Curves and surfaces

Explicit representation

• The curve (or surface) is described as the graph of a function $y=f(x)$ (or $z=f(x,y)$).
• (Note: Not all curves/surfaces can be represented explicitly.)

Implicit representation

• The curve (or surface) is described as the solution set of an equation:
  • (reacangular equation) $F(x,y)=0$ (or $F(x,y,z)=0$).
  • (polar equation) $r=g(\theta )$ (or $r=g(\theta ,\varphi )$).

Parametric representation

Parametric curve

This is a curve in ${\mathbb{R}}^2$ (plane curve). A space curve in ${\mathbb{R}}^3$ can be defined similarly by $\mathbf{r}(t)=(x(t),y(t),z(t))$, $x,y,z:I\to \mathbb{R}$.

• A set $\mathcal{C}=\{\mathbf{r}(t):t\in I\}$, where:

  • $I\subseteq \mathbb{R}$ is the parameter interval
    • If $I=[a,b]$, then $(x(a),y(a))$ is the initial point and $(x(b),y(b))$ is the terminal point of the curve.
  • $x,y:I\to \mathbb{R}$
    • $x=x(t)$ and $y=y(t)$ are the parametric equations of the curve.
  • $\mathbf{r}:I\to {\mathbb{R}}^2$ and $\mathbf{r}(t)=(x(t),y(t))$
  • $t$ is the parameter of the curve.

• A parameterization $\mathbf{r}(t)$ is smooth on $I$ if ${\mathbf{r}}^{\prime }(t)$ is continuous on $I$ and ${\mathbf{r}}^{\prime }(t)\ne 0$ for all $t\in I$.

  • A curve is smooth if it has a smooth parameterization

• A curve is simple if $\mathbf{r}(t_1)\ne \mathbf{r}(t_2)$ for all $t_1,t_2\in I$ with $t_1\ne t_2$.

• A curve is closed if $\mathbf{r}(a)=\mathbf{r}(b)$.

• A curve is regular if ${\mathbf{r}}^{\prime }(t)\ne 0$ for all $t\in I$.

• The arc length of a $\mathcal{C}$ is given by $L={\int }_a^b\Vert {\mathbf{r}}^{\prime }(t)\Vert dt$

  • (plane curve) $L={\int }_a^b\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$.
  • (space curve) $L={\int }_a^b\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2+{\left(\frac{dz}{dt}\right)}^2}dt$.

• The tangent vector to $\mathcal{C}$ at the point $\mathbf{r}(t)$ is ${\mathbf{r}}^{\prime }(t)$ (if ${\mathbf{r}}^{\prime }(t)\ne 0$).

  • The unit tangent vector is $\mathbf{T}(t)=\frac{{\mathbf{r}}^{\prime }(t)}{\Vert {\mathbf{r}}^{\prime }(t)\Vert }$
  • If $\forall t\in I$, $\Vert \mathbf{r}(t)\Vert =c$ (constant), then $\forall t\in I$, $\mathbf{r}(t)\perp {\mathbf{r}}^{\prime }(t)$. .

• The curvature of a smooth curve $\mathcal{C}$ at the point $\mathbf{r}(t)$ is $\kappa (t)=\frac{\Vert {\mathbf{T}}^{\prime }(t)\Vert }{\Vert {\mathbf{r}}^{\prime }(t)\Vert }$.

Parametric surface

• A set $S=\{\mathbf{r}(u,v):(u,v)\in D\}$, where:

  • $D\subseteq {\mathbb{R}}^2$ is the parameter domain
  • $x,y,z:D\to \mathbb{R}$
    • $x=x(u,v)$, $y=y(u,v)$, and $z=z(u,v)$ are the parametric equations of the surface
  • $\mathbf{r}:D\to {\mathbb{R}}^3$ and $\mathbf{r}(u,v)=(x(u,v),y(u,v),z(u,v))$
  • $u,v$ are the parameters of the surface.

• The surface area of $S$ is given by $A(S)={\iint }_D\Vert \frac{\partial \mathbf{r}}{\partial u}\times \frac{\partial \mathbf{r}}{\partial v}\Vert dA$.

• The tangent plane to $S$ at the point $\mathbf{r}(u_0,v_0)$ is given by the point $\mathbf{r}(u_0,v_0)$ and the normal vector $\frac{\partial \mathbf{r}}{\partial u}\times \frac{\partial \mathbf{r}}{\partial v}$ (if $\frac{\partial \mathbf{r}}{\partial u}\times \frac{\partial \mathbf{r}}{\partial v}\ne 0$).

• (The parametric equations are collectively called a parametric representation or parametrization of the curve/surface)

• (Remark: A curve/surface can have multiple parameterizations.)

• The process of finding a rectangular equation from a parametric representation is called elimination of the parameter.

Examples

Curve	Explicit	Implicit	Parametric
Circle (radius $R$)	Not a function	$x^2+y^2-R^2=0$	$\mathbf{r}(t)=(R\cos t,R\sin t),t\in [0,2\pi ]$
Ellipse	Not a function	$\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0$	$\mathbf{r}(t)=(a\cos t,b\sin t),t\in [0,2\pi ]$
Parabola	$y=a{x}^2+bx+c$	$y-a{x}^2-bx-c=0$	$\mathbf{r}(t)=(t,a{t}^2+bt+c),t\in \mathbb{R}$
Hyperbola	Not a function	$\frac{x^2}{a^2}-\frac{y^2}{b^2}-1=0$	$\mathbf{r}(t)=(a\sec t,b\tan t),t\in (-\frac{\pi }{2},\frac{\pi }{2})$
Line	$y=mx+b$	$Ax+By+C=0$	$\mathbf{r}(t)=(x_0+t,y_0+mt),t\in \mathbb{R}$