[metapost] Re: [metafont] Re: Intersections of NURBs

[L. Nobre G.]

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