# Funciones de forma triangulares

## Elemento de tres nodos

\begin{equation*} \begin{bmatrix} & 1 & \\ x & & y \end{bmatrix} \end{equation*}

campo de desplazamientos

\begin{align*} u &= \alpha_{1} + \alpha_{2} x + \alpha_{3} y \\ v &= \alpha_{4} + \alpha_{5} x + \alpha_{6} y \end{align*}

en forma matricial

\begin{equation*} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} 1 & x & y & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x & y \end{bmatrix} \begin{bmatrix} \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \\ \alpha_{6} \end{bmatrix} \end{equation*}

reemplazando valores

\begin{align*} \alpha_{1} + \alpha_{2} x_{1} + \alpha_{3} y_{1} &= u_{1} \\ \alpha_{4} + \alpha_{5} x_{1} + \alpha_{6} y_{1} &= v_{1} \\ \alpha_{1} + \alpha_{2} x_{2} + \alpha_{3} y_{2} &= u_{2} \\ \alpha_{4} + \alpha_{5} x_{2} + \alpha_{6} y_{2} &= v_{2} \\ \alpha_{1} + \alpha_{2} x_{3} + \alpha_{3} y_{3} &= u_{3} \\ \alpha_{4} + \alpha_{5} x_{3} + \alpha_{6} y_{3} &= v_{3} \end{align*}

en forma matricial

\begin{equation*} \begin{bmatrix} 1 & x_{1} & y_{1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x_{1} & y_{1} \\ 1 & x_{2} & y_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x_{2} & y_{2} \\ 1 & x_{3} & y_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x_{3} & y_{3} \end{bmatrix} \begin{bmatrix} \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \\ \alpha_{6} \end{bmatrix} = \begin{bmatrix} u_{1} \\ v_{1} \\ u_{2} \\ v_{2} \\ u_{3} \\ v_{3} \end{bmatrix} \end{equation*}

resolviendo el sistema

\begin{equation*} \begin{bmatrix} \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \\ \alpha_{6} \end{bmatrix} = \begin{bmatrix} \frac{x_{2} y_{3} - x_{3} y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{1} x_{3} - x_{1} y_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} y_{2} - y_{1} x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 \\ \frac{y_{2} - y_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{3} - y_{1}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{1} - y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 \\ \frac{x_{3} - x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} - x_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{2} - x_{1}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 \\ 0 & \frac{x_{2} y_{3} - x_{3} y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{1} x_{3} - x_{1} y_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} y_{2} - y_{1} x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} \\ 0 & \frac{y_{2} - y_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{3} - y_{1}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{1} - y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} \\ 0 & \frac{x_{3} - x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} - x_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{2} - x_{1}} {y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} \end{bmatrix} \begin{bmatrix} u_{1} \\ v_{1} \\ u_{2} \\ v_{2} \\ u_{3} \\ v_{3} \end{bmatrix} \end{equation*}

reemplazando

\begin{equation*} d = \begin{bmatrix} 1 & x & y & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x & y \end{bmatrix} \begin{bmatrix} \frac{x_{2} y_{3} - x_{3} y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{1} x_{3} - x_{1} y_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} y_{2} - y_{1} x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 \\ \frac{y_{2} - y_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{3} - y_{1}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{1} - y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 \\ \frac{x_{3} - x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} - x_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{2} - x_{1}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 \\ 0 & \frac{x_{2} y_{3} - x_{3} y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{1} x_{3} - x_{1} y_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} y_{2} - y_{1} x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} \\ 0 & \frac{y_{2} - y_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{3} - y_{1}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{y_{1} - y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} \\ 0 & \frac{x_{3} - x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} - x_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{2} - x_{1}} {y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} \end{bmatrix} \begin{bmatrix} u_{1} \\ v_{1} \\ u_{2} \\ v_{2} \\ u_{3} \\ v_{3} \end{bmatrix} \end{equation*}

multiplicando

\begin{equation*} d = \begin{bmatrix} \frac{x_{2} y_{3} - x_{3} y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(y_{2} - y_{3}) x}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(x_{3} - x_{2}) y}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} y_{1} - y_{3} x_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(y_{3} - y_{1}) x}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(x_{1} - x_{3}) y}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} y_{2} - y_{1} x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(y_{1} - y_{2}) x}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(x_{2} - x_{1}) y}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 \\ 0 & \frac{x_{2} y_{3} - x_{3} y_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(y_{2} - y_{3}) x}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(x_{3} - x_{2}) y}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} y_{1} - y_{3} x_{3}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(y_{3} - y_{1}) x}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(x_{1} - x_{3}) y}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} & 0 & \frac{x_{1} y_{2} - y_{1} x_{2}}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(y_{1} - y_{2}) x}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} + \frac{(x_{2} - x_{1}) y}{y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}} \end{bmatrix} \begin{bmatrix} u_{1} \\ v_{1} \\ u_{2} \\ v_{2} \\ u_{3} \\ v_{3} \end{bmatrix} \end{equation*}

El área de un triángulo con vértices $(x_{1}, y_{1})$, $(x_{2}, y_{2})$ y $(x_{3}, y_{3})$ es

\begin{equation*} A = \frac{1}{2} \begin{vmatrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{vmatrix} = \frac{1}{2} (y_{1} x_{3} - y_{1} x_{2} - x_{1} y_{3} + x_{1} y_{2} + x_{2} y_{3} - x_{3} y_{2}) \end{equation*}

reemplazando

\begin{equation*} d = \frac{1}{2 A} \begin{bmatrix} (x_{2} y_{3} - x_{3} y_{2}) + (y_{2} - y_{3}) x + (x_{3} - x_{2}) y & 0 & (x_{1} y_{1} - y_{3} x_{3}) + (y_{3} - y_{1}) x + (x_{1} - x_{3}) y & 0 & (x_{1} y_{2} - y_{1} x_{2}) + (y_{1} - y_{2}) x + (x_{2} - x_{1}) y & 0 \\ 0 & (x_{2} y_{3} - x_{3} y_{2}) + (y_{2} - y_{3}) x + (x_{3} - x_{2}) y & 0 & (x_{1} y_{1} - y_{3} x_{3}) + (y_{3} - y_{1}) x + (x_{1} - x_{3}) y & 0 & (x_{1} y_{2} - y_{1} x_{2}) + (y_{1} - y_{2}) x + (x_{2} - x_{1}) y \end{bmatrix} \begin{bmatrix} u_{1} \\ v_{1} \\ u_{2} \\ v_{2} \\ u_{3} \\ v_{3} \end{bmatrix} \end{equation*}

En forma matricial

\begin{equation*} d = \begin{bmatrix} N_{1} & 0 & N_{2} & 0 & N_{3} & 0 \\ 0 & N_{1} & 0 & N_{2} & 0 & N_{3} \end{bmatrix} \begin{bmatrix} u_{1} \\ v_{1} \\ u_{2} \\ v_{2} \\ u_{3} \\ v_{3} \end{bmatrix} \end{equation*}

En forma matricial reducida

\begin{equation*} d = \mathbf{N} \ \mathbf{d} \end{equation*}