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1. I'm analyzing the transformation g(x) = a·f(x) as a vertical stretch/compression.2. I want to express this transformation using homogeneous coordinates.3. What is the correct 3×3 transformation matrix A_stretch that maps (x, f(x), 1) to (x, a·f(x), 1)?Please show the matrix and explain briefly why it works.

Answer

this should answer your question fully.

Answer

hi,
please see the 2 attachments for the detailed answer to your posted question. - attachment no. 1 shows the steps from 1 to 4.- attachment no. 2 shows step 5 and the final matrix.
the answer provided shows the full details with explanations. all the best,

Answer

dude, i wrote everything down in a file.

Answer

1. representation in homogeneous coordinates in 2d, a point

(x, y)

can be represented in homogeneous coordinates as:

\mapsto (x, y, 1)

a linear transformation in homogeneous coordinates is represented by a 3×3 matrix

a

acting on

x, y, 1)^t

. 2. vertical stretch/compression the transformation is:

\cdot f(x)

so geometrically:

\mapsto (x, a \cdot f(x))

x-coordinate: unchanged → factor 1 y-coordinate: scaled by

a

→ factor

a

3. forming the homogeneous transformation matrix a general 2d homogeneous transformation matrix is:

\begin{bmatrix} m_{11} & m_{12} & m_{13} \ m_{21} & m_{22} & m_{23} \ m_{31} & m_{32} & m_{33} \end{bmatrix}

m11m21m31m12m22m32m13m23m33 we want it to satisfy:

[x′y′1]=a⋅[xf(x)1]=[xaf(x)1]\begin{bmatrix} x' \ y' \ 1 \end{bmatrix} = a \cdot \begin{bmatrix} x \ f(x) \ 1 \end{bmatrix} = \begin{bmatrix} x \ a f(x) \ 1 \end{bmatrix}

x′y′1 =a⋅ xf(x)1 = xaf(x)1

\cdot x + 0 \cdot y + 0 \cdot 1

\cdot x + a \cdot y + 0 \cdot 1

\cdot x + 0 \cdot y + 1 \cdot 1

thus the matrix is:

astretch=[1000a0001]\boxed{a_ ext{stretch} = \begin{bmatrix}1 & 0 & 0 \0 & a & 0 \0 & 0 & 1\end{bmatrix}}

1000a0001 4. why this works the first row leaves

x

unchanged. the second row multiplies the

y

-coordinate by

a

, producing the vertical stretch/compression. the third row keeps the homogeneous coordinate at 1, which is necessary for affine transformations. ✅ this is exactly the homogeneous matrix representation of a vertical scaling by

a

.

Answer

practicalexplanationofverticalstretch-compression851.pdf

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