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Given the three points $P_{1}=(10,-8,3), P_{2}=(8,-4,3.2)$ and $P_{3}=(8,4,3.2),$ Derive the equation of the triangle defined by them.
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Solution:

$P_{1}=(10,-8,3), P_{2}=(8,-4,3.2)$ and $P_{3}=(8,4,3.2)$

Equation of surface is given by

$P(u, v)=P_{1}(1-u-v)+P_{2} u+P_{3} v$

The values for $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ coordinates can be determined as,

$P_{x}(u, v)=10(1-u-v)+8 u+8 v$

$\begin{aligned} &=10-10 u-10 v+8 u+8 v \\ &=10-2 u-2 v \\ P_{y}(u, v) &=-8(1-u-v)-4 u+4 v \\ &=-8+8 u+8 v-4 u+4 v \\ &=-8+4 u+12 v \\ P_{z}(u,v) &=-3(1-u-v)+3.2(u)+3.2(v) \\ &=3-3 u-3 v+3.2 u+3.2 v \\ &=3+0.2 u+0.2 v \end{aligned}$

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