Comments on: Better Sampling

By: rory

rory — Thu, 14 Jul 2011 16:02:43 +0000

Ok, ok! Thanks for the clear explanation. I changed the post.

By: Anon (same as before - I finally got it ;))

Anon (same as before - I finally got it ;)) — Thu, 14 Jul 2011 11:11:44 +0000

To make it a bit more explicit:

1 – x * x – y * y

1 – (r*cos(t))^2 – (r*sin(t))^2

1 – r^2*cos^2(t) – r^2*sin^2(t)

1 – r^2*(cos^2(t) + sin^2(t)) // Took out -r^2 as a factor

1 – (r^2*(1)) // Trig identities!

1 – r^2

1 – u1

Yay!

By: flemming

flemming — Tue, 26 Apr 2011 12:42:19 +0000

Agree with Boris and others. it should be
return Vector3(x, y, Sqrt(1.0f – u1))
since 1-(r*cos(t))^2-(r*sin(t))^2 =>
1 – r^2 =>
1-u1

By: Anon

Anon — Sun, 17 Apr 2011 15:41:52 +0000

Hmm, maybe I’m wrong… bah, I’m too tired to work it out.

By: Anon

Anon — Sun, 17 Apr 2011 13:25:46 +0000

ShinZo, I’m pretty sure (x * x – y * y) is not u1, it’s u1*cos(2*theta), or 2*x*x-u1 (which is just as costly as the original formula).

By: Cosine weighted hemisphere | pathtracing

Cosine weighted hemisphere | pathtracing — Thu, 03 Mar 2011 20:44:17 +0000

[...] same topic is discussed here: http://www.rorydriscoll.com/2009/01/07/better-sampling/ This entry was posted in Uncategorized. Bookmark the permalink. ← A simple [...]

By: Boris

Boris — Sun, 27 Feb 2011 05:59:23 +0000

return Vector3(x, y, Sqrt(1.0f – u1));

actualy

By: ShinZo

ShinZo — Fri, 04 Jun 2010 22:50:42 +0000

You should change:

return Vector3(x, y, Sqrt(Max(0.0f, 1 – x * x – y * y)));

return Vector3(x, y, Sqrt(Max(0.0f, 1 – u1)));

because (x * x – y * y) = u1 in your example

By: rory

rory — Thu, 11 Jun 2009 16:16:50 +0000

This function just generates the ray direction in tangent space. It assumes that you will rotate the direction into world space afterwards.

I just use the normal and tangent vectors from the mesh (generating the bitangent from the cross product) and generate a basis (3×3 matrix) which transforms the ray from tangent space into world space.

By: Freeman

Freeman — Thu, 11 Jun 2009 08:03:35 +0000

Hi,

Nice article

I just have a question about the randomly generated point on the hemisphere.
In your code, you never use the normal to generate it. How do you know that the hemisphere is centered on the normal ?