---- > [!definition] Definition. ([[parameterized curve]]) > Let $I \subset \mathbb{R}$ be an [[open interval]]. A **path** in $\mathbb{R}^{n}$ is a [[continuous]] function $\alpha: I \to \mathbb{R}^{n}$. The pair $(\alpha, \im \alpha)$ is called a **curve** in $\mathbb{R}^{n}$. > \ > \ > When context is apparent we may just write $\alpha, \im \alpha$, or $\alpha(I)$. ^885256 > [!definition] Definition. ([[path]]) > Let $X$ be a [[topological space]]. Given $x,y \in X$, a **path in $X$ from $x$ to $y$** is a [[continuous]] function $\gamma:[0,1] \to X$ of the [[closed interval]] $[0,1] \subset \mathbb{R}$ into $X$, such that $\gamma(0)=x$ and $\gamma(1)=y$. (A [[parameterized curve|curve]] starting at $x$ and ending at $y$.) > \ > If in particular $x=y$, then $\gamma$ is called a **loop based at $x$**. > \ > The path $\overline{\gamma}:[0,1] \to X$ given by $\overline{\gamma}(t):=\gamma(1-t)$ is called the **reverse path of $\gamma$**. > ![[CleanShot 2022-09-17 at [email protected]]] > [!basicexample] > **1.** The curve given by $\alpha(t):=(a \cos t, a \sin t, bt), \ t \in \mathbb{R}$ is a [[parameterized curve]] tracing out a helix of pitch (height of one complete turn) $2\pi b$ on the cylinder $x^{2} + y^{2} = a ^{2}$. > > ![[CleanShot 2024-01-11 at 12.40.15.jpg]] > > **2.** The map $\alpha: \mathbb{R} \to \mathbb{R}^{2}$ given by $\alpha(t)=(t^{3}, t^{2}), t\in \mathbb{R}$ is parameterized differentiable curve with the below trace: > > ![[CleanShot 2024-01-11 at 12.44.18.jpg]] > > **3.** The straight [[line]] connecting $(1,2,3)$ to $(4,5,6)$: $\alpha(t)=(1,2,3)+(3,3,3)t, \ t \in [0,1].$ > A reparameterization, used e.g. in the definition of [[convex set]], would be $\alpha(t)=(1,2,3)+ (4,5,6)t - (1,2,3)t = (1,2,3)(1-t)+ (4,5,6)t.$ > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` ---- ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```