Hi Benjamin,

I’ve looked at your explanation, and after you put it like that,
I was all “doh!”… The problem is way simpler to solve in that form (extending
the main line (center to red) so the projected point has the same
characteristics of the purple points). After that it was pretty simple to get
right…

Thanks a lot for the help!

Diogo

**From:** Benjamin Rouveyrol
[mailto:pthiben77@yahoo.fr]

**Sent:** sexta-feira, 8 de Janeiro de 2010 16:36

**To:** Game Development Algorithms; drizzt.du@libero.it

**Subject:** [Algorithms] Re : Re : R: Sphere to rectangle

Hi Diogo,

You pretty much spotted the problem with http://www.spellcasterstudios.com/sphere_problem.jpg

What I wanted to do is to keep the old direction (ie center of the sphere
towards the red dots), but change the vector's magnitude so that the point is
on the line made by the camera position and the pink dot.

So you compute first an angle (alpha
= asin(R/D) ) on the triangle made by light center, camera position and
pink dot.

Then you apply Thales (tan(alpha)
= x / D <=> x = D * tan(alpha) ) to get your final point
outside the circle, and which is on the pink line. And yes, I computed the
right/up vectors the same way you did (with a normalize for Right ;-) )

When I look at my source code, I see that the 2D bounding rectangle checks all
the xy coordinates of all the points to get the left/right/top/bottom.

I can't remember why I did that, but I believe this was to solve the problem
you're showing.

Ben

**De :** Diogo de Andrade
<diogo.andrade@netvisao.pt>

**À :** Game Development Algorithms
<gdalgorithms-list@lists.sourceforge.net>; drizzt.du@libero.it

**Envoyé le :** Ven 8 Janvier 2010, 10 h 51 min 22 s

**Objet :** Re: [Algorithms] Re : R: Sphere to rectangle

Hi Benjamin,

Your solution almost works for me…

It works when the camera is head on to the center of the sphere:

http://www.spellcasterstudios.com/sphere_rect_problem01.jpg

But when I turn left or right, I get:

http://www.spellcasterstudios.com/sphere_rect_problem02.jpg

http://www.spellcasterstudios.com/sphere_rect_problem03.jpg

This might be related to the projection part, I’ll look into it…

What do you mean by: “Up and right vectors based on the camera
and light positions (and the camera Up).”? I’m just accounting for camera
orientation to calculate the up and right vectors… Are you suggesting something
like:

View=LightPos-CameraPos

Right=cross(CameraUp,View)

Up=cross(View,Right)

?

From the more practical standpoint, from what I understand,
you’re applying a scale factor to the radius to account for distance… How did
you came about this formula?

You didn’t happen to need something similar for
spotlights/cones?

Thanks,

Diogo

**From:** Benjamin Rouveyrol
[mailto:pthiben77@yahoo.fr]

**Sent:** sexta-feira, 8 de Janeiro de 2010 15:03

**To:** Game Development Algorithms; drizzt.du@libero.it

**Subject:** [Algorithms] Re : R: Sphere to rectangle

Hi,

The way I solved this (but a bit math heavy):

If R is the light Radius and D the distance between the camera and the light:

NewR = D *
tan(asin(R/D))

Then you compute Up and right vectors based on the camera and light positions
(and the camera Up).

You end up with 4 corners: LightCenter
+|- Up * NewR +|- Right * NewR

You can project them and this should give you your bounding rectangle.
If someone has a simpler solution, I'd be most interested ^_^

Ben

**De
:**
Diogo de Andrade <diogo.andrade@netvisao.pt>

**À :** drizzt.du@libero.it; Game Development Algorithms <gdalgorithms-list@lists.sourceforge.net>

**Envoyé le :** Ven 8 Janvier 2010, 9 h 28 min 54 s

**Objet :** Re: [Algorithms] R: Sphere to rectangle

Hi!

That's basically what I'm trying to do with the "wrong" solution,
although I don't tell it like that...

It was a good thought... I had to think about it for 2 mins before I understood
that was exactly what I was trying to do...

Visually, you can see the problem at http://www.spellcasterstudios.com/sphere_problem.jpg

With your solution (and my initial one), I'd get the red points (and their
projection, the yellow line), which is smaller than the correct solution (the
purple points/projection), which has the correct size...

Thanks!

Diogo

-----Original Message-----

From: drizzt.du@libero.it
[mailto:drizzt.du@libero.it]

Sent: sexta-feira, 8 de Janeiro de 2010 13:54

To: gdalgorithms-list@lists.sourceforge.net

Subject: [Algorithms] R: Sphere to rectangle

----Messaggio originale----

Da: diogo.andrade@netvisao.pt

Data: 08/01/2010 13.45

> Hi all…

>

> I’ve been wrestling with a problem for some days now, and I

> honestly can’t seem to figure an elegant solution…

>

> I want to find the rectangle that bounds a omni light source,

> so the problem is: given a sphere with center C=(x0,y0,z0) and radius

RADIUS, what is the bounding rectangle rect=(rx1,ry1,rx2,ry2) on screen?

Ok, I read this ML only for educational purpose, so...well, I'm sure I will

suggest the wrong solution :-D

Arent'you simply trying to do screen-aligned billboarding?

I mean...the bounding rectangle around a sphere of a given radius, on

screen...is not always the same rectangle simply billboarded?

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