Section: 17 | Geometry |
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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:

$\begin{array}{cc}\hfill & V={V}_{1}+r{V}_{2}\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& (V-{V}_{1})=r{V}_{2}\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& (V-{V}_{1})\times {V}_{2}=0\hfill \\ \multicolumn{1}{c}{}\end{array}$ - The equation of the the plane through
$A$
perpendicular to
${V}_{2}$
is:

$(V-{V}_{1})\xb7{V}_{2}=0$ - 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:

$V=r(\frac{{V}_{1}}{\left|{V}_{1}\right|}\pm \frac{{V}_{2}}{\left|{V}_{2}\right|})\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{or}\hspace{0.17em}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}V=r({\stackrel{\wedge}{V}}_{1}\pm {\stackrel{\wedge}{V}}_{2})$ - The perpendicular from
$C$
to the line through
$A$
parallel to
${V}_{2}$
has as its equation:

$V={V}_{1}-{V}_{3}-{\stackrel{\wedge}{V}}_{2}\xb7({V}_{1}-{V}_{3}){\stackrel{\wedge}{V}}_{2}.$ - 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

$[(V-{V}_{1}){V}_{3}({V}_{3}\times {V}_{4})]=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\hspace{0.17em}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}[(V-{V}_{2}){V}_{4}({V}_{3}\times {V}_{4})]=0$ - 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 <i>ABC</i> 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
$V\xb7{V}_{2}=r$
defines the plane which is perpendicular to
${V}_{2}$
, and the perpendicular distance from
$A$
to this plane is

$\frac{r-{V}_{1}\xb7{V}_{2}}{\left|{V}_{2}\right|}$ - The distance from $A$ , measured along a line parallel to ${V}_{3}$ , is $\frac{r-{V}_{1}\xb7{V}_{2}}{{V}_{2}\xb7{\stackrel{\wedge}{v}}_{3}}$ or $\frac{r-{V}_{1}\xb7{V}_{2}}{{v}_{2}cos\theta}$ where $\theta $ is the angle between ${V}_{2}$ and ${V}_{3}$ . (If this plane contains the point $C$ then $r={V}_{3}\xb7{V}_{2}$ and if it passes through the origin then $r=0$ .)
- For two given planes $\{V\xb7{V}_{1}=r,V\xb7{V}_{2}=s\}$ any plane through the line of intersection of these planes is given by $V\xb7({V}_{1}+\lambda {V}_{2})=r+\lambda s$ where $\lambda $ is a scalar parameter. In particular, using $\lambda =\pm \frac{\left|{V}_{1}\right|}{\left|{V}_{2}\right|}$ 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

$\begin{array}{cc}\hfill & V={V}_{1}+r{V}_{2}+s{V}_{3}\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& (V-{V}_{1})\xb7{V}_{2}\times {V}_{3}=0\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& [{\mathrm{VV}}_{2}{V}_{3}]-[{V}_{1}{V}_{2}{V}_{3}]=0\hfill \\ \multicolumn{1}{c}{}\end{array}$

so that the expansion in rectangular Cartesian coordinates yields (where $V\equiv xi+yj+zk$ ):

$\left|\begin{array}{ccc}\left(x-{a}_{1}\right)& \left(y-{b}_{1}\right)& \left(z-{c}_{1}\right)\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}\right|=0$

which is the usual linear equation in $x$ , $y$ , and $z$ . - The plane through <i>AB</i> parallel to
${V}_{3}$
is given by
$[(V-{V}_{1})({V}_{1}-{V}_{2}){V}_{3}]=0$
or

$[{\mathrm{VV}}_{2}{V}_{3}]-[{\mathrm{VV}}_{1}{V}_{3}]-[{V}_{1}{V}_{2}{V}_{3}]=0.$ - The plane through the three points
$A$
,
$B$
, and
$C$
is

$\begin{array}{cc}\hfill & V={V}_{1}+s({V}_{2}-{V}_{1})+t({V}_{3}-{V}_{1})\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& V=r{V}_{1}+s{V}_{2}+t{V}_{3}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}(\text{with}\phantom{\rule{0.2em}{0ex}}r+s+t\equiv 1)\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& [(V-{V}_{1})({V}_{1}-{V}_{2})({V}_{2}-{V}_{3})]=0\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& [{\mathrm{VV}}_{1}{V}_{2}]+[{\mathrm{VV}}_{2}{V}_{3}]+[{\mathrm{VV}}_{3}{V}_{1}]-[{V}_{1}{V}_{2}{V}_{3}]=0\hfill \\ \multicolumn{1}{c}{}\end{array}$ - 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
$OA$
is

