Re: bad bezier behavior

From: Douglas G. Danforth <"Douglas>
Date: Mon, 20 Dec 2004 12:05:14 -0500

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Fyodor,
You didn't mention the program you used to generate the anomolous Bezier
curve. I would think the curve should not be discontinuous it is first
derivative. From below it mentions that the "order of continuity" is
controlable.

-Doug

 From Hyperdictionary
Definition:BEZIER CURVE

A type of curve defined by mathematical formulae, used in computer
graphics. A curve with coordinates P(u), where u varies from 0 at one
end of the curve to 1 at the other, is defined by a set of n+1 "control
points" (X(i), Y(i), Z(i)) for i = 0 to n.

         P(u) = Sum i=0..n [(X(i), Y(i), Z(i)) * B(i, n, u)]
         B(i, n, u) = C(n, i) * u^i * (1-u)^(n-i)
         C(n, i) = n!/i!/(n-i)!

A Bezier curve (or surface) is defined by its control points, which
makes it invariant under any affine mapping (translation, rotation,
parallel projection), and thus even under a change in the axis system.
You need only to transform the control points and then compute the new
curve. The control polygon defined by the points is itself affine invariant.

Bezier curves also have the variation-diminishing property. This makes
them easier to split compared to other types of curve such as Hermite or
B-spline.

Other important properties are multiple values, global and local
control, versatility, and order of continuity.



Fyodor Tkachov wrote:
>
> Could anyone explain the fact that a segment of a Bezier curve is sometimes incorrectly drawn as a straight line?

> The picture is attached: it also visualizes the array of 10 Ports.point's which was passed to the Views.Frame.DrawPath procedure.
>
> The second picture shows that the Bezier segment for the anomalous four points is well-defined (the middle red curve segment was produces in an independent drawing application using the four points).
>
> The effect is stable with respect to sufficiently small deformations of the View's parameters -- but disappears for larger deformations, and reappears (with another segment drawn incorrectly) for another set of parameters.

>
> Perhaps there is a way to detect and prevent this condition beforehand?
>
> Thanks in advance.

>
> Cheers
> Fyodor.Tkachov{([at]})nowhere.xy
>
> Attachments: badbezier.gif, badbezier2.gif
>
> Powered by the E-mail PIM - Info Select - www.miclog.com
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Received on Mon Dec 20 2004 - 18:05:14 UTC

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