an encyclopedia of finite element definitions

# Degree 1 rotated Buffa–Christiansen on a dual polygon

◀ Back to rotated Buffa–Christiansen definition page
In this example:
• $$R$$ is the reference dual polygon. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{2}{5}\\\displaystyle \tfrac{- 2 \sqrt{2} x - 3 \sqrt{2} - \sqrt{10}}{5 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{3 \sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{25} - \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{1}{5}\\\displaystyle \tfrac{- 6 \sqrt{2} x - 2 \sqrt{10} x + 3 \sqrt{10} + 11 \sqrt{2}}{5 \left(\sqrt{5} + 3\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{- \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{y \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{\sqrt{5}}{2} + \tfrac{1}{2}}{-10 - 2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 5 \sqrt{5}}\\\displaystyle \tfrac{- \tfrac{3 x \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{3 x \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{x \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{x \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{9 \sqrt{10 - 2 \sqrt{5}}}{40} - \tfrac{\sqrt{2 \sqrt{5} + 10}}{40} + \tfrac{3 \sqrt{10 \sqrt{5} + 50}}{40} + \tfrac{\sqrt{50 - 10 \sqrt{5}}}{8}}{-10 - 2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 5 \sqrt{5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{16 \sqrt{10} y}{5} + 8 \sqrt{2} y - 4 \sqrt{\sqrt{5} + 5} - \tfrac{8 \sqrt{5 \sqrt{5} + 25}}{5}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 40 \sqrt{2} x - 16 \sqrt{10} x + 10 \sqrt{2} + 6 \sqrt{10}}{- 25 \sqrt{\sqrt{5} + 5} + 25 \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y + 2 \sqrt{10} y - \sqrt{5} \sqrt{5 - \sqrt{5}} - \sqrt{5 - \sqrt{5}}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{\sqrt{2} \left(- 4 x - \sqrt{5} - 1\right)}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y + 2 \sqrt{10} y + \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{5 - \sqrt{5}}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{- 4 \sqrt{2} x - \sqrt{10} - \sqrt{2}}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{7 y \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{7 y \sqrt{2 \sqrt{5} + 10}}{10} + 4 + \tfrac{9 \sqrt{5}}{5}}{2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 5 \sqrt{5}}\\\displaystyle \tfrac{- \tfrac{7 x \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{3 x \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{7 x \sqrt{10 - 2 \sqrt{5}}}{10} - \tfrac{3 x \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{\sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{\sqrt{50 - 10 \sqrt{5}}}{20} + \tfrac{11 \sqrt{2 \sqrt{5} + 10}}{40} + \tfrac{\sqrt{10 \sqrt{5} + 50}}{8}}{2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 5 \sqrt{5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{16 \sqrt{10} y}{5} + 8 \sqrt{2} y - 4 \sqrt{5 - \sqrt{5}} - \tfrac{8 \sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{16 \sqrt{5 \sqrt{5} + 25}}{5} + 8 \sqrt{\sqrt{5} + 5}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 40 \sqrt{2} x - 16 \sqrt{10} x + 50 \sqrt{2} + 26 \sqrt{10}}{- 25 \sqrt{\sqrt{5} + 5} + 25 \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{1}{5}\\\displaystyle \tfrac{- 2 \sqrt{10} x + 2 \sqrt{2} x - 7 \sqrt{2} + 5 \sqrt{10}}{5 \left(-1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} - \tfrac{2}{5}\\\displaystyle \tfrac{- 2 \sqrt{2} x - 3 \sqrt{2} - \sqrt{10}}{5 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} - \tfrac{2}{5}\\\displaystyle \tfrac{- 2 \sqrt{2} x - 3 \sqrt{2} + \sqrt{10}}{5 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{3 \sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{25} - \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{2}{5}\\\displaystyle \tfrac{2 \left(- 3 \sqrt{2} x - \sqrt{10} x - 7 \sqrt{2} - 3 \sqrt{10}\right)}{5 \left(\sqrt{5} + 3\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{- 4 \sqrt{10} y + 4 \sqrt{2} y - 4 \sqrt{5} \sqrt{\sqrt{5} + 5} - 4 \sqrt{5} \sqrt{5 - \sqrt{5}} + 4 \sqrt{5 - \sqrt{5}} + 4 \sqrt{\sqrt{5} + 5}}{- 85 \sqrt{\sqrt{5} + 5} - 65 \sqrt{5 - \sqrt{5}} + 25 \sqrt{5} \sqrt{5 - \sqrt{5}} + 35 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 20 \sqrt{2} x + 4 \sqrt{10} x - 4 \sqrt{10} + 20 \sqrt{2}}{- 500 \sqrt{\sqrt{5} + 5} - 325 \sqrt{5 - \sqrt{5}} + 155 \sqrt{5} \sqrt{5 - \sqrt{5}} + 230 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{16 \sqrt{10} y}{5} + 8 \sqrt{2} y - 8 \sqrt{\sqrt{5} + 5} - \tfrac{16 \sqrt{5 \sqrt{5} + 25}}{5} + \tfrac{8 \sqrt{25 - 5 \sqrt{5}}}{5} + 4 \sqrt{5 - \sqrt{5}}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 40 \sqrt{2} x - 16 \sqrt{10} x + 50 \sqrt{2} + 26 \sqrt{10}}{- 25 \sqrt{\sqrt{5} + 5} + 25 \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \cdot \left(2 \sqrt{2} y \left(1 + \sqrt{5}\right) - 3 \sqrt{5} \sqrt{5 - \sqrt{5}} - 3 \sqrt{5 - \sqrt{5}}\right)}{5 \left(1 + \sqrt{5}\right)^{2} \sqrt{5 - \sqrt{5}}}\\\displaystyle \tfrac{\sqrt{2} \left(- 4 x - \sqrt{5} - 1\right)}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y + 2 \sqrt{10} y - \sqrt{5} \sqrt{5 - \sqrt{5}} - \sqrt{5 - \sqrt{5}}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{- 4 \sqrt{2} x - \sqrt{10} - \sqrt{2}}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{12 \sqrt{10} y + 28 \sqrt{2} y + 6 \sqrt{5} \sqrt{5 - \sqrt{5}} + 14 \sqrt{5 - \sqrt{5}}}{45 \sqrt{5 - \sqrt{5}} + 30 \sqrt{\sqrt{5} + 5} + 25 \sqrt{5} \sqrt{5 - \sqrt{5}} + 20 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- \tfrac{6 \sqrt{10} x}{5} - 2 \sqrt{2} x - 2 \sqrt{2} - \tfrac{4 \sqrt{10}}{5}}{\sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 5 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{16 \sqrt{10} y}{5} + 8 \sqrt{2} y + \tfrac{8 \sqrt{5 \sqrt{5} + 25}}{5} + 4 \sqrt{\sqrt{5} + 5}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 12 \sqrt{2} x - 4 \sqrt{10} x + 2 \sqrt{2} + 2 \sqrt{10}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 15 \sqrt{5} \sqrt{5 - \sqrt{5}} + 15 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{2 \left(- \sqrt{10} x + \sqrt{2} x - \sqrt{2} + \sqrt{10}\right)}{5 \left(-1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} - \tfrac{1}{5}\\\displaystyle \tfrac{- 4 \sqrt{2} x - \sqrt{10} + 9 \sqrt{2}}{10 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} - \tfrac{1}{5}\\\displaystyle \tfrac{- 4 \sqrt{2} x - \sqrt{2} + \sqrt{10}}{10 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{3 \sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{25} - \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{2}{5}\\\displaystyle \tfrac{2 \left(- 3 \sqrt{2} x - \sqrt{10} x - 2 \sqrt{2}\right)}{5 \left(\sqrt{5} + 3\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{- \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{y \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} - 1 + \tfrac{2 \sqrt{5}}{5}}{-10 - 2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 5 \sqrt{5}}\\\displaystyle \tfrac{- \tfrac{3 x \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{3 x \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{x \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{x \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{3 \sqrt{10 \sqrt{5} + 50}}{40} - \tfrac{\sqrt{50 - 10 \sqrt{5}}}{10} - \tfrac{\sqrt{2 \sqrt{5} + 10}}{40} + \tfrac{3 \sqrt{10 - 2 \sqrt{5}}}{20}}{-10 - 2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 5 \sqrt{5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{44 \sqrt{10} y}{5} + 20 \sqrt{2} y + \tfrac{44 \sqrt{25 - 5 \sqrt{5}}}{5} + 20 \sqrt{5 - \sqrt{5}} + \tfrac{44 \sqrt{5 \sqrt{5} + 25}}{5} + 20 \sqrt{\sqrt{5} + 5}}{- 15 \sqrt{\sqrt{5} + 5} + 15 \sqrt{5 - \sqrt{5}} + 29 \sqrt{5} \sqrt{5 - \sqrt{5}} + 29 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 7 \sqrt{2} x - 3 \sqrt{10} x + 3 \sqrt{10} + 7 \sqrt{2}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 10 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{4 \sqrt{2} y}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\\\displaystyle \tfrac{4 \sqrt{2} \left(- x + 1 + \sqrt{5}\right)}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y + 2 \sqrt{10} y - 3 \sqrt{5} \sqrt{5 - \sqrt{5}} - 3 \sqrt{5 - \sqrt{5}}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{- 4 \sqrt{2} x - \sqrt{10} - \sqrt{2}}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{7 y \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{7 y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{2 \sqrt{5}}{5} + 1}{2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 5 \sqrt{5}}\\\displaystyle \tfrac{- \tfrac{7 x \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{3 x \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{7 x \sqrt{10 - 2 \sqrt{5}}}{10} - \tfrac{3 x \sqrt{50 - 10 \sqrt{5}}}{10} - \tfrac{31 \sqrt{10 \sqrt{5} + 50}}{40} - \tfrac{69 \sqrt{2 \sqrt{5} + 10}}{40} - \tfrac{3 \sqrt{50 - 10 \sqrt{5}}}{10} - \tfrac{13 \sqrt{10 - 2 \sqrt{5}}}{20}}{2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 5 \sqrt{5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{16 \sqrt{10} y}{5} + 8 \sqrt{2} y + \tfrac{8 \sqrt{25 - 5 \sqrt{5}}}{5} + 4 \sqrt{5 - \sqrt{5}}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 40 \sqrt{2} x - 16 \sqrt{10} x - 14 \sqrt{10} - 30 \sqrt{2}}{- 25 \sqrt{\sqrt{5} + 5} + 25 \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{1}{5}\\\displaystyle \tfrac{- 2 \sqrt{10} x + 2 \sqrt{2} x - \sqrt{10} + 3 \sqrt{2}}{5 \left(-1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{2 \sqrt{2} \cdot \left(1 - x\right)}{5 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{2 \sqrt{2} \cdot \left(1 - x\right)}{5 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{3 \sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{25} - \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{1}{5}\\\displaystyle \tfrac{- 6 \sqrt{2} x - 2 \sqrt{10} x + \sqrt{2} + \sqrt{10}}{5 \left(\sqrt{5} + 3\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{- \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{y \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{1}{2} + \tfrac{\sqrt{5}}{10}}{-10 - 2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 5 \sqrt{5}}\\\displaystyle \tfrac{- \tfrac{3 x \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{3 x \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{x \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{x \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{\sqrt{10 \sqrt{5} + 50}}{40} - \tfrac{\sqrt{50 - 10 \sqrt{5}}}{40} - \tfrac{\sqrt{2 \sqrt{5} + 10}}{40} + \tfrac{\sqrt{10 - 2 \sqrt{5}}}{40}}{-10 - 2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 5 \sqrt{5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{16 \sqrt{10} y}{5} + 8 \sqrt{2} y - 8 \sqrt{5 - \sqrt{5}} - \tfrac{16 \sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{8 \sqrt{5 \sqrt{5} + 25}}{5} + 4 \sqrt{\sqrt{5} + 5}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 40 \sqrt{2} x - 16 \sqrt{10} x - 34 \sqrt{10} - 70 \sqrt{2}}{- 25 \sqrt{\sqrt{5} + 5} + 25 \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \cdot \left(2 \sqrt{2} y \left(1 + \sqrt{5}\right) + 3 \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}}\right)}{5 \left(1 + \sqrt{5}\right)^{2} \sqrt{5 - \sqrt{5}}}\\\displaystyle \tfrac{\sqrt{2} \left(- 4 x - \sqrt{5} - 1\right)}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\left(\begin{array}{c}\displaystyle \tfrac{8 y}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{2} + \sqrt{10}\right)}\\\displaystyle \tfrac{4 \left(- \sqrt{2} x + \sqrt{2} + \sqrt{10}\right)}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{44 \sqrt{2} y}{5} + 4 \sqrt{10} y - 4 \sqrt{5 \sqrt{5} + 25} - \tfrac{44 \sqrt{\sqrt{5} + 5}}{5} - 4 \sqrt{25 - 5 \sqrt{5}} - \tfrac{44 \sqrt{5 - \sqrt{5}}}{5}}{7 \sqrt{5} \sqrt{5 - \sqrt{5}} + 17 \sqrt{5 - \sqrt{5}} + 5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 13 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 260 \sqrt{2} x - 116 \sqrt{10} x + 116 \sqrt{10} + 260 \sqrt{2}}{500 \sqrt{5 - \sqrt{5}} + 230 \sqrt{5} \sqrt{5 - \sqrt{5}} + 325 \sqrt{\sqrt{5} + 5} + 155 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{16 \sqrt{10} y}{5} + 8 \sqrt{2} y - 4 \sqrt{\sqrt{5} + 5} - \tfrac{8 \sqrt{5 \sqrt{5} + 25}}{5} + \tfrac{16 \sqrt{25 - 5 \sqrt{5}}}{5} + 8 \sqrt{5 - \sqrt{5}}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 40 \sqrt{2} x - 16 \sqrt{10} x - 34 \sqrt{10} - 70 \sqrt{2}}{- 25 \sqrt{\sqrt{5} + 5} + 25 \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{5 - \sqrt{5}} + 55 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{2}{5}\\\displaystyle \tfrac{2 \left(- \sqrt{10} x + \sqrt{2} x - 2 \sqrt{10} + 4 \sqrt{2}\right)}{5 \left(-1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{1}{5}\\\displaystyle \tfrac{- 4 \sqrt{2} x - \sqrt{2} + \sqrt{10}}{10 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \boldsymbol{\phi}_{4} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{1}{5}\\\displaystyle \tfrac{- 4 \sqrt{2} x - \sqrt{10} + 9 \sqrt{2}}{10 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{2 \left(- 3 \sqrt{2} x - \sqrt{10} x + \sqrt{10} + 3 \sqrt{2}\right)}{5 \left(\sqrt{5} + 3\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{8 \sqrt{2} y - 4 \sqrt{\sqrt{5} + 5}}{- 45 \sqrt{\sqrt{5} + 5} - 30 \sqrt{5 - \sqrt{5}} + 20 \sqrt{5} \sqrt{5 - \sqrt{5}} + 25 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{\tfrac{4 \sqrt{10} x}{5} - \sqrt{2} + \tfrac{\sqrt{10}}{5}}{- 5 \sqrt{\sqrt{5} + 5} - 5 \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{16 \sqrt{10} y}{5} + 8 \sqrt{2} y - 4 \sqrt{5 - \sqrt{5}} - \tfrac{8 \sqrt{25 - 5 \sqrt{5}}}{5}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{5 - \sqrt{5}} + 11 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 12 \sqrt{2} x - 4 \sqrt{10} x - 4 \sqrt{10} - 8 \sqrt{2}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 15 \sqrt{5} \sqrt{5 - \sqrt{5}} + 15 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y + 2 \sqrt{10} y + \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{5 - \sqrt{5}}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{\sqrt{2} \left(- 4 x - \sqrt{5} - 1\right)}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y + 2 \sqrt{10} y + 3 \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{- 4 \sqrt{2} x - \sqrt{10} - \sqrt{2}}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{7 y \sqrt{10 - 2 \sqrt{5}}}{10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{7 y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{11}{2} + \tfrac{5 \sqrt{5}}{2}}{2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 5 \sqrt{5}}\\\displaystyle \tfrac{- \tfrac{7 x \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{3 x \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{7 x \sqrt{10 - 2 \sqrt{5}}}{10} - \tfrac{3 x \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{19 \sqrt{10 - 2 \sqrt{5}}}{40} + \tfrac{9 \sqrt{50 - 10 \sqrt{5}}}{40} + \tfrac{51 \sqrt{2 \sqrt{5} + 10}}{40} + \tfrac{23 \sqrt{10 \sqrt{5} + 50}}{40}}{2 \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 5 \sqrt{5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{\tfrac{44 \sqrt{10} y}{5} + 20 \sqrt{2} y - 20 \sqrt{\sqrt{5} + 5} - \tfrac{44 \sqrt{5 \sqrt{5} + 25}}{5} - 20 \sqrt{5 - \sqrt{5}} - \tfrac{44 \sqrt{25 - 5 \sqrt{5}}}{5}}{- 15 \sqrt{\sqrt{5} + 5} + 15 \sqrt{5 - \sqrt{5}} + 29 \sqrt{5} \sqrt{5 - \sqrt{5}} + 29 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 7 \sqrt{2} x - 3 \sqrt{10} x + 3 \sqrt{10} + 7 \sqrt{2}}{- 5 \sqrt{\sqrt{5} + 5} + 5 \sqrt{5 - \sqrt{5}} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 10 \sqrt{5} \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{2}{5}\\\displaystyle \tfrac{2 \left(- \sqrt{10} x + \sqrt{2} x - \sqrt{10} - \sqrt{2}\right)}{5 \left(-1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{2}{5}\\\displaystyle \tfrac{- 2 \sqrt{2} x - 3 \sqrt{2} + \sqrt{10}}{5 \sqrt{\sqrt{5} + 5}}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$