Section: 17 | Geometry |

How to Cite this Reference

3 GEOMETRY

3.1 GEOMETRY OF THE PLANE, STRAIGHT LINE, AND SPHERE

Assume the position vectors of the fixed points $A$ , $B$ , $C$ , $D$ relative to an origin $O$ are $V_{1}$ , $V_{2}$ , $V_{3}$ , $V_{4}$ and the position vector of the variable point $P$ is $V$ .

The equation of the straight line through $A$ parallel to $V_{2}$ is:
$\begin{matrix} V = V_{1} + r V_{2} \\ or & (V - V_{1}) = r V_{2} \\ or & (V - V_{1}) \times V_{2} = 0 \end{matrix}$
The equation of the the plane through $A$ perpendicular to $V_{2}$ is:
$(V - V_{1}) \cdot V_{2} = 0$
The equation of the line AB is:
$V = r V_{1} + (1 - r) V_{2}$
The equations of the bisectors of the angles between $V_{1}$ and $V_{2}$ are:
$V = r (\frac{V_{1}}{| V_{1} |} \pm \frac{V_{2}}{| V_{2} |}) or V = r ({\overset{\land}{V}}_{1} \pm {\overset{\land}{V}}_{2})$
The perpendicular from $C$ to the line through $A$ parallel to $V_{2}$ has as its equation:
$V = V_{1} - V_{3} - {\overset{\land}{V}}_{2} \cdot (V_{1} - V_{3}) {\overset{\land}{V}}_{2} .$
The condition for the intersection of the two lines ( $V = V_{1} + r V_{3}$ ) and ( $V = V_{2} + s V_{4}$ ) is:
$[(V_{1} - V_{2}) V_{3} V_{4}] = 0 .$

The common perpendicular to the above two lines is the line of intersection of the two planes
$[(V - V_{1}) V_{3} (V_{3} \times V_{4})] = 0 and [(V - V_{2}) V_{4} (V_{3} \times V_{4})] = 0$
The length of this perpendicular is
$\frac{[(V_{1} - V_{2}) V_{3} V_{4}]}{| V_{3} \times V_{4} |} .$

The equation of the line perpendicular to the plane ABC is
$V = V_{1} \times V_{2} + V_{2} \times V_{3} + V_{3} \times V_{1}$
and the distance of the plane from the origin is
$\frac{[V_{1} V_{2} V_{3}]}{| (V_{2} - V_{1}) \times (V_{3} - V_{1}) |}$
In general the vector equation $V \cdot V_{2} = r$ defines the plane which is perpendicular to $V_{2}$ , and the perpendicular distance from $A$ to this plane is
$\frac{r - V_{1} \cdot V_{2}}{| V_{2} |}$
The distance from $A$ , measured along a line parallel to $V_{3}$ , is $\frac{r - V_{1} \cdot V_{2}}{V_{2} \cdot {\overset{\land}{v}}_{3}}$ or $\frac{r - V_{1} \cdot V_{2}}{v_{2} cos θ}$ where $θ$ is the angle between $V_{2}$ and $V_{3}$ . (If this plane contains the point $C$ then $r = V_{3} \cdot V_{2}$ and if it passes through the origin then $r = 0$ .)
For two given planes ${V \cdot V_{1} = r, V \cdot V_{2} = s}$ any plane through the line of intersection of these planes is given by $V \cdot (V_{1} + λ V_{2}) = r + λ s$ where $λ$ is a scalar parameter. In particular, using $λ = \pm \frac{| V_{1} |}{| V_{2} |}$ gives the two equations for the two planes bisecting the angle between the given planes.
The plane through $A$ parallel to the plane of $V_{2}$ , $V_{3}$ is
$\begin{matrix} V = V_{1} + r V_{2} + s V_{3} \\ or & (V - V_{1}) \cdot V_{2} \times V_{3} = 0 \\ or & [{VV}_{2} V_{3}] - [V_{1} V_{2} V_{3}] = 0 \end{matrix}$
so that the expansion in rectangular Cartesian coordinates yields (where $V \equiv x i + y j + z k$ ):
$| \begin{matrix} (x - a_{1}) & (y - b_{1}) & (z - c_{1}) \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} | = 0$
which is the usual linear equation in $x$ , $y$ , and $z$ .
The plane through AB parallel to $V_{3}$ is given by $[(V - V_{1}) (V_{1} - V_{2}) V_{3}] = 0$ or
$[{VV}_{2} V_{3}] - [{VV}_{1} V_{3}] - [V_{1} V_{2} V_{3}] = 0 .$
The plane through the three points $A$ , $B$ , and $C$ is
$\begin{matrix} V = V_{1} + s (V_{2} - V_{1}) + t (V_{3} - V_{1}) \\ or & V = r V_{1} + s V_{2} + t V_{3} (with r + s + t \equiv 1) \\ or & [(V - V_{1}) (V_{1} - V_{2}) (V_{2} - V_{3})] = 0 \\ or & [{VV}_{1} V_{2}] + [{VV}_{2} V_{3}] + [{VV}_{3} V_{1}] - [V_{1} V_{2} V_{3}] = 0 \end{matrix}$
For four points $A$ , $B$ , $C$ , $D$ to be coplanar, then
$r V_{1} + s V_{2} + t V_{3} + u V_{4} \equiv 0 \equiv r + s + t + u$
The following formulae relate to a sphere when the vectors are taken to lie in three-dimensional space and to a circle when the space is two-dimensional. For a circle in three dimensions take the intersection of the sphere with a plane.

