Is my 4d perspective projection equation really true?

[achinapong]

I've build the 4d perspective projection by using homogenous co-ordinate.

Then It's come out like this

4d-> 3d

x3d = x4d / w4d
y3d = y4d / w4d
z3d = z4d / w4d

3d->2d same way (I've use the rest of 3d perspective projection by using OpenGL Libary)


Why the picture come out is the usual cube? It's not like a tesseract that I've seen usually

Best Regards,
Chinapong

[alkaline]

i'm not exactly sure what those equations mean. have you seen this paper on visualization of 4d objects?

http://research.microsoft.com/~hollasch/thesis/

you might be doing an orthogonal projection, and your camera may be facing the fourth dimension face on - you need to face it at an angle.

[Keiji]

i want a 4d raytracer...

[alkaline]

go to pat's page, he has an n-d raytracer:

http://nklein.com/products/rt/

[BClaw]

You need a picture plane (realm) and an observation point to calculate perspective:

pt3.x = pt4.x-((pt4.w-ppdist)*(pt4.x-obspt.x)/(pt4.w-obspt.w))
pt3.y = pt4.y-((pt4.w-ppdist)*(pt4.y-obspt.y)/(pt4.w-obspt.w))
pt3.z = pt4.z-((pt4.w-ppdist)*(pt4.z-obspt.z)/(pt4.w-obspt.w))

ppdist = the w value of the picture realm. Make it less than the smallest point on your object (my tesseract's points are 1 or -1 in each directionm, and so I set this value at -2..) Then the observation point (mine is set at 0,0,0,-4) What these formulas are doing is to find out where a line crosses the picture realm when drawn from a point on the object to the observer's position. If you think about it, this is why we see things the way we do. Everything we see is gathered to a point (our eye). :-) To further reduce it to 2D (for display on a monitor...)

pt2.x = pt3.x-((pt3.z-ppdist)*(pt3.x-obspt.x)/(pt3.z-obspt.z))
pt2.y = pt3.y-((pt3.z-ppdist)*(pt3.y-obspt.y)/(pt3.z-obspt.z))

You *could* have a different ppdist and obspt for this operation...

Does that make sense? It's what I used for my animations:

4D Measure 1
4D Measure 2
4D Measure 2 (alt)

5D Measure 1
5D Measure 2
5D Measure 2 (alt)

[Aale de Winkel]

Wow, those animations are great, do you allow me to use one/some of them on my http://home.wanadoo.nl/aaledewinkel/Encyclopedia/ , where exactly I don't know yet, probably on a few places. In case please send my privately some details to introduce them.

On the Geometry thread these kind of animations might be helpfull on objects such as the duocircle (4d) the duocylinder (5d), I don't know wether your software allows easy creation of these?
.

[pat]

Those animations are great... Do you happen to know what sequences of rotations you applied (particularly in the second of each set)? I'm interested in seeing what they look like in my raytracer's stacked view as opposed to the projected view.

I'm assuming that you created these by making your own tool to rotate and project down to three-dimensions and then create a POVray input file for each frame. Is that correct?

[BClaw]

Thanks for your compliments Aale and Pat. Feel free to post those on your site, Aale. I ask only that you place a copy on your own web-server, because I don't have a lot of room on the net, and I'll be taking them off at some point. :?

I'm using 3D Studio Max, and creating the geometry frame by frame with a Max Script program. So, all I need to know is what a duocircle and duocylinder are! I'm planning to do all six basic polytopes, and I'm in the middle of a render of a 4-sphere. As for the rotations, Pat, I can tell you exactly what the rotations were, frame by frame if you like. Tell me what kind of a format would be useful, and I'll email them to you.

[pat]

As for the rotations, I can tell you exactly what the rotations were, frame by frame if you like. What kind of a format, and I'll email them to you.

Hmmm.... well, I'd prefer something less-detailed than the frame-by-frame orientation matrixes, though that would do. I was assuming you had some sort of motion plan like:

• rotate for 15 frames parallel to the z-w plane at 45-degrees per 30 frames.
• rotate for 30 frames parallel to the x-w plane at 45-degrees per 30 frames while rotating parallel to the y-z plane at 30-degrees per 30 frames.
• etc.

If you just want to send me the snippets of the script that calculate the orientation, that'd be fine too... I'm fairly sure I can decipher most programming languages (be they not too insane).

However it's convenient would be wonderful.

Thanks,
Patrick

[PWrong]

I don't understand any of these equations, but they look interesting. Is there a website where I can get a good introduction?