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 The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

# 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 $V$ .

1. The equation of the straight line through $A$ parallel to ${V}_{2}$ is:
$\begin{array}{cc}\hfill & V={V}_{1}+r{V}_{2}\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& \left(V-{V}_{1}\right)=r{V}_{2}\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& \left(V-{V}_{1}\right)×{V}_{2}=0\hfill \\ \multicolumn{1}{c}{}\end{array}$
2. The equation of the the plane through $A$ perpendicular to ${V}_{2}$ is:
$\left(V-{V}_{1}\right)·{V}_{2}=0$
3. The equation of the line AB is:
$V=r{V}_{1}+\left(1-r\right){V}_{2}$
4. The equations of the bisectors of the angles between ${V}_{1}$ and ${V}_{2}$ are:
$V=r\left(\frac{{V}_{1}}{|{V}_{1}|}±\frac{{V}_{2}}{|{V}_{2}|}\right)\mathrm{ }\text{or} \mathrm{ }V=r\left({\stackrel{\wedge }{V}}_{1}±{\stackrel{\wedge }{V}}_{2}\right)$
5. The perpendicular from $C$ to the line through $A$ parallel to ${V}_{2}$ has as its equation:
$V={V}_{1}-{V}_{3}-{\stackrel{\wedge }{V}}_{2}·\left({V}_{1}-{V}_{3}\right){\stackrel{\wedge }{V}}_{2}.$
6. The condition for the intersection of the two lines ( $V={V}_{1}+r{V}_{3}$ ) and ( $V={V}_{2}+s{V}_{4}$ ) is:
$\left[\left({V}_{1}-{V}_{2}\right){V}_{3}{V}_{4}\right]=0.$
1. The common perpendicular to the above two lines is the line of intersection of the two planes
$\left[\left(V-{V}_{1}\right){V}_{3}\left({V}_{3}×{V}_{4}\right)\right]=0\mathrm{ }\text{and} \mathrm{ }\left[\left(V-{V}_{2}\right){V}_{4}\left({V}_{3}×{V}_{4}\right)\right]=0$
2. The length of this perpendicular is
$\frac{\left[\left({V}_{1}-{V}_{2}\right){V}_{3}{V}_{4}\right]}{|{V}_{3}×{V}_{4}|}.$
7. The equation of the line perpendicular to the plane <i>ABC</i> is
$V={V}_{1}×{V}_{2}+{V}_{2}×{V}_{3}+{V}_{3}×{V}_{1}$
and the distance of the plane from the origin is
$\frac{\left[{V}_{1}{V}_{2}{V}_{3}\right]}{|\left({V}_{2}-{V}_{1}\right)×\left({V}_{3}-{V}_{1}\right)|}$
8. In general the vector equation $V·{V}_{2}=r$ defines the plane which is perpendicular to ${V}_{2}$ , and the perpendicular distance from $A$ to this plane is
$\frac{r-{V}_{1}·{V}_{2}}{|{V}_{2}|}$
9. The distance from $A$ , measured along a line parallel to ${V}_{3}$ , is $\frac{r-{V}_{1}·{V}_{2}}{{V}_{2}·{\stackrel{\wedge }{v}}_{3}}$ or $\frac{r-{V}_{1}·{V}_{2}}{{v}_{2}cos\theta }$ where $\theta$ is the angle between ${V}_{2}$ and ${V}_{3}$ . (If this plane contains the point $C$ then $r={V}_{3}·{V}_{2}$ and if it passes through the origin then $r=0$ .)
10. For two given planes $\left\{V·{V}_{1}=r,V·{V}_{2}=s\right\}$ any plane through the line of intersection of these planes is given by $V·\left({V}_{1}+\lambda {V}_{2}\right)=r+\lambda s$ where $\lambda$ is a scalar parameter. In particular, using $\lambda =±\frac{|{V}_{1}|}{|{V}_{2}|}$ gives the two equations for the two planes bisecting the angle between the given planes.
11. The plane through $A$ parallel to the plane of ${V}_{2}$ , ${V}_{3}$ is
$\begin{array}{cc}\hfill & V={V}_{1}+r{V}_{2}+s{V}_{3}\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& \left(V-{V}_{1}\right)·{V}_{2}×{V}_{3}=0\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& \left[{\mathrm{VV}}_{2}{V}_{3}\right]-\left[{V}_{1}{V}_{2}{V}_{3}\right]=0\hfill \\ \multicolumn{1}{c}{}\end{array}$
