I am reading the irradiance gradient on Greg Ward Radiance book and also this web page:

Quoting from this web page:

For normalized spherical coordinates the gradient is:

enter image description here

The irradiance contribution from a direction on the hemisphere about the surface normal, specified using spherical coordinates φ [0, 2π) and θ [0, π / 2) is:

enter image description here

Where L is the incident radiance in the supplied direction. So the gradient vector at any point on the hemisphere:

enter image description here

The vector v is a unit vector on the plane of the hemisphere pointing in the perpendicular direction to the angle φ.

The page author further concludes the gradient vector of irradiance on local hemisphere(after dividing by cosine pdf) is as below

enter image description here

The z coordinate is 0 so I assume the gradient vector here lies on the plain of the hemisphere.

However I don't understand specifically why enter image description here should lie on "the plane of the hemisphere". The two spherical unit vectors(phi and theta) are tangent to the sphere surface. My understanding is that this vector is a spherical unit vector(theta) pointing opposite the z axis.

enter image description here

Is it not correct to say that the gradient of the cosine is: enter image description here

and the vector theta: with spherical coordinate as(explained here):

enter image description here

If this is correct I can't get my head around why the grad vector should lie on the plane of the hemisphere.


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