Re: [Algorithms] Up Vector in Projection Matrix

[Ron L. <ro...@do...>]

----- Original Message -----
From: "Kelmenson, Dan" <Dan...@br...>
To: <gda...@li...>
Sent: Wednesday, August 09, 2000 2:34 PM
Subject: RE: [Algorithms] Up Vector in Projection Matrix


> > -----Original Message-----
> > From: Pai-Hung Chen [mailto:pa...@ac...]
> >
> > >
> > > When you specify a point or direction to "look at",  you have only
> > specified
> > > two of the degrees of freedom of the system.  That object
> > could be in the
> > > middle of the screen - either upside-down - or right way up
> > - or at any
> > > 'roll' angle in between.
> > >
> > > The 'up' vector only has to be enough to resolve that issue
> > - and only one
> > > of its degrees of freedom is needed.  Hence, there are a
> > wide variety of
> > > vectors that will do the job.  The only constraint is that
> > the up vector
> > > and the 'look direction' are not parallel.
> > >
> > > Hence, so long as your character can't look absolutely
> > vertically upwards
> > > or absolutely vertically downwards, an 'up' vector of
> > (0,1,0) will do
> > fine.
> >
> > That explains everything (though I don't know why/how the
> > underlying math
> > works).  An ever-confusing question got explained!  Thanks a lot! :-)
> >
>
> The most likely reason the math worked (the reason the math worked in my
> program, and most likely in yours...) is that the main use of the 'up'
> vector is crossing it with the forward/view direction.  As long as your
> character/camera doesn't lean to the side, the true (0,1,0) up vector and
> the actual perpendicular-to-forward up vector will both be coplanar with
the
> forward vector, and therefore give you the same cross product.
>

Another way to think about it is that the "look-at" functions in the various
APIs and toolkits do not require that the up vector argument be
perpendicular to the look-at vector argument, but only that they be linearly
independent.  Then it's a fact that there is a unique unit up vector that IS
perpendicular to the look-at vector argument, in the plane determined by the
two vector arguments (why they must be linearly independent) and pointing,
in that plane, to the same side of the argument look-at vector as points the
argument up vector.  The API look-at routine computes that perpendicular up
vector from the argument vectors, either by iterated cross products or, more
likely, by the Gram-Schmidt formula, to use in constructing the correct view
matrix.