07. Parametric Representation of Curves and Surfaces

7.1 Standard Curves and Surfaces

7.1.1 Conics

Circle

Equation:

$$x^2+y^2=a^2$$

Parametrisation: $(x,y)=(a\cos \theta ,a\sin \theta )$>
Area $\pi a^2$

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Ellipse

Equation:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(b<a)$$

Parametrisation: $(x,y)=(a\cos \theta ,b\sin \theta )$
Eccentricity: $e=\sqrt{1-\frac{b^2}{a^2}}$
Foci: $(\pm ae,0)$
Directrices: $x=\pm \frac{a}{e}$
Area: $\pi ab$

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Parabola

Equation:

$$y^2=4ax$$

Parametrisation: $(x,y)=(a{t}^2,2at)$
Eccentricity: $e=1$
Focus: $a=0$
Directrix: $x=-a$

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Hyperbola

Equation:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

Parametrisation: $(x,y)=(a\sec t,b\tan t)$
Eccentricity: $e=\sqrt{1+\frac{b^2}{a^2}}$
Focus: $(\pm ae,0)$
Directrix: $x=\pm \frac{a}{e}$
Asymptotes: $y=\pm \frac{bx}{a}$

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7.1.2 Quadrics

Quadrics

Sphere:

$$x^2+y^2+z^2+a^2$$

Ellipsoid

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$

Hyperboloid of One Sheet

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$$

Hyperboloid of Two Sheets

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$$

Paraboloid

$$z=x^2+y^2$$

Hyperbolic Paraboloid

$$z=x^2-y^2$$

Cone

$$z^2=x^2+y^2$$

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Smooth Parametrised Surface

Map $\mathbf{r}$ given by parametrisation

$$\mathbf{r}:U-{\mathbb{R}}^3:(u,v)\mapsto (x(u,v),y(u,v),z(u,v))$$

from an open subset $U\subseteq {\mathbb{R}}^2$ to ${\mathbb{R}}^3$ such that

• $x,y,z$ have continuous partial derivatives with respect to $u,v$ of all orders
• $\mathbf{r}$ is a bijection, with both $\mathbf{r}$ and ${\mathbf{r}}^{-1}$ being continuous
• the vectors

$$\frac{\partial \mathbf{r}}{\partial u}\text{ and }\frac{\partial \mathbf{r}}{\partial v}\text{ are linearly independent}$$

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Tangent Plane

Let $\mathbf{r}:U\to {\mathbb{R}}^3$ be a smooth parametrised surface
Let $\mathbf{p}$ be a point on the surface

The plane containing $\mathbf{p}$ which is also parallel to vectors

$$\frac{\partial \mathbf{r}}{\partial u}(\mathbf{p})\text{ and }\frac{\partial \mathbf{r}}{\partial v}(\mathbf{p})$$

is known as the tangent plane to $\mathbf{r}(U)$ at $\mathbf{p}$

Note that it is well defined as the vectors are linearly independent

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Transclude of 10---Surfaces#^f990d4

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7.2 Scalar line integrals

Scalar Line Integral

Let $\mathbf{F}(r)$ be a vector field

Integral of a vector field $\mathbf{F}(\mathbf{r})$ along a path $C$ defined by $\mathbf{r}=\mathbf{r}(t)$
from $\mathbf{r}(t_0)$ to $\mathbf{r}(t_1)$ is

$${\int }_C\mathbf{F}(\mathbf{r})\cdot d\mathbf{r}={\int }_{t_0}^{t_1}\mathbf{F}(t)\frac{d\mathbf{r}}{dt}dt$$

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7.3 The length of a curve

Length of a Curve

Let $\mathbf{F}(\mathbf{r})$ be a vector field then
Integral of a vector field $\mathbf{F}(\mathbf{r})$ along a path $C$ defined by $\mathbf{r}=\mathbf{r}(t)$
from $\mathbf{r}(t_0)$ to $\mathbf{r}(t_1)$ is

$${\int }_C\mathbf{F}(\mathbf{r})\cdot d\mathbf{r}={\int }_{t_0}^{t_1}|\dot{\mathbf{r}}(t)|dt={\int }_{t_0}^{t_1}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$$

Proof

Let $t$ represent time then
$r(t)$ represents the point on curve $C$ at time $t$ hence ${\mathbf{r}}^{\prime }(t)$ represents the velocity of the point as it moves along curve $C$

Suppose

$$\mathbf{F}(t)=\frac{\dot{\mathbf{r}}(t)}{|\dot{\mathbf{r}}(t)|}$$

Then $\mathbf{F}(r)$ is a unit vector in the direction of vector $\dot{\mathbf{r}}(t)$

Hence

$$\mathbf{F}(t)\cdot \frac{d\mathbf{r}}{dt}=\frac{\dot{\mathbf{r}}(t)}{|\dot{\mathbf{r}}(t)|}\cdot \dot{\mathbf{r}}(t)=|\dot{\mathbf{r}}(t)|$$

So the Scalar Line Integral becomes

$${\int }_C\mathbf{F}(\mathbf{r})\cdot d\mathbf{r}={\int }_{t_0}^{t_1}|\dot{\mathbf{r}}(t)|dt={\int }_{t_0}^{t_1}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$$

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