[metapost] Re: [metafont] Re: Intersections of NURBs
L. Nobre G.
nobre at lince.cii.fc.ul.pt
Sun Jan 30 22:26:52 CET 2005
On Sun, 30 Jan 2005, Laurence Finston wrote:
> Larry Siebenmann wrote:
>
> > Here is a simplest example where a difficulty arises. I stand
> > before an (infinite) blackboard and project bezier curves on the
> > blackboard from my eye onto the (infinite) floor. Consider in
> > particular the bezier cubic bb with its controll trilateral
> > A--B--C--D on the blackboard thus:
> >
> >
> > B o-------------o C
> > / \
> > - - - -/ - - - - - -\ - - - H horizon of floor
> > / x \ projected to blackboard
> > / x x \
> > / \
> > / x x \
> > / \ (paper = blackboard)
> > /x x\
> > / \
> > / \
> > A o o D
> >
>
> I've taken another look at this, and there are a couple of things I don't
> understand. When I perform the perspective transformation in 3DLDF, I have a
> `Focus' with `Points' representing the position and the direction of view, and
> a `real' value representing the distance of the position from the plane of
> projection. In order to simplify things, the `Focus' is transformed such that
>
> `position' and `position + direction' come to lie on the negative z-axis, (in
> a left-handed Cartesian coordinate system), and this transformation is applied
> to all of the objects on the `Picture' prior to applying the perspective
> transformation. Any objects whose z-coordinates are less than or equal to
> that of `Focus.position' are not projected. According to my understanding,
> there is no concept of a "horizon" when using this method.
I think the following "visualization" agrees with both Laurence and Larry
views:
Imagine that you are in center of the earth looking towards the North
Pole. There is a blackboard laying on the earth surface somewhere over the
equator on your left. This blackboard has a Bezier curve and its control
points drawn on it as Larry drawn but instead of the Horizon it is the
equator. Accordingly to Laurence you don't see the control points B and C
(they are on the southern hemisphere, z-coordinates are less than or
equal to that of `Focus.position') but you should see the bezier curve on
your left. If control points B and C were projected as Larry explained they
would show up on your right. Right?
Lu\'{\i}s Nobre Gon\c{c}alves - http://matagalatlante.org
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