$\begin{array}{cc}\hfill & V\xb7V={v}_{1}^{2}\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& (V-{V}_{1})\xb7(V+{V}_{1})=0\hfill \\ \multicolumn{1}{c}{}\end{array}$ - Note that in two-dimensional polar coordinates this is simply

$r=2a\phantom{\rule{0.2em}{0ex}}cos\theta$ - While in three-dimensional Cartesian coordinates it is

${x}^{2}+{y}^{2}+{z}^{2}-2\phantom{\rule{0.2em}{0ex}}({a}_{1}x+{b}_{1}y+{c}_{1}x)=0.$ - The equation of a sphere having the points
$A$
and
$B$
as the extremities of a diameter is

$(V-{V}_{1})\xb7(V-{V}_{2})=0.$ - The square of the length of the tangent from
$C$
to the sphere with center
$B$
and radius
${V}_{1}$
is given by

$({V}_{3}-{V}_{2})\xb7({V}_{3}-{V}_{2})={v}_{1}^{2}$ - The condition that the plane
$V\xb7{V}_{3}=s$
is tangential to the sphere
$(V-{V}_{2})\xb7(V-{V}_{2})={v}_{1}^{2}$
is

$(s-{V}_{3}\xb7{V}_{2})\xb7(s-{V}_{3}\xb7{V}_{2})={v}_{1}^{2}{v}_{3}^{2}.$ - The equation of the tangent plane at
$D$
, on the surface of sphere
$(V-{V}_{2})\xb7(V-{V}_{2})={v}_{1}^{2}$
, is

$\begin{array}{cc}\hfill & (V-{V}_{4})\xb7({V}_{4}-{V}_{2})=0\hfill \\ \multicolumn{1}{c}{\text{or}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}}& V\xb7{V}_{4}-{V}_{2}\xb7(V+{V}_{4})={v}_{1}^{2}-{v}_{2}^{2}\hfill \\ \multicolumn{1}{c}{}\end{array}$ - The condition that the two circles
$(V-{V}_{2})\xb7(V-{V}_{2})={v}_{1}^{2}$
and
$(V-{V}_{4})\xb7(V-{V}_{4})={v}_{3}^{2}$
intersect orthogonally is

$({V}_{2}-{V}_{4})\xb7({V}_{2}-{V}_{4})={v}_{1}^{2}+{v}_{3}^{2}$

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

1. | Binormal line |
$y=\lambda b(s)+x$ |

2. | Curvature |
$\kappa (s)=n(s)\xb7k(s)$ |

3. | Curvature vector |
$k(s)=\stackrel{\xb7}{t}(s)$ |

4. | Normal plane |
$(y-x)\xb7t(s)=0$ |

5. | Osculating plane |
$(y-x)\xb7b(s)=0$ |

6. | Principal normal line |
$y=\lambda n(s)+x$ |

7. | Principal normal unit vector |
$n(s)=\pm \frac{k(s)}{|k(s)|}$ for $k(s)\ne 0$ |

8. | Radius of curvature |
$\rho (s)=\frac{1}{|\kappa (s)|}$ when $\kappa (s)\ne 0$ |

9. | Rectifying plane |
$(y-x)\xb7n(s)=0$ |

10. | Tangent line |
$y=\lambda t(s)+x$ |

11. | Torsion |
$\tau (s)=-n(s)\xb7\stackrel{\xb7}{b}(s)$ |

12. | Unit binormal vector |
$b(s)=t(s)\times n(s)$ |

13. | Unit tangent vector |
$t(s)=\stackrel{\xb7}{g}(s)$ with $(\stackrel{\xb7}{g}(s)=\frac{dg}{ds})$ |

And the osculating sphere is $(y-c)\xb7(y-c)={r}^{2}$ where $c=x+\rho (s)n(s)-\frac{\stackrel{\xb7}{\kappa}(s)}{{\kappa}^{2}(s)\tau (s)}b(s)$ and ${r}^{2}={\rho}^{2}(s)+\frac{{\kappa}^{2}(s)}{{\kappa}^{4}(s){\tau}^{2}(s)}$ 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
$C$
:

