📗 Translation

If $\color{#5F8}v⃗ ∈ 𝕏⃗$, we define the transformation:

$$\color{#5F5} 𝜏_{v⃗} : 𝕏⃗ ∋ x⃗ ⟶ x⃗ + v⃗ ∈ 𝕏⃗$$

This is called the translation by vector $\color{#5F6}v⃗$.

A translation is an affine transformation, meaning it preserves parallelism and relative distances between points:

$$\color{#5F7} 𝜏_{v⃗} : 𝕏⃗ ⟶ 𝕏⃗$$

---

📘 Homothety

Given a point $c$ in the affine space $\color{#58F}X$ and a scaling factor $\color{#55F}𝜆 ∈ 𝕂 |｛0,1｝$, we define the affine homothety centered at $\color{#56F}c$ with ratio $\color{#57F}𝜆$ as the transformation:

$$\color{#58F} h : 𝕏⃗ ∋ x⃗ ⟶ c⃗ + 𝜆 (x⃗ - c⃗) ∈ 𝕏⃗ $$

Where:

• $\color{#59F}c$: Center of homothety.
• $\color{#5AF}𝜆$: Scaling factor.
  • If $\color{#5BF}𝜆 > 1$, the shape expands.
  • If $\color{#5CF}0 < 𝜆 < 1$, the shape contracts.

---

⚜️ Both translations and homotheties are fundamental transformations in affine geometry. Translations shift objects without altering their size or shape, while homotheties scale objects relative to a fixed center. These transformations are widely used in computer graphics, physics, and architectural modeling.