Re: [help-3dldf] Re: button-hole problem

[Martijn van Manen]

Here is my eternal sorry again for not participating
a lot. I really am sorry, but I just don't have the time.
Quickly scanning through the e-mail below I noticed the following 
phrase:
 
> What I do is the following:  A `Focus' contains a `Point' representing the
> position of the "camera" in space and another representing the direction of
> view.  The "up" direction is determined somehow, I don't remember how, and can
> be modified by using a `real' value for an angle of rotation. 

Here is a crucial point in my opinion. What is the "Up" direction?
I wanted very much to use 3DLDF for an article. So I did some
experiments. But I could not get it right. I tried stuff with
the focus and the distance, but to no avail. In what source file
do I find how the "Up" direction is determined?
Line whereabouts?

Pascals theorem by the way roughly says that through any five points in
the plane there is a unique ellipse.
An ellipse is given by an equation

a x2 + bxy + cy2 + d x + e y = 1 

So if you have five points, you get five unknowns a,b,c,d,e and
five equations. So you can determine a solution. 
Ofcourse, a mathematician would know say that I'm retarded. He/she
would be right, but this I think is the computational essence.


GRTZ,

Martijn


----- Original Message -----
From: "Laurence Finston" <address@hidden>
To: "Larry Siebenmann" <address@hidden>
Subject: [help-3dldf] Re: button-hole problem
Date: Fri, 22 Apr 2005 23:07:53 +0200

> 
> From: Larry Siebenmann <address@hidden>
> 
> >>> [is there] an algebraic formula for the curve on a plane
> >>> of projection representing the perspective projection of a
> >>> circle?
> 
> > I answered "off-the-cuff":
> >
> >  > On the image plane it is a conic section, ie an ellipse, a
> >  > parabola, or hyperbola.
> >
> > If you almost understood this answer, read on; otherwise stop
> > right here.
> 
> After I posted my question, I was somewhat mortified when I remembered
> something about circles in perspective always being ellipses.  When I thought
> about it some more, I wasn't sure whether this was perhaps only true of
> circles in planes parallel to one of the main planes (x-y, x-z, or z-y).  I
> have a couple of plastic templates of perspective ellipses---they are still
> available in stores, but are probably destined to go the way of the slide
> rule.  I believe that a circle that is coplanar with the ray of vision and
> perpendicular to the plane of projection must be a straight line.  Perhaps
> others are really all elliptical; I wish I knew how to figure this out.  At
> any rate, I regretted not having asked about conic sections in the first
> place.  I'd also be interested in knowing about the effect of non-affine
> transformations on them other than the perspective transformation, if I
> haven't exhausted your patience already.  I suspect this is probably something
> that I won't be able to understand, though.
> 
> > For example, if the
> > projection point is on the target plane, which happens to be
> > disjoint from the circle, then the projected circle is the
> > EMPTY SET.
> 
> Wouldn't it be a point, namely the projection point, or what I call the focus?
>   And wouldn't the projections of all other objects in the 3D space be that
> same point?  In GNU 3DLDF, this would imply that the `distance' element of the
> `Focus' object would be 0, which would probably fail anyway, probably because
> of a division by 0 somewhere.
> 
> >
> > However, from an answer that is generically true one can
> > usually (with some work)  deduce the general answer -- which
> > is often rather complicated and correspondingly OBSCURE.  For
> > example, the generic (and stable) affine classification of
> > complete bezier cubic loci (forgetting parameters) is just
> > twofold: doublepoint versus inflexion pair. But the full
> > classification is a bit of a mess. See the MP list for
> > details
> 
> I'm working on non-arbitrary geometric figures because they're easier
> than Bezier curves.  Consider the task of dividing a polyhedron using a plane.
>   From a mathematician's point of view, this is trivial, but programming it
> will certainly take me quite a few hours.  In fact, I have already spent many
> hours working on the "preliminaries", which are features in their own right.
> Once it works, it should be possible to extend it to dividing a polyhedron by
> another polyhedron.  I'm looking forward to making a sequence of pictures with
> two polyhedra passing through each other.  However, I still have a couple of
> unsolved problems involving the occlusion of two polygons, so I'll have to
> solve them first.
> 
> I have also given quite a bit of thought to representing what I am calling
> "ellipse slices" and "circle slices".  These are the figures resulting from
> dividing ellipses and circles by straight lines and/or other ellipses
> and circles, once or multiple times.  Someone has been kind enough to explain
> to me how one can find the intersections of two ellipses algebraically, but
> unfortunately I don't have the mathematical background to understand the
> explanation.  This is by way of explaining why I'm not in such a hurry to get
> stuck into Bezier curves.
> 
> >
> > Incidentally, the parabolic case above is not generic or
> > stable *unless* one considers movies, or one states that the
> > union
> >
> > (elliptic case) \cup (parabolic case) \cup (hyperbolic) case)
> >
> > is stable and generic.
> >
> 
> Would it be possible to express these ideas in words of one syllable?
> 
> > I meant the intersection, with that sphere, of a cone having
> > center the eye's center, and with base an ellipse in a well
> > chosen plane far from the eye.
> >
> 
> This is interesting, but I think it may not be of practical importance for
> 3DLDF.  So far, I've been satisifed with projection onto a plane.  Should I be
> concerned with the intersection of spheres and cones and the center of the
> eye?  This is a genuine question.
> 
> >  ----------------
> >
> > As for algebraic formulae, I assume you are happy with
> > pointwise execution
> 
> I'm not sure.  I've been thinking about this lately.  In some ways, it's
> useful to have the individual points available.  In other ways, it might be
> useful to just store the center of an object and a transformation matrix.  I
> haven't made a decision about this yet.  If you have an opinion, I'd be quite
> interested to hear it.
> 
> > so I suggest you just compose two
> > perspectivities in R^3:
> 
> >
> > (1) the given perspsectivity projecting from the given circle
> > to the target plane in 3-space.
> >
> > (2) the perspectivity with center the eye of the (imaginary)
> > television camera, that maps from the mentioned target
> > plane to the computer screen.
> >
> > This lets you move any number of points from your original
> > circle to the computer screen in such a way that you "see"
> > on the screen the given projection of the circle.
> 
> What I do is the following:  A `Focus' contains a `Point' representing the
> position of the "camera" in space and another representing the direction of
> view.  The "up" direction is determined somehow, I don't remember how, and can
> be modified by using a `real' value for an angle of rotation.  These values
> are used to determine a transformation which would place the position point
> and the direction point on the z-axis and the plane of projection into the x-y
> plane.  This transformation is then applied to all of the objects in the space
> before applying the perspective transformation.
> 
> 
> >
> > Once you have 5 points of the screen image, you can (if you
> > wish) get all other points on screen by programming
> > 2-dimensional Pascal's theorem in MP.
> 
> I'll have to look up Pascal's theorem.  I'm afraid my reach exceeds my grasp
> when it comes to math.
> 
> Thank you; I appreciate your help very much.
> 
> Laurence
> 
> 
> 
> _______________________________________________
> help-3dldf mailing list
> address@hidden
> http://lists.gnu.org/mailman/listinfo/help-3dldf



Mieulx est de ris que de larmes escripre,
Pour ce que rire est le propre de l'homme.


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