an encyclopedia of finite element definitions

Degree 1 Buffa–Christiansen on a dual polygon

◀ Back to 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{18 \sqrt{2} x - 3 \sqrt{2} - \sqrt{10}}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{18 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} - \tfrac{2}{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 \left(- 7 \sqrt{2} x - 3 \sqrt{10} x + 5 \sqrt{10} + 12 \sqrt{2}\right)}{5 \sqrt{\sqrt{5} + 5} \left(3 \sqrt{5} + 7\right)}\\\displaystyle - \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{3 \sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{25} - \tfrac{1}{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{2} x + 4 \sqrt{10} x - 14 \sqrt{2} - 2 \sqrt{10}}{- 40 \sqrt{\sqrt{5} + 5} - 35 \sqrt{5 - \sqrt{5}} + 5 \sqrt{5} \sqrt{5 - \sqrt{5}} + 10 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{4 \sqrt{2} y + 4 \sqrt{10} y - 4 \sqrt{5} \sqrt{\sqrt{5} + 5} - 4 \sqrt{\sqrt{5} + 5} + 2 \sqrt{5 - \sqrt{5}} + 2 \sqrt{5} \sqrt{5 - \sqrt{5}}}{- 40 \sqrt{\sqrt{5} + 5} - 35 \sqrt{5 - \sqrt{5}} + 5 \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)/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{- 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}}\\\displaystyle \tfrac{- \tfrac{12 \sqrt{2} y}{5} - \tfrac{4 \sqrt{10} y}{5} + \tfrac{2 \sqrt{5 \sqrt{5} + 25}}{5} + \tfrac{6 \sqrt{\sqrt{5} + 5}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 3 \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{10} x - 2 \sqrt{2} x - 3 \sqrt{2} - \sqrt{10}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{- \tfrac{4 \sqrt{10} y}{5} + \tfrac{2 \sqrt{25 - 5 \sqrt{5}}}{5}}{\sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 5\right)}\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{\sqrt{10} x \sqrt{5 - \sqrt{5}}}{25} - \tfrac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{20} - \tfrac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{100}\\\displaystyle \tfrac{2 \left(- 2 \sqrt{2} y - \sqrt{5 - \sqrt{5}}\right)}{5 \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{- 8 \sqrt{10} x - 16 \sqrt{2} x + 2 \sqrt{10} + 6 \sqrt{2}}{5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5 - \sqrt{5}} + 35 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 8 \sqrt{10} y - 16 \sqrt{2} y - 4 \sqrt{5} \sqrt{\sqrt{5} + 5} - 8 \sqrt{\sqrt{5} + 5}}{5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5 - \sqrt{5}} + 35 \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{- 12 \sqrt{2} x - 4 \sqrt{10} x + 12 \sqrt{2} + 8 \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}}\\\displaystyle \tfrac{- \tfrac{12 \sqrt{2} y}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{12 \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 \sqrt{5 \sqrt{5} + 25}}{5} + \tfrac{2 \sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{6 \sqrt{5 - \sqrt{5}}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 3 \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 \left(- \sqrt{10} x - \sqrt{2} x + \sqrt{2} + 2 \sqrt{10}\right)}{5 \left(1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{1}{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{18 \sqrt{2} x - 3 \sqrt{2} - \sqrt{10}}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{18 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{2}{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} x - 3 \sqrt{2} + \sqrt{10}}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{2}{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 \left(27 \sqrt{10} x + 63 \sqrt{2} x - 18 \sqrt{2} - 8 \sqrt{10}\right)}{5 \sqrt{\sqrt{5} + 5} \left(3 \sqrt{5} + 7\right)}\\\displaystyle - \tfrac{27 \sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{25} - \tfrac{9 \sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{9 y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{27 y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{2}{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{- 108 \sqrt{10} x - 180 \sqrt{2} x - 12 \sqrt{10} - 20 \sqrt{2}}{- 150 \sqrt{\sqrt{5} + 5} - 125 \sqrt{5 - \sqrt{5}} + 