2^2 = 27 * 4 -% 216 = 3^3 * 2^3 = 27 * 8 -% -% 3*18*18 = 3^5 * 2^2 = 27 * 36 -% 3*36*18 = 3^5 * 2^3 = 27 * 72 -% 3*36*36 = 3^5 * 2^4 = 27 * 144 -% - -%% >>> - -% >>> - \end{document} % vim:foldmethod=marker:foldmarker=<<<,>>> This was sent by the SourceForge.net collaborative development platform, the world's largest Open Source development site. 
 [PyX-checkins] SF.net SVN: pyx: [2627] trunk/pyx/design/beziers.tex From: - 2006-04-27 16:16:52 Revision: 2627 Author: wobsta Date: 2006-04-27 09:16:41 -0700 (Thu, 27 Apr 2006) ViewCVS: http://svn.sourceforge.net/pyx/?rev=2627&view=rev Log Message: ----------- another set of minor updates to our beziers tricks ... :-) Modified Paths: -------------- trunk/pyx/design/beziers.tex Modified: trunk/pyx/design/beziers.tex =================================================================== --- trunk/pyx/design/beziers.tex 2006-04-27 14:53:11 UTC (rev 2626) +++ trunk/pyx/design/beziers.tex 2006-04-27 16:16:41 UTC (rev 2627) @@ -281,12 +281,12 @@ However, although the geometric changes are limited to distances of $\epsilon$, the parametrization $t$ of the B\'ezier curve might be mistakenly represented by the straight line on a much larger scale. -In the shown example, the point $X$ on the straight line and $Y$ on -the B\'ezier curve are both taken at the parameter value $t=0.5$, but +In the shown example, the point $X$ on the B\'ezier curve and $Y$ on +the straight line are both taken at the parameter value $t=0.5$, but clearly are more separated from each other than one would expect from the geometric distance of the two paths. While the parametrization on a line is proportional to the arc length, a non-linear behaviour is -found on a B\'ezier curve. This non-linearity has is originated in +found on a B\'ezier curve. This non-linearity is originated in considerably different lengths $l_1$, $l_2$ and $l_3$ and the mapping of the non-linear parameter to a linear parametrization (in terms of the arc length) can be reduced to a one-dimensional problem upon an @@ -298,7 +298,7 @@ % In this one-dimensional approximation the parameter $t'$ performs a linear mapping as for any straight line while $t$ represents the usual -B\'ezier curve parametrization. It now becomes a matter of expression +B\'ezier curve parametrization. It now becomes a matter of expressing $t$ by $t'$. The polynomial in $t$ to be solved is: % This was sent by the SourceForge.net collaborative development platform, the world's largest Open Source development site.