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r is the radius of the sphere. r sin(theta) is the radius of the upper horizontal circular cross-section. A circle arc length is angle_in_radians * r. To get lengths of sides of the patch, we find lengths of the corresponding arcs (one with radius r sin(theta) and angle dphi and another with radius r and angle dtheta).

w is the position vector of the patch. dw is the unit solid angle.