# Deformable Templates

## Generators for 2-D Shapes

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A template is any set of generators. In the case where a template is a line, the generators are the endpoints. In the case where a template is a polygon, the generators are the vectors that define the vertices of the polygon.

A template can be changed or "deformed" into an endless amount of shapes. The importance of a template is that all of the resulting shapes can be related to each other based on the amount of variance each one has from the template. There are many ways to deform a template. One way is to scale the template and another is to rotate the template. To rotate the generators by an angle $$\theta$$ and scale them by a factor $$\rho$$, we perform the following operation:

\left[\begin{matrix} \rho\cos\theta & \rho\sin\theta \\ -\rho\sin\theta & \rho\cos\theta \end{matrix}\right] \left[\begin{matrix} x_0 \\ y_0 \end{matrix}\right] = \left[\begin{matrix} x \\ y \end{matrix}\right]

This page is an introduction to the scaling and rotation of a vector. The generator in this case is the vector (1,0) in black.

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Rotate:
Scale:

$$\theta$$: 0 $$\rho$$: 1

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