Parametrizations

Line segments
Given two points $a,b$ in $\Bbb C$,the line segment $ab$ has parametric equation $$\color{blue}{z(t)=(1-t)a+tb},\quad t\in [0,1].$$
Circles
We know the rectangular equation for a circle with center at the origin: $$x^2+y^2=r^2.$$ The equality also holds when we substitute $$\color{blue}{x=r\cos t,\quad y=r\sin t}.$$ What about parametrizing a circle with center $(a,b)$? We have $$(x-a)^2+(y-b)^2=r^2.$$ We can put $$\color{blue}{x=r\cos t+a,\quad y=r\sin t+b}.$$ In the complex plane, we have $$z(t)=re^{it},\quad t\in [0,2\pi]$$ for circles with center at the origin and $$\color{blue}{z(t)=z_0+re^{it}},\quad t\in [0,2\pi]$$ for circles with center $z_0$.

Ellipses
A ellipse is the set of points $P$ such that the sum of whose distances from two fixed points (the foci $F_1$ and $F_2$) separated by a distance $2c$ is a given positive constant $2a$. It is given by $$\{P:|d(P,F_1)+d(P,F_2)|=2a\}.$$ When $F_1=(-c,0)$ and $F_2=(c,0)$, we have $$\bigg\{(x,y) \;\bigg|\; \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\bigg\}, \quad b=\sqrt{c^2-a^2}$$ with parametrization $$\color{blue}{x=a\cos t, \quad y=b\sin t},\quad t\in [0,2\pi]$$
Hyperbolas
A hyperbola is the set of points $P$ in a plane that the difference of whose distances from two fixed points (the foci $F_1$ and $F_2$) separated by a distance $2c$ is a given positive constant $2a$. It is given by $$\{P:|d(P,F_1)-d(P,F_2)|=2a\}.$$ When $F_1=(-c,0)$ and $F_2=(c,0)$, we have $$\bigg\{(x,y)\;\bigg| \;\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\bigg\}, \quad b=\sqrt{c^2-a^2}.$$ Recall that $\cosh^2 t-\sinh^2 t=1$. The right branch is parametrized by $$\color{blue}{x=a\cosh t,\quad y=b\sinh t},\quad t\in (-\infty,\infty).$$ The left branch is parametrized by $$\color{blue}{x=-a\cosh t,\quad y=b\sinh t},\quad t\in (-\infty,\infty).$$
More to know
Lemniscates
A lemniscate is the set of points $P$ in a plane that the product of whose distances from two fixed points (the foci $F_1$ and $F_2$) a distance $2c$ away is the constant $c^2$.

Minimal surfaces of revolution
Let $\gamma:I\to \Bbb R^2$ be a parametrized curve of the form $\gamma(t)=(t,y(t))$ for some smooth function $y:I\to \Bbb R$ with $y(t)>0$ for all $t\in I$. The surface obtained by rotating $\gamma $ about the $x$-axis is given by $$\phi(t,\theta)=(t,y(t)\cos\theta,y(t)\sin\theta).$$ Reference
Parametric curves