$\begin{array}{cc}\left|\kappa \right|\hfill & =\frac{|{x}^{\u2033}\times {x}^{\prime}|}{\left|{x}^{\prime}\right|{}^{3}}=\frac{\sqrt{({z}^{\u2033}{y}^{\prime}-{y}^{\u2033}{z}^{\prime}){}^{2}+({x}^{\u2033}{z}^{\prime}-{z}^{\u2033}{x}^{\prime}){}^{2}+({y}^{\u2033}{x}^{\prime}-{x}^{\u2033}{y}^{\prime}){}^{2}}}{({x}^{\prime 2}+{y}^{\prime 2}+{z}^{\prime 2}){}^{3/2}}\hfill \\ \multicolumn{1}{c}{\tau}& =\frac{det({x}^{\prime},{x}^{\u2033},{x}^{\prime})}{|{x}^{\prime}\times {x}^{\u2033}|{}^{2}}=\frac{({x}^{\prime}\times {x}^{\u2033})\xb7{x}^{\prime})}{|{x}^{\prime}\times {x}^{\u2033}|{}^{2}}=\frac{{z}^{\u2033\prime}({x}^{\prime}{y}^{\u2033}-{y}^{\prime}{x}^{\u2033})+{z}^{\u2033}({x}^{\u2033\prime}{y}^{\prime}-{x}^{\prime}{y}^{\u2033\prime})+{z}^{\prime}({x}^{\u2033}{y}^{\u2033\prime}-{x}^{\u2033\prime}{y}^{\u2033})}{({x}^{\prime 2}+{y}^{\prime 2}+{z}^{\prime 2})({x}^{\prime \prime 2}+{y}^{\prime \prime 2}+{z}^{\prime \prime 2})}\hfill \\ \multicolumn{1}{c}{}\end{array}$ - The vectors of the moving trihedron satisfy the
*Serret-Frenet*equations

$\stackrel{\xb7}{t}=\kappa n,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\stackrel{\xb7}{n}=-\kappa t+\tau b,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\stackrel{\xb7}{b}=-\tau n.$ - For any planar curve represented parametrically by
$x=d(t)=(t,f(t),0)$
,

$\left|\kappa \right|=\frac{\left|\frac{{d}^{2}x}{{dt}^{2}}\right|}{{(1+{\left(\frac{dx}{dt}\right)}^{2})}^{3/2}}.$ - Expressions for the curvature vector and curvature of a
*plane curve*corresponding to different representations are:Representation $x=f(t),y=g(t)$ $y=f(x)$ $r=f(\theta )$ Curvature vector $k$ $\frac{(\stackrel{\xb7}{x}\stackrel{\mathrm{\xb7\xb7}}{y}-\stackrel{\xb7}{y}\stackrel{\mathrm{\xb7\xb7}}{x})}{({\stackrel{\xb7}{x}}^{2}+{\stackrel{\xb7}{y}}^{2}){}^{2}}(-\stackrel{\xb7}{y},\stackrel{\xb7}{x})$ $\frac{{y}^{\u2033}}{(1+{{y}^{\prime}}^{2}){}^{2}}(-{y}^{\prime},1)$ $\frac{({r}^{2}+2{{r}^{\prime}}^{2}-{rr}^{\u2033})}{({r}^{2}+{{r}^{\prime}}^{2}){}^{2}}(-\stackrel{\xb7}{r}sin\theta -rcos\theta ,\stackrel{\xb7}{r}cos\theta -rsin\theta )$ Curvature $\left|\kappa \right|={\rho}^{-1}$ $\frac{|\stackrel{\xb7}{x}\stackrel{\mathrm{\xb7\xb7}}{y}-\stackrel{\xb7}{y}\stackrel{\mathrm{\xb7\xb7}}{x}|}{({\stackrel{\xb7}{x}}^{2}+{\stackrel{\xb7}{y}}^{2}){}^{3/2}}$ $\frac{\left|{y}^{\u2033}\right|}{(1+{{y}^{\prime}}^{2}){}^{3/2}}$ $\frac{{r}^{2}+2{{r}^{\prime}}^{2}-{rr}^{\u2033}}{({r}^{2}+{{r}^{\prime}}^{2}){}^{3/2}}$ - For a plane curve, the equation of the <i>osculating circle</i> is
$(y-c)\xb7(y-c)={\rho}^{2}$
, where
$c=x+{\rho}^{2}k$
is the
*center of curvature*.