so that the expansion in rectangular Cartesian coordinates yields (where $V\equiv xi+yj+zk$ ):
$|\begin{array}{ccc}\left(x-{a}_{1}\right)& \left(y-{b}_{1}\right)& \left(z-{c}_{1}\right)\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}|=0$
which is the usual linear equation in $x$ , $y$ , and $z$ .
12. The plane through <i>AB</i> parallel to ${V}_{3}$ is given by $\left[\left(V-{V}_{1}\right)\left({V}_{1}-{V}_{2}\right){V}_{3}\right]=0$ or
$\left[{\mathrm{VV}}_{2}{V}_{3}\right]-\left[{\mathrm{VV}}_{1}{V}_{3}\right]-\left[{V}_{1}{V}_{2}{V}_{3}\right]=0.$
13. The plane through the three points $A$ , $B$ , and $C$ is
$\begin{array}{cc}\hfill & V={V}_{1}+s\left({V}_{2}-{V}_{1}\right)+t\left({V}_{3}-{V}_{1}\right)\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& V=r{V}_{1}+s{V}_{2}+t{V}_{3}\mathrm{ }\mathrm{ }\left(\text{with}\phantom{\rule{0.2em}{0ex}}r+s+t\equiv 1\right)\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& \left[\left(V-{V}_{1}\right)\left({V}_{1}-{V}_{2}\right)\left({V}_{2}-{V}_{3}\right)\right]=0\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& \left[{\mathrm{VV}}_{1}{V}_{2}\right]+\left[{\mathrm{VV}}_{2}{V}_{3}\right]+\left[{\mathrm{VV}}_{3}{V}_{1}\right]-\left[{V}_{1}{V}_{2}{V}_{3}\right]=0\hfill \\ \multicolumn{1}{c}{}\end{array}$
14. For four points $A$ , $B$ , $C$ , $D$ to be coplanar, then
$r{V}_{1}+s{V}_{2}+t{V}_{3}+u{V}_{4}\equiv 0\equiv r+s+t+u$
15. 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.
1. The equation of a sphere with center $O$ and radius $OA$ is
$\begin{array}{cc}\hfill & V·V={v}_{1}^{2}\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& \left(V-{V}_{1}\right)·\left(V+{V}_{1}\right)=0\hfill \\ \multicolumn{1}{c}{}\end{array}$
2. Note that in two-dimensional polar coordinates this is simply
$r=2a\phantom{\rule{0.2em}{0ex}}cos\theta$
3. While in three-dimensional Cartesian coordinates it is
${x}^{2}+{y}^{2}+{z}^{2}-2\phantom{\rule{0.2em}{0ex}}\left({a}_{1}x+{b}_{1}y+{c}_{1}x\right)=0.$
16. The equation of a sphere having the points $A$ and $B$ as the extremities of a diameter is
$\left(V-{V}_{1}\right)·\left(V-{V}_{2}\right)=0.$
17. The square of the length of the tangent from $C$ to the sphere with center $B$ and radius ${V}_{1}$ is given by
$\left({V}_{3}-{V}_{2}\right)·\left({V}_{3}-{V}_{2}\right)={v}_{1}^{2}$
18. The condition that the plane $V·{V}_{3}=s$ is tangential to the sphere $\left(V-{V}_{2}\right)·\left(V-{V}_{2}\right)={v}_{1}^{2}$ is
$\left(s-{V}_{3}·{V}_{2}\right)·\left(s-{V}_{3}·{V}_{2}\right)={v}_{1}^{2}{v}_{3}^{2}.$
19. The equation of the tangent plane at $D$ , on the surface of sphere $\left(V-{V}_{2}\right)·\left(V-{V}_{2}\right)={v}_{1}^{2}$ , is
$\begin{array}{cc}\hfill & \left(V-{V}_{4}\right)·\left({V}_{4}-{V}_{2}\right)=0\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{ }}& V·{V}_{4}-{V}_{2}·\left(V+{V}_{4}\right)={v}_{1}^{2}-{v}_{2}^{2}\hfill \\ \multicolumn{1}{c}{}\end{array}$
20. The condition that the two circles $\left(V-{V}_{2}\right)·\left(V-{V}_{2}\right)={v}_{1}^{2}$ and $\left(V-{V}_{4}\right)·\left(V-{V}_{4}\right)={v}_{3}^{2}$ intersect orthogonally is
$\left({V}_{2}-{V}_{4}\right)·\left({V}_{2}-{V}_{4}\right)={v}_{1}^{2}+{v}_{3}^{2}$

