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Mathematical representation: $T(x, y) = (x+dx, y+dy)$; Application: Moving objects in a 2D space without altering orientation.

Mathematical representation: Maps texture coordinates $(u, v)$ to object coordinates $(x, y, z)$; Application: Applying 2D image textures to the surface of a 3D model.

Mathematical representation: $S(x, y) = (k_x x, k_y y)$ where $k_x$ and $k_y$ are the scaling factors; Application: Altering the width and height of an object independently.

Mathematical representation: $Sh_x(x, y) = (x+ky, y)$ for shearing in X-direction, and $Sh_y(x, y) = (x, y+kx)$ for shearing in Y-direction; Application: Skewing an object along a particular axis.

Mathematical representation: Combination of linear transformations and translations; Application: Maintains lines and parallelism in transformations.

Mathematical representation: $R(x, y) = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta)$; Application: Rotating an object about a pivot point.

Mathematical representation: $S(x, y) = (kx, ky)$ where $k$ is the scaling factor; Application: Changing the size of an object uniformly.

Mathematical representation: $Ref_x(x, y) = (-x, y)$ for reflection across the Y-axis, and $Ref_y(x, y) = (x, -y)$ for reflection across the X-axis; Application: Creating a mirror image of an object.

Mathematical representation: $P(x, y, z) = (x, y)$; Application: Projecting a 3D object onto a 2D plane by ignoring the z-component.

Mathematical representation: $P(x, y, z) = (\frac{fx}{z}, \frac{fy}{z})$ where $f$ is the focal length; Application: Simulating realistic 3D views by projecting onto a 2D plane with a point of convergence.

Mathematical representation: Representation of a point $(x, y)$ as $(x, y, w)$; Application: Facilitates transformations like translation as matrix operations in graphics.