The equation of a sphere with center $O$ and radius $O A$ is
$\begin{matrix} V \cdot V = v_{1}^{2} \\ or & (V - V_{1}) \cdot (V + V_{1}) = 0 \end{matrix}$
Note that in two-dimensional polar coordinates this is simply
$r = 2 a cos θ$
While in three-dimensional Cartesian coordinates it is
$x^{2} + y^{2} + z^{2} - 2 (a_{1} x + b_{1} y + c_{1} x) = 0 .$

The equation of a sphere having the points $A$ and $B$ as the extremities of a diameter is
$(V - V_{1}) \cdot (V - V_{2}) = 0 .$
The square of the length of the tangent from $C$ to the sphere with center $B$ and radius $V_{1}$ is given by
$(V_{3} - V_{2}) \cdot (V_{3} - V_{2}) = v_{1}^{2}$
The condition that the plane $V \cdot V_{3} = s$ is tangential to the sphere $(V - V_{2}) \cdot (V - V_{2}) = v_{1}^{2}$ is
$(s - V_{3} \cdot V_{2}) \cdot (s - V_{3} \cdot V_{2}) = v_{1}^{2} v_{3}^{2} .$
The equation of the tangent plane at $D$ , on the surface of sphere $(V - V_{2}) \cdot (V - V_{2}) = v_{1}^{2}$ , is
$\begin{matrix} (V - V_{4}) \cdot (V_{4} - V_{2}) = 0 \\ or & V \cdot V_{4} - V_{2} \cdot (V + V_{4}) = v_{1}^{2} - v_{2}^{2} \end{matrix}$
The condition that the two circles $(V - V_{2}) \cdot (V - V_{2}) = v_{1}^{2}$ and $(V - V_{4}) \cdot (V - V_{4}) = v_{3}^{2}$ intersect orthogonally is
$(V_{2} - V_{4}) \cdot (V_{2} - V_{4}) = v_{1}^{2} + v_{3}^{2}$

3.2 GEOMETRY OF CURVES IN SPACE

Let $g$ be a natural representation of a regular curve $C$ . The arc length is $L = \int_{a}^{b} | g^{'} (u) | d u$ . At each point

1.	Binormal line	$y = λ b (s) + x$
2.	Curvature	$κ (s) = n (s) \cdot k (s)$
3.	Curvature vector	$k (s) = \overset{\cdot}{t} (s)$
4.	Normal plane	$(y - x) \cdot t (s) = 0$
5.	Osculating plane	$(y - x) \cdot b (s) = 0$
6.	Principal normal line	$y = λ n (s) + x$
7.	Principal normal unit vector	$n (s) = \pm \frac{k (s)}{\| k (s) \|}$ for $k (s) \neq 0$
8.	Radius of curvature	$ρ (s) = \frac{1}{\| κ (s) \|}$ when $κ (s) \neq 0$
9.	Rectifying plane	$(y - x) \cdot n (s) = 0$
10.	Tangent line	$y = λ t (s) + x$
11.	Torsion	$τ (s) = - n (s) \cdot \overset{\cdot}{b} (s)$
12.	Unit binormal vector	$b (s) = t (s) \times n (s)$
13.	Unit tangent vector	$t (s) = \overset{\cdot}{g} (s)$ with $(\overset{\cdot}{g} (s) = \frac{d g}{d s})$