## 3.2 GEOMETRY OF CURVES IN SPACE

Let $g$ be a natural representation of a regular curve $C$ . The <i>arc length</i> is $L={\int }_{a}^{b}|{g}^{\prime }\left(u\right)|\phantom{\rule{0.2em}{0ex}}du$ . At each point

 1 Binormal line $y=\lambda b\left(s\right)+x$ 2 Curvature $\kappa \left(s\right)=n\left(s\right)·k\left(s\right)$ 3 Curvature vector $k\left(s\right)=\stackrel{·}{t}\left(s\right)$ 4 Normal plane $\left(y-x\right)·t\left(s\right)=0$ 5 Osculating plane $\left(y-x\right)·b\left(s\right)=0$ 6 Principal normal line $y=\lambda n\left(s\right)+x$ 7 Principal normal unit vector $n\left(s\right)=±\frac{k\left(s\right)}{|k\left(s\right)|}$ for $k\left(s\right)\ne 0$ 8 Radius of curvature $\rho \left(s\right)=\frac{1}{|\kappa \left(s\right)|}$ when $\kappa \left(s\right)\ne 0$ 9 Rectifying plane $\left(y-x\right)·n\left(s\right)=0$ 10 Tangent line $y=\lambda t\left(s\right)+x$ 11 Torsion $\tau \left(s\right)=-n\left(s\right)·\stackrel{·}{b}\left(s\right)$ 12 Unit binormal vector $b\left(s\right)=t\left(s\right)×n\left(s\right)$ 13 Unit tangent vector $t\left(s\right)=\stackrel{·}{g}\left(s\right)$ with $\left(\stackrel{·}{g}\left(s\right)=\frac{dg}{ds}\right)$

And the osculating sphere is $\left(y-c\right)·\left(y-c\right)={r}^{2}$ where $c=x+\rho \left(s\right)n\left(s\right)-\frac{\stackrel{·}{\kappa }\left(s\right)}{{\kappa }^{2}\left(s\right)\tau \left(s\right)}b\left(s\right)$ and ${r}^{2}={\rho }^{2}\left(s\right)+\frac{{\kappa }^{2}\left(s\right)}{{\kappa }^{4}\left(s\right){\tau }^{2}\left(s\right)}$ Then the moving trihedron is $\left\{t\left(s\right),n\left(s\right),b\left(s\right)\right\}$ and

