Find a transformation from u,v space to x,y,z space that takes the triangle U=[001001] to the triangle T=[(1,0),−2),(−1,2,0),(1,1,2)]

Key to "Find a transformation from u,v space to x,y,z..."

aortiH

2021-01-02

Find a transformation from u,v space to x,y,z space that takes the triangle

U = [\begin{array}{cc} 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{array}]

to the triangle

T = [(1, 0), - 2), (- 1, 2, 0), (1, 1, 2)]

averes8

Skilled2021-01-03Added 92 answers

Step 1
The triangle

U = [\begin{array}{cc} 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{array}]

and the triangle

T = [\begin{array}{ccc} 1 & 0 & - 2 \\ - 1 & 2 & 0 \\ 1 & 1 & 2 \end{array}]

Step 2
Consider

T (u, v) = (a_{1} u + b_{1} v + c_{1}, a_{2} u + b_{2} v + c_{2}, a_{3} u + b_{3} v = c_{3})

.Obtain the transformation from u, v to x, y, z as

T (0, 0) = (a_{1} (0) + b_{1} (0) + c_{1}, a_{2} (0) + b_{2} (0) + c_{2}, a_{3} (0) + b_{3} (0) + c_{3})

T (0, 0) = (c_{1}, c_{2}, c_{3})

T (0, 0) = (1, 0, - 2)

\Rightarrow c_{1} = 1, c_{2} = 0, c_{3} = - 2

Step 3
Further evaluate as,

T (1, 0) = (a_{1} (1)) + b_{1} (0) + c_{1}, a_{2} (1) + b_{2} (0) + c_{2}, a_{3} (1) + b_{2} (0) + c_{3})

T (0, 0) = (a_{1} + c_{1}, a_{2} + c_{2}, a_{3} + c_{3})

T (0, 0) = (- 1, 2, 0)

\Rightarrow a_{1} = c_{1} = - 1, a_{2} + c_{2} = 2, a_{3} + c_{3} = 0

\Rightarrow a_{1} = - 2, a_{2} = 2, a_{3} = 2

Step 4
Also evaluate,

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