And the osculating sphere is $(y - c) \cdot (y - c) = r^{2}$ where $c = x + ρ (s) n (s) - \frac{\overset{\cdot}{κ} (s)}{κ^{2} (s) τ (s)} b (s)$ and $r^{2} = ρ^{2} (s) + \frac{κ^{2} (s)}{κ^{4} (s) τ^{2} (s)}$ Then the moving trihedron is ${t (s), n (s), b (s)}$ and

If $x = (x (t), y (t), z (t)) = d (t)$ is a regular representation of a regular curve $C$ , then the following hold at a point $d (t)$ of $C$ :
$\begin{matrix} | κ | & = \frac{| x^{″} \times x^{'} |}{| x^{'} |^{3}} = \frac{\sqrt{(z^{″} y^{'} - y^{″} z^{'})^{2} + (x^{″} z^{'} - z^{″} x^{'})^{2} + (y^{″} x^{'} - x^{″} y^{'})^{2}}}{(x^{' 2} + y^{' 2} + z^{' 2})^{3 / 2}} \\ τ & = \frac{det (x^{'}, x^{″}, x^{'})}{| x^{'} \times x^{″} |^{2}} = \frac{(x^{'} \times x^{″}) \cdot x^{'})}{| x^{'} \times x^{″} |^{2}} = \frac{z^{″'} (x^{'} y^{″} - y^{'} x^{″}) + z^{″} (x^{″'} y^{'} - x^{'} y^{″'}) + z^{'} (x^{″} y^{″'} - x^{″'} y^{″})}{(x^{' 2} + y^{' 2} + z^{' 2}) (x^{'' 2} + y^{'' 2} + z^{'' 2})} \end{matrix}$
The vectors of the moving trihedron satisfy the Serret-Frenet equations
$\overset{\cdot}{t} = κ n, \overset{\cdot}{n} = - κ t + τ b, \overset{\cdot}{b} = - τ n .$
For any planar curve represented parametrically by $x = d (t) = (t, f (t), 0)$ ,
$| κ | = \frac{| \frac{d^{2} x}{{d t}^{2}} |}{{(1 + {(\frac{d x}{d t})}^{2})}^{3 / 2}} .$

Expressions for the curvature vector and curvature of a plane curve corresponding to different representations are:

Representation	$x = f (t), y = g (t)$	$y = f (x)$	$r = f (θ)$
Curvature vector $k$	$\frac{(\overset{\cdot}{x} \overset{··}{y} - \overset{\cdot}{y} \overset{··}{x})}{({\overset{\cdot}{x}}^{2} + {\overset{\cdot}{y}}^{2})^{2}} (- \overset{\cdot}{y}, \overset{\cdot}{x})$	$\frac{y^{″}}{(1 + {y^{'}}^{2})^{2}} (- y^{'}, 1)$	$\frac{(r^{2} + 2 {r^{'}}^{2} - {r r}^{″})}{(r^{2} + {r^{'}}^{2})^{2}} (- \overset{\cdot}{r} sin θ - r cos θ, \overset{\cdot}{r} cos θ - r sin θ)$
Curvature $\| κ \| = ρ^{- 1}$	$\frac{\| \overset{\cdot}{x} \overset{··}{y} - \overset{\cdot}{y} \overset{··}{x} \|}{({\overset{\cdot}{x}}^{2} + {\overset{\cdot}{y}}^{2})^{3 / 2}}$	$\frac{\| y^{″} \|}{(1 + {y^{'}}^{2})^{3 / 2}}$	$\frac{r^{2} + 2 {r^{'}}^{2} - {r r}^{″}}{(r^{2} + {r^{'}}^{2})^{3 / 2}}$

For a plane curve, the equation of the osculating circle is $(y - c) \cdot (y - c) = ρ^{2}$ , where $c = x + ρ^{2} k$ is the center of curvature.

Entry Display

This is where the entry will be displayed

Other ChemNetBase Products

Cookie Policy

We have developed this cookie policy (the “Cookie Policy”) in order to explain how we use cookies and similar technologies (together, “Cookies”) on this website (the “Website”) and to demonstrate our firm commitment to the privacy of your personal information.

The first time that you visit our Website, we notify you about our use of Cookies through a notification banner. By continuing to use the Website, you consent to our use of Cookies as described in this Cookie Policy. However, you can choose whether or not to continue accepting Cookies at any later time. Information on how to manage Cookies is set out later in this Cookie Policy.