1. If $x=\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)=d\left(t\right)$ is a regular representation of a regular curve $C$ , then the following hold at a point $d\left(t\right)$ of $C$ :
$\begin{array}{cc}|\kappa |\hfill & =\frac{|{x}^{″}×{x}^{\prime }|}{|{x}^{\prime }|{}^{3}}=\frac{\sqrt{\left({z}^{″}{y}^{\prime }-{y}^{″}{z}^{\prime }\right){}^{2}+\left({x}^{″}{z}^{\prime }-{z}^{″}{x}^{\prime }\right){}^{2}+\left({y}^{″}{x}^{\prime }-{x}^{″}{y}^{\prime }\right){}^{2}}}{\left({x}^{\prime 2}+{y}^{\prime 2}+{z}^{\prime 2}\right){}^{3/2}}\hfill \\ \multicolumn{1}{c}{\tau }& =\frac{det\left({x}^{\prime },{x}^{″},{x}^{\prime }\right)}{|{x}^{\prime }×{x}^{″}|{}^{2}}=\frac{\left({x}^{\prime }×{x}^{″}\right)·{x}^{\prime }\right)}{|{x}^{\prime }×{x}^{″}|{}^{2}}=\frac{{z}^{″\prime }\left({x}^{\prime }{y}^{″}-{y}^{\prime }{x}^{″}\right)+{z}^{″}\left({x}^{″\prime }{y}^{\prime }-{x}^{\prime }{y}^{″\prime }\right)+{z}^{\prime }\left({x}^{″}{y}^{″\prime }-{x}^{″\prime }{y}^{″}\right)}{\left({x}^{\prime 2}+{y}^{\prime 2}+{z}^{\prime 2}\right)\left({x}^{\prime \prime 2}+{y}^{\prime \prime 2}+{z}^{\prime \prime 2}\right)}\hfill \\ \multicolumn{1}{c}{}\end{array}$
2. The vectors of the moving trihedron satisfy the Serret-Frenet equations
$\stackrel{·}{t}=\kappa n,\mathrm{ }\stackrel{·}{n}=-\kappa t+\tau b,\mathrm{ }\stackrel{·}{b}=-\tau n.$
3. For any planar curve represented parametrically by $x=d\left(t\right)=\left(t,f\left(t\right),0\right)$ ,
$|\kappa |=\frac{|\frac{{d}^{2}x}{{dt}^{2}}|}{{\left(1+{\left(\frac{dx}{dt}\right)}^{2}\right)}^{3/2}}.$
4. Expressions for the curvature vector and curvature of a plane curve corresponding to different representations are:
 Representation $x=f\left(t\right),y=g\left(t\right)$ $y=f\left(x\right)$ $r=f\left(\theta \right)$ Curvature vector $k$ $\frac{\left(\stackrel{·}{x}\stackrel{\mathrm{··}}{y}-\stackrel{·}{y}\stackrel{\mathrm{··}}{x}\right)}{\left({\stackrel{·}{x}}^{2}+{\stackrel{·}{y}}^{2}\right){}^{2}}\left(-\stackrel{·}{y},\stackrel{·}{x}\right)$ $\frac{{y}^{″}}{\left(1+{{y}^{\prime }}^{2}\right){}^{2}}\left(-{y}^{\prime },1\right)$ $\frac{\left({r}^{2}+2{{r}^{\prime }}^{2}-{rr}^{″}\right)}{\left({r}^{2}+{{r}^{\prime }}^{2}\right){}^{2}}\left(-\stackrel{·}{r}sin\theta -rcos\theta ,\stackrel{·}{r}cos\theta -rsin\theta \right)$ Curvature $|\kappa |={\rho }^{-1}$ $\frac{|\stackrel{·}{x}\stackrel{\mathrm{··}}{y}-\stackrel{·}{y}\stackrel{\mathrm{··}}{x}|}{\left({\stackrel{·}{x}}^{2}+{\stackrel{·}{y}}^{2}\right){}^{3/2}}$ $\frac{|{y}^{″}|}{\left(1+{{y}^{\prime }}^{2}\right){}^{3/2}}$ $\frac{{r}^{2}+2{{r}^{\prime }}^{2}-{rr}^{″}}{\left({r}^{2}+{{r}^{\prime }}^{2}\right){}^{3/2}}$
5. For a plane curve, the equation of the <i>osculating circle</i> is $\left(y-c\right)·\left(y-c\right)={\rho }^{2}$ , where $c=x+{\rho }^{2}k$ is the center of curvature.
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