Metric Geometry

This is a sub-page of our page on Geometry.

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The sub-pages of this page are:

Euclidean Geometry
Non-Euclidean Geometry

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Related sources of information:

Projective metrics
Metric Projective Geometry
Cayley-Klein Metric

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Below we make the following summation convention (due to Einstein):

\, {\omega}^i x_i \stackrel {\mathrm{def}}{=}{\sum\limits_{1 \le k \le n+1}^{ \text {} } {\omega}^k x_k} \, .

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Let \, X \, and \, \omega \, denote a point and a hyperplane in \, P^3 \, , and let

\,{\lbrack \, X \, \rbrack}_{C.T.U} = ( x_1 : x_2 : x_3 : x_4 ) = (x : y : z : t) \, , and

\,{\lbrack \, \omega \, \rbrack}_{c.t.u} = ( {\omega}^1 : {\omega}^2 : {\omega}^3 : {\omega}^4 ) = (u : v : w : p) \,

denote their respective coordinates in two dually unified systems.

Then the point \, X \, and the plane \, \omega \, are incident if and only if

\, {\omega}^i x_i = 0 \, .

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The Poncelét-Gergonne (bilinear form) duality can be expressed as

\, ux+vy+wz+pt=0 \,

or, using index notation:

\, {\omega}^1 x_1+{\omega}^2 x_2+{\omega}^3 x_3 + {\omega}^4 x_4 = 0 \, .

Centrally symmetric quadratic forms in point coordinates:

\, \frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c} = t^2 \,

The same quadratic form in plane coordinates:

\, au^2+bv^2+cw^2 = p^2 \,

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\, \begin{matrix} {\begin{matrix} \\ \text{Point coordinates:} \\ \\ {x^2/a+y^2/b+z^2/c = t^2} \\ \\ {x^2+y^2+z^2-\frac{1}{\epsilon}t^2 = 0} \\ \\ \begin{matrix} {\epsilon \rightarrow 0 \text{ leads to a logical collapse:}} \\ \\ {x^2+y^2+z^2 = 0} \\ {t^2 = 0} \end{matrix} \end{matrix}} & {\; \; \; \; \; \; \; \; \; \;} & {\begin{matrix} \text{Plane coordinates:} \\ \\ {au^2+bv^2+cw^2 = p^2} \\ \\ {u^2+v^2+w^2-{\epsilon}p^2 = 0} \\ \\ {\epsilon \rightarrow 0 \text{ leads to a value collapse:}} \\ \\ {u^2+v^2+w^2 = 0} \end{matrix}} \end{matrix} \,

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Distance metric: \, d_{istance}(A,B) \stackrel {\mathrm{def}}{=} k \log(AB|O_1 O_2) .

Distance metric
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Angle metric: \, a_{ngle}(a,b) \stackrel {\mathrm{def}}{=} k \log(ab|o_1 o_2) .

Angle metric

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Laguerre’s angle formula:

Laguerre’s angle formula(diagram):
Laguerre's angle formula(diagram)

Laguerre’s angle formula(computation):

\, (PQ|RS) = \frac{RP/RQ}{SP/SQ} = \frac{\frac{i-tan{\theta}_1}{i-tan{\theta}_2}}{\frac{-i-tan{\theta}_1}{-i-tan{\theta}_2}} = \frac{(i cos{\theta}_1 - sin{\theta}_1)(-i cos{\theta}_2 - sin{\theta}_2)}{(-i cos{\theta}_1 - sin{\theta}_1)(i cos{\theta}_2 - sin{\theta}_2)} = \,

\, = \frac{(cos{\theta}_1 + i sin{\theta}_1)(cos{\theta}_2 - i sin{\theta}_2)}{( cos{\theta}_1 - i sin{\theta}_1)(cos{\theta}_2 + i sin{\theta}_2)} = \frac{e^{i{{\theta}_1}} e^{-i{{\theta}_2}}}{e^{-i{{\theta}_1}} e^{i{{\theta}_2}}} = \frac{e^{i({{\theta}_1 - {\theta}_2})}}{e^{-i({{\theta}_1 - {\theta}_2})}} = e^{2 i({{\theta}_1 - {\theta}_2})} .

We see that if we choose the constant \, k = \frac{1}{2i} \, this formula is indeed expressing the familiar euclidean angle. This calculation was discovered by Laguerre in 1853, and it is referred to as Laguerre’s angle formula:

\, \frac{1}{2i} \log(PQ|RS) = {{\theta}_1 - {\theta}_2} .

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