Please note that our use of any personal information we collect about you is subject to our Privacy Policy.

What are Cookies?

Cookies are small text files containing user IDs that are automatically placed on your computer or other device by when you visit a website. The Cookies are stored by the internet browser. The browser sends the Cookies back to the website on each subsequent visit, allowing the website to recognise your computer or device. This recognition enables the website provider to observe your activity on the website, deliver a personalised, responsive service and improve the website.

Cookies can be ‘Session Cookies’ or ‘Persistent Cookies’. Session Cookies allow a website to link a series of your actions during one browser session, for example to remember the items you have added to a shopping basket. Session Cookies expire after a browser session and are therefore not stored on your computer or device afterwards. Persistent Cookies are stored on your computer or device between browser sessions and can be used when you make subsequent visits to the website, for example to remember your website preferences, such as language or font size.

Cookies We Use and Their Purpose

We use three types of Cookies - ‘Strictly Necessary’ Cookies, ‘Performance’ Cookies and ‘Functionality’ Cookies. Each type of Cookie and the purposes for which we use them are described in this section. To learn about the specific Cookies we use, please see our List of Cookies.

1. Strictly Necessary Cookies

‘Strictly Necessary’ Cookies enable you to move around the Website and use essential features. For example, if you log into the Website, we use a Cookie to keep you logged in and allow you to access restricted areas, without you having to repeatedly enter your login details. If you are registering for or purchasing a product or service, we will use Cookies to remember your information and selections, as you move through the registration or purchase process.

Strictly Necessary Cookies are necessary for our Website to provide you with a full service. If you disable them, certain essential features of the Website will not be available to you and the performance of the Website will be impeded.

2. Performance Cookies

‘Performance’ Cookies collect information about how you use our Website, for example which pages you visit and if you experience any errors. These Cookies don’t collect any information that could identify you – all the information collected is anonymous. We may use these Cookies to help us understand how you use the Website and assess how well the Website performs and how it could be improved.

3. Functionality Cookies

‘Functionality’ Cookies enable a website to provide you with specific services or a customised experience. We may use these Cookies to provide you with services such as watching a video or adding user comments. We may also use such Cookies to remember changes you make to your settings or preferences (for example, changes to text size or your choice of language or region) or offer you time-saving or personalised features.

You can control whether or not Functionality Cookies are used, but disabling them may mean we are unable to provide you with some services or features of the Website.

First and Third Party Cookies

The Cookies placed on your computer or device include ‘First Party’ Cookies, meaning Cookies that are placed there by us, or by third party service providers acting on our behalf. Where such Cookies are being managed by third parties, we only allow the third parties to use the Cookies for our purposes, as described in this Cookie Policy, and not for their own purposes.

The Cookies placed on your computer or device may also include ‘Third Party’ Cookies, meaning Cookies that are placed there by third parties. These Cookies may include third party advertisers who display adverts on our Website and/or social network providers who provide ‘like’ or ‘share’ capabilities (see the above section on Targeting or Advertising Cookies). They may also include third parties who provide video content which is embedded on our Website (such as YouTube). Please see the website terms and policies of these third parties for further information on their use of Cookies.

To learn about the specific First Party and Third Party Cookies used by our, please see our List of Cookies.

Managing Cookies

You always have a choice over whether or not to accept Cookies. When you first visit the Website and we notify you about our use of Cookies, you can choose not to consent to such use. If you continue to use the Website, you are consenting to our use of Cookies for the time being. However, you can choose not to continue accepting Cookies at any later time. In this section, we describe ways to manage Cookies, including how to disable them.

You can manage Cookies through the settings of your internet browser. You can choose to block or restrict Cookies from being placed on your computer or device. You can also review periodically review the Cookies that have been placed there and disable some or all of them.

You can learn more about how to manage Cookies on the following websites: www.allaboutcookies.org and www.youronlinechoices.com.

Please be aware that if you choose not to accept certain Cookies, it may mean we are unable to provide you with some services or features of the Website.

Changes to Cookie Policy

In order to keep up with changing legislation and best practice, we may revise this Cookie Policy at any time without notice by posting a revised version on this Website. Please check back periodically so that you are aware of any changes.

Questions or Concerns

If you have any questions or concerns about this Cookie Policy or our use of Cookies on the Website, please contact us by email to [email protected]

You can also contact the Privacy Officer for the Informa PLC group at [email protected].