## Derivation of a 2D trigonometric function - the pseudo-elliptic cosine

For the surface normalisation factor of the pseudo-elliptic cone, the integral from equation 16.51b on page  is

 dS = 1 - cosθmax dφ dσ. (16.41)

When combined with the pseudo-ellipse of equation 16.39, this becomes the intractable integral

 dS = 1 - cos dφ dσ, (16.42)

Instead the cosine series expansion will be used

 cosθmax = 1 - + - + - + ... , (16.43) = θmax2n. (16.44)

Integrating each element of the sum over the φ parameter and using the assumption that θx, θy≥ 0 gives

 dφ = πθxθy, dφ = θx2 + θy2, dφ = 3θx4 +2θx2θy2 +3θy4, dφ = 5θx6 +3θx4θy2 +3θx2θy4 +5θy6, dφ = 35θx8 +20θx6θy2 +18θx4θy4 +20x2θy6 +35θy8, ... (16.45)

Therefore a new two dimension trigonometric function, the pseudo-elliptic cosine, can be defined as

Let

 a = θx2, b = θy2, (16.47)

then the first of the fn(θx, θy) equations are

 f0 = 1, f1 = 2a + 2b, f2 = 6a2 +4ab + 6b2, f3 = 20a3 +12a2b + 12ab2 +20b3, f4 = 70a4 +40a3b + 36a2b2 +40ab3 +70b4, f5 = 252a5 +140a4b + 120a3b2 +120a2b3 +140ab4 +252b5, f6 = 924a6 +504a5b + 420a4b2 +400a3b3 +420a2b4 +504ab5 +924b6, f7 = 3432a7 +1848a6b + 1512a5b2 +1400a4b3 +1400a3b4 +1512a2b5 +1848ab6 +3432b7, ... (16.48)

Or

 f0 = 1, f1 = 2(a + b), f2 = 6(a2 + b2) + 4ab, f3 = 20(a3 + b3) + 12(a2b + ab2), f4 = 70(a4 + b4) + 40(a3b + ab3) + 36a2b2, f5 = 252(a5 + b5) + 140(a4b + ab4) + 120(a3b2 + a2b3), f6 = 924(a6 + b6) + 504(a5b + ab5) + 420(a4b2 + a2b4) + 400a3b3, f7 = 3432(a7 + b7) + 1848(a6b + ab6) + 1512(a5b2 + a2b5) + 1400(a4b3 + a3b4), f8 = 12870(a8 + b8) + 6864(a7b + ab7) + 5544(a6b2 + a2b6) + 5040(a5b3 + a3b5) + 4900a4b4, f9 = 48620(a9 + b9) + 25740(a8b + ab8) + 20592(a7b2 + a2b7) + 18480(a6b3 + a3b6) + 17640(a5b4 + a4b5), f10 = 184756(a10 + b10) + 97240(a9b + ab9) + 77220(a8b2 + a2b8) + 68640(a7b3 + a3b7) + 64680(a6b4 + a4b6) + 63504a5b5, ... (16.49)

This series expansion up to n = 10 is sufficient for writing a fast and accurate pec function implementation in computer code. The numerical representation of this function is shown in figure 16.14.

Figure 16.14: The pseudo-ellipse cosine 2D trigonometric function. This is the surface area on a unit sphere bounded by the pseudo-elliptic cone.

The relax user manual (PDF), created 2016-10-28.