From: Dr. Olaf Hoffmann <Dr.O.Hoffmann@gmx.de>

Date: Fri, 10 Apr 2009 12:36:35 +0100

To: www-style@w3.org

Message-Id: <200904101336.35969.Dr.O.Hoffmann@gmx.de>

Date: Fri, 10 Apr 2009 12:36:35 +0100

To: www-style@w3.org

Message-Id: <200904101336.35969.Dr.O.Hoffmann@gmx.de>

For transitions Chebyshev polynomials (or similar related functions) could be used as an additional alternative to provide some bouncing effects - for example one could simple note the number of the intended polynom. Because they have a norm, they are not outside of the box of allowed values and they represent an explicite time dependency, in case this is simpler to understand for some authors. They have to be shifted a little bit to meet the intended box [0;1], but they are often used to approximate/interpolate an arbitrary curve - therefore there should exist already many modules to handle those polynomials effectively. http://en.wikipedia.org/wiki/Chebyshev_polynomials Another option could be obviously trigonometric functions - sinus, cosinus or the square of them. They can be used to approximate arbitrary explicite time dependent functions, if this is really required (I do not assume, that this is necessary for transitions). Giovanni Campagna: >> In general such equation does not exist for the Cubic-Bezier - is ambiguous >> in our "space-time" ((C) Dr. O.Hoffmann). >Not it is ambiguous, rather the equation X(t) = >(1-t)^2*t*A+(1-t)*t^2*B+t^3 (parametric form for the Bezier Curve, >given P0 = 0 and P3 = 1) is a cubic, so it can have (at most) three >solutions, ie three values of the parameter, given a point of time. >That is, you need to ensure that X(t) is bijective, so: >1) there must exists at least a value for t in [0;1] so that X(t) >= 1 >and a value of t in [0;1] so that >X(t) <= 0 (this ensures surjectivity) >2) X(t) must be monotonic strictly increasing or strictly decreasing, >that is it's first derivative must be never 0 >(Dr O.Hoffman said it must be strictly increasing, he was wrong >because he assumed that the independent variable is t, the parameter, >while really it is the time) I noted the opposite ;o) The time function (the time as a function of the parameter running from 0 to 1) must have a monotonic behaviour; for animation or transitions we do not really need strictly monotonic, what allows discrete jumps, what is not possible of course in a classical space-time too, but because viewers typically have only a display with discrete time steps this can be neglected. For example keyTimes in SMIL/SVG allow such a mixture of continuous and discrete interpolation. With the current CSS drafts this is not possible (but can be approximated within an animation, not within a transition). And I think, the arguments are almost the same, just from another point of view. For a trajectory in space-time you need something like a blackhole to get in an area with a classically strange time behaviour, but as far as we know, there is no practical possibility of time travels (this means more than one solution for one moment in time for a classical trajectory - position in space as a function of time). However superposition principles and quantum mechanics may not exclude, that an object is at the same time at different positions - but one needs again a projection to a classical observable for display, what means something like a random choice, if there is more than one possible solution - I think, typical authors are not interested in such kind of behaviour, most of them are still restricted to classical behaviour in their thinking ;o)Received on Friday, 10 April 2009 11:46:43 UTC

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