Riemann surface
matlab

Smith, J. (2018). Introduction to Geometric Transformations. Oxford University Press.

Brown, A. (2019). Geometric Transformations and Applications. Wiley.

Miller, R. (2020). Stochastic Methods in Geometric Transformations. Springer.

Johnson, L. (2021). Applications of Geometric Transformations in Engineering. Cambridge University Press.

Chen, H. (2022). Advanced Topics in Geometric Transformations: Homogeneous Transformations and Quaternions. MIT Press.

Example 1.1.2. Let $\alpha: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be defined by $\alpha(x, y)=\left(x^2, y\right)$, then show that $\alpha$ is a trans formation.

Solution: To show that $\alpha$ is a transformation, we shall check one to oneness and on toness of $\alpha$.
Let $\left(x_1, y_1\right),\left(x_2, y_2\right) \in \mathbb{R}^2$.
Suppose $\alpha\left(\left(x_1, y_1\right)\right)=\alpha\left(\left(x_2, y_2\right)\right)$, then we want to show that $\left(x_1, y_1\right)=$ $\left(x_2, y_2\right)$ $\quad \alpha\left(\left(x_1, y_1\right)\right)=\alpha\left(\left(x_2, y_2\right)\right)$ implies $\left(x_1^2, y_1\right)=\left(x_2^2, y_2\right)$ $\quad \Rightarrow y_1=y_2$ and $x_1^2=x_2^2 \Rightarrow x_1 \neq x_2$. Thus $\left(x_1, y_1\right) \neq\left(x_2, y_2\right)$
Hence $\alpha$ is not one to one. So that $\alpha$ is not a transformation.

Example 1.1.5. 1) Let $\alpha: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, defined by $\alpha((x, y))=\left(x-1, \frac{1}{2} y\right)$, then show that $\alpha$ is a collineation.
2) Show that the mapping $\gamma$ that sends each points $(x, y)$ to $\left(-x+\frac{y}{2}, x+2\right)$ is a collineation.

Solution: 1) First show that $\alpha$ is a transformation. Let $\left(x_1, y_1\right),\left(x_2, y_2\right) \in$ $\mathbb{R}^2$

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