Solid Angle of Conical Surfaces, Polyhedral Cones, and Intersecting Spherical Caps

Molecular theory for the phase equilibria and cluster distribution of associating fluids with small bond angles

“…is similar to the two patch case 1 and is given by (14) The constant d I is the probability that two colloids are oriented such that double bonding at patches A and B occurs and is given as 1 (15) Where AB S is the solid angle of the overlap of the patches A and B and is given by 28 (16)…”

“…The condition that 0  SP c for AB SP  allows for simplification of the Q function obtained from Eq. ( 5) as (28) Which using the results above can be further simplified as (29) The S c are obtained from Eq (18) as (30) where…”

Temporally Reliable Motion Vectors for Real‐time Ray Tracing

“…4.3, one of the key technical dependencies is the calculation of the overlapping solid angle between each pair of cones on a unit hemisphere. We refer to the solution by Mazonka et al [Maz12]. Specifically, for a pair of cones with same apex angle π 6 , the overlapping solid angle Ω can be written as:…”

Thermodynamic perturbation theory for associating fluids with small bond angles: Effects of steric hindrance, ring formation, and double bonding

“…This solid angle AB S is simply the solid angle of the intersection of 2 cones of apex angle c  which share a common origin whose axes are at an angle of AB  to each other. For association to occur both colloids must be oriented correctly, so we square the single colloid probability to obtain (12) The solid angle AB S has been solved elsewhere 28 and is given by…”

“…6 Using Eqns. ( 4) -( 5) and ( 25) we obtain the density of colloids bonded at patch A as (28) The density of colloids bonded twice is given by Eqns. ( 4) -( 5), ( 11), ( 18) and ( 26)…”