15 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- \tfrac{108 \sqrt{10} y}{5} - 36 \sqrt{2} y + 4 \sqrt{5 - \sqrt{5}} + \tfrac{12 \sqrt{25 - 5 \sqrt{5}}}{5} + 4 \sqrt{\sqrt{5} + 5} + \tfrac{12 \sqrt{5 \sqrt{5} + 25}}{5}}{- 30 \sqrt{\sqrt{5} + 5} - 25 \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 8 \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{- 12 \sqrt{2} x - 4 \sqrt{10} x + 12 \sqrt{2} + 8 \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}}\\\displaystyle \tfrac{- \tfrac{12 \sqrt{2} y}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{6 \sqrt{5 - \sqrt{5}}}{5} - \tfrac{2 \sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{4 \sqrt{5 \sqrt{5} + 25}}{5} + \tfrac{12 \sqrt{\sqrt{5} + 5}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 3 \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{10} x - 2 \sqrt{2} x - 3 \sqrt{2} - \sqrt{10}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{- \tfrac{4 \sqrt{10} y}{5} + \tfrac{6 \sqrt{25 - 5 \sqrt{5}}}{5}}{\sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 5\right)}\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{\sqrt{10} x \sqrt{5 - \sqrt{5}}}{25} - \tfrac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{20} - \tfrac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{100}\\\displaystyle \tfrac{2 \left(- 2 \sqrt{2} y + \sqrt{5 - \sqrt{5}}\right)}{5 \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{- 8 \sqrt{10} x - 16 \sqrt{2} x - 14 \sqrt{2} - 6 \sqrt{10}}{5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5 - \sqrt{5}} + 35 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 8 \sqrt{10} y - 16 \sqrt{2} y - 4 \sqrt{5} \sqrt{5 - \sqrt{5}} - 8 \sqrt{5 - \sqrt{5}}}{5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5 - \sqrt{5}} + 35 \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{- 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}}\\\displaystyle \tfrac{- \tfrac{12 \sqrt{2} y}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{6 \sqrt{\sqrt{5} + 5}}{5} - \tfrac{2 \sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 3 \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 \left(- \sqrt{10} x - \sqrt{2} x + \sqrt{2} + \sqrt{10}\right)}{5 \left(1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{2 y \left(- \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{- 4 \sqrt{2} x - \sqrt{10} + 9 \sqrt{2}}{10 \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{1}{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{- 4 \sqrt{2} x - \sqrt{2} + \sqrt{10}}{10 \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{1}{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 \left(- 7 \sqrt{2} x - 3 \sqrt{10} x - 3 \sqrt{2} - \sqrt{10}\right)}{5 \sqrt{\sqrt{5} + 5} \left(3 \sqrt{5} + 7\right)}\\\displaystyle - \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{3 \sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{25} + \tfrac{2}{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{2} x + 4 \sqrt{10} x + 4 \sqrt{10} + 16 \sqrt{2}}{- 40 \sqrt{\sqrt{5} + 5} - 35 \sqrt{5 - \sqrt{5}} + 5 \sqrt{5} \sqrt{5 - \sqrt{5}} + 10 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{4 \sqrt{2} y + 4 \sqrt{10} y - 4 \sqrt{5} \sqrt{5 - \sqrt{5}} - 4 \sqrt{5 - \sqrt{5}} + 2 \sqrt{\sqrt{5} + 5} + 2 \sqrt{5} \sqrt{\sqrt{5} + 5}}{- 40 \sqrt{\sqrt{5} + 5} - 35 \sqrt{5 - \sqrt{5}} + 5 \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)/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{27 \sqrt{10} x + 63 \sqrt{2} 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}}\\\displaystyle \tfrac{\tfrac{27 \sqrt{10} y}{5} + \tfrac{63 \sqrt{2} y}{5} - \tfrac{7 \sqrt{\sqrt{5} + 5}}{5} - \tfrac{3 \sqrt{5 \sqrt{5} + 25}}{5} - \tfrac{7 \sqrt{5 - \sqrt{5}}}{5} - \tfrac{3 \sqrt{25 - 5 \sqrt{5}}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 2 \sqrt{5} \sqrt{5 - \sqrt{5}} + 2 \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 \left(9 \sqrt{2} x + 9 \sqrt{10} x + 2 \sqrt{10} + 6 \sqrt{2}\right)}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{9 y \sqrt{50 - 10 \sqrt{5}}}{25}\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{\sqrt{10} x \sqrt{5 - \sqrt{5}}}{25} - \tfrac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{20} - \tfrac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{100}\\\displaystyle \tfrac{2 \left(- 2 \sqrt{2} y + 3 \sqrt{5 - \sqrt{5}}\right)}{5 \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{- 8 \sqrt{10} x - 16 \sqrt{2} x - 34 \sqrt{2} - 14 \sqrt{10}}{5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5 - \sqrt{5}} + 35 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 8 \sqrt{10} y - 16 \sqrt{2} y - 8 \sqrt{5} \sqrt{5 - \sqrt{5}} - 16 \sqrt{5 - \sqrt{5}} + 8 \sqrt{\sqrt{5} + 5} + 4 \sqrt{5} \sqrt{\sqrt{5} + 5}}{5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5 - \sqrt{5}} + 35 \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{- 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}}\\\displaystyle \tfrac{- \tfrac{12 \sqrt{2} y}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{6 \sqrt{5 - \sqrt{5}}}{5} - \tfrac{2 \sqrt{25 - 5 \sqrt{5}}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 3 \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 \left(- \sqrt{10} x - \sqrt{2} x + \sqrt{2}\right)}{5 \left(1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{50} - \tfrac{1}{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} \left(1 - x\right)}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{2 \sqrt{2} y}{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} \left(1 - x\right)}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{2 \sqrt{2} y}{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{2 \left(- 7 \sqrt{2} x - 3 \sqrt{10} x + 2 \sqrt{2} + \sqrt{10}\right)}{5 \sqrt{\sqrt{5} + 5} \left(3 \sqrt{5} + 7\right)}\\\displaystyle - \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{10} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{3 \sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{25} + \tfrac{1}{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{2} x + 4 \sqrt{10} x + 2 \sqrt{10} + 6 \sqrt{2}}{- 40 \sqrt{\sqrt{5} + 5} - 35 \sqrt{5 - \sqrt{5}} + 5 \sqrt{5} \sqrt{5 - \sqrt{5}} + 10 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{4 \sqrt{2} y + 4 \sqrt{10} y - 2 \sqrt{5} \sqrt{5 - \sqrt{5}} - 2 \sqrt{5 - \sqrt{5}}}{- 40 \sqrt{\sqrt{5} + 5} - 35 \sqrt{5 - \sqrt{5}} + 5 \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)/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{- 12 \sqrt{2} x - 4 \sqrt{10} x - 10 \sqrt{10} - 18 \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}}\\\displaystyle \tfrac{- \tfrac{12 \sqrt{2} y}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{6 \sqrt{\sqrt{5} + 5}}{5} - \tfrac{2 \sqrt{5 \sqrt{5} + 25}}{5} + \tfrac{4 \sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{12 \sqrt{5 - \sqrt{5}}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 3 \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{10} x - 2 \sqrt{2} x - 3 \sqrt{2} - \sqrt{10}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{- \tfrac{4 \sqrt{10} y}{5} - \tfrac{6 \sqrt{25 - 5 \sqrt{5}}}{5}}{\sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 5\right)}\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{4 \left(9 \sqrt{2} x + \sqrt{2} + \sqrt{10}\right)}{5 \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}\\\displaystyle \tfrac{72 y}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{2} + \sqrt{10}\right)}\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{540 \sqrt{2} x + 252 \sqrt{10} x + 60 \sqrt{2} + 28 \sqrt{10}}{15 \sqrt{5} \sqrt{\sqrt{5} + 5} + 40 \sqrt{5} \sqrt{5 - \sqrt{5}} + 150 \sqrt{5 - \sqrt{5}} + 125 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{108 \sqrt{2} y + \tfrac{252 \sqrt{10} y}{5} + 12 \sqrt{5 - \sqrt{5}} + \tfrac{28 \sqrt{25 - 5 \sqrt{5}}}{5} + 12 \sqrt{\sqrt{5} + 5} + \tfrac{28 \sqrt{5 \sqrt{5} + 25}}{5}}{3 \sqrt{5} \sqrt{\sqrt{5} + 5} + 8 \sqrt{5} \sqrt{5 - \sqrt{5}} + 30 \sqrt{5 - \sqrt{5}} + 25 \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{- 12 \sqrt{2} x - 4 \sqrt{10} x - 10 \sqrt{10} - 18 \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}}\\\displaystyle \tfrac{- \tfrac{12 \sqrt{2} y}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{12 \sqrt{5 - \sqrt{5}}}{5} - \tfrac{4 \sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{2 \sqrt{5 \sqrt{5} + 25}}{5} + \tfrac{6 \sqrt{\sqrt{5} + 5}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 3 \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 \left(- \sqrt{10} x - \sqrt{2} x - \sqrt{10} + \sqrt{2}\right)}{5 \left(1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{50} - \tfrac{2}{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{- 4 \sqrt{2} x - \sqrt{2} + \sqrt{10}}{10 \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} - \tfrac{1}{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{- 4 \sqrt{2} x - \sqrt{10} + 9 \sqrt{2}}{10 \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} - \tfrac{1}{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 \left(- 7 \sqrt{2} x - 3 \sqrt{10} x + 3 \sqrt{10} + 7 \sqrt{2}\right)}{5 \sqrt{\sqrt{5} + 5} \left(3 \sqrt{5} + 7\right)}\\\displaystyle - \tfrac{2 \sqrt{2} y}{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), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\left(\begin{array}{c}\displaystyle \tfrac{4 \sqrt{2} x + 4 \sqrt{10} x - 4 \sqrt{2}}{- 40 \sqrt{\sqrt{5} + 5} - 35 \sqrt{5 - \sqrt{5}} + 5 \sqrt{5} \sqrt{5 - \sqrt{5}} + 10 \sqrt{5} \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{4 \sqrt{2} y + 4 \sqrt{10} y - 2 \sqrt{5} \sqrt{\sqrt{5} + 5} - 2 \sqrt{\sqrt{5} + 5}}{- 40 \sqrt{\sqrt{5} + 5} - 35 \sqrt{5 - \sqrt{5}} + 5 \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)/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{- 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}}\\\displaystyle \tfrac{- \tfrac{12 \sqrt{2} y}{5} - \tfrac{4 \sqrt{10} y}{5} + \tfrac{2 \sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{6 \sqrt{5 - \sqrt{5}}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 3 \sqrt{5} \sqrt{5 - \sqrt{5}} + 3 \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{10} x - 2 \sqrt{2} x - 3 \sqrt{2} - \sqrt{10}}{5 \sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 3\right)}\\\displaystyle \tfrac{- \tfrac{4 \sqrt{10} y}{5} - \tfrac{2 \sqrt{25 - 5 \sqrt{5}}}{5}}{\sqrt{5 - \sqrt{5}} \left(\sqrt{5} + 5\right)}\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{\sqrt{10} x \sqrt{5 - \sqrt{5}}}{25} - \tfrac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{20} - \tfrac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{100}\\\displaystyle \tfrac{2 \left(- 2 \sqrt{2} y - 3 \sqrt{5 - \sqrt{5}}\right)}{5 \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{- 8 \sqrt{10} x - 16 \sqrt{2} x + 10 \sqrt{10} + 26 \sqrt{2}}{5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5 - \sqrt{5}} + 35 \sqrt{\sqrt{5} + 5}}\\\displaystyle \tfrac{- 8 \sqrt{10} y - 16 \sqrt{2} y - 8 \sqrt{5} \sqrt{\sqrt{5} + 5} - 16 \sqrt{\sqrt{5} + 5} + 8 \sqrt{5 - \sqrt{5}} + 4 \sqrt{5} \sqrt{5 - \sqrt{5}}}{5 \sqrt{5} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5} \sqrt{5 - \sqrt{5}} + 40 \sqrt{5 - \sqrt{5}} + 35 \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{27 \sqrt{10} x + 63 \sqrt{2} 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}}\\\displaystyle \tfrac{\tfrac{27 \sqrt{10} y}{5} + \tfrac{63 \sqrt{2} y}{5} + \tfrac{3 \sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{7 \sqrt{5 - \sqrt{5}}}{5} + \tfrac{3 \sqrt{5 \sqrt{5} + 25}}{5} + \tfrac{7 \sqrt{\sqrt{5} + 5}}{5}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5 - \sqrt{5}} + 2 \sqrt{5} \sqrt{5 - \sqrt{5}} + 2 \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 \left(9 \sqrt{2} x + 9 \sqrt{10} x - 2 \sqrt{10} - 4 \sqrt{2}\right)}{5 \left(1 + \sqrt{5}\right) \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{9 y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{9 y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{2}{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} x - 3 \sqrt{2} + \sqrt{10}}{5 \sqrt{\sqrt{5} + 5}}\\\displaystyle - \tfrac{2 \sqrt{2} y}{5 \sqrt{\sqrt{5} + 5}} - \tfrac{2}{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}$$