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| − | ''double ratio, anharmonic ratio, of four points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271304.png" /> on a straight line'' | + | {{TEX|done}} |
| | + | ''double ratio, anharmonic ratio, of four points $M_1$, $M_2$, $M_3$, $M_4$ on a straight line'' |
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| − | A number denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271305.png" /> and equal to | + | A number denoted by the symbol $(M_1M_2M_3M_4)$ and equal to |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271306.png" /></td> </tr></table>
| + | $$\frac{M_1M_3}{M_3M_2}:\frac{M_1M_4}{M_4M_2}.$$ |
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| − | Here, the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271307.png" /> is considered to be positive if the directions of the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c0271309.png" /> coincide, and is considered to be negative if these directions are opposite. The cross ratio depends on the numbering of the points, which may or may not be the same as the order of their appearance on the straight line. As well as the cross ratio of four points, one may consider the cross ratio of four straight lines passing through a point. This ratio, which is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713010.png" />, is equal to | + | Here, the ratio $M_1M_3/M_3M_2$ is considered to be positive if the directions of the segments $M_1M_3$ and $M_3M_2$ coincide, and is considered to be negative if these directions are opposite. The cross ratio depends on the numbering of the points, which may or may not be the same as the order of their appearance on the straight line. As well as the cross ratio of four points, one may consider the cross ratio of four straight lines passing through a point. This ratio, which is denoted by the symbol $(m_1m_2m_3m_4)$, is equal to |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713011.png" /></td> </tr></table>
| + | $$\frac{\sin(m_1m_3)}{\sin(m_3m_2}:\frac{\sin(m_1m_4)}{\sin(m_4m_2)},$$ |
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| − | and the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713012.png" /> between the straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713014.png" /> is considered together with its sign. If the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713018.png" /> lie on the straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713022.png" />, one has | + | and the angle $(m_im_j)$ between the straight lines $m_i$ and $m_j$ is considered together with its sign. If the points $M_1$, $M_2$, $M_3$, $M_4$ lie on the straight lines $m_1$, $m_2$, $m_3$, $m_4$, one has |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713023.png" /></td> </tr></table>
| + | $$(M_1M_2M_3M_4)=(m_1m_2m_3m_4).$$ |
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| − | If the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713031.png" /> are obtained by the intersection of the same quadruple of straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713035.png" />, then | + | If the points $M_1$, $M_2$, $M_3$, $M_4$ and $M_1'$, $M_2'$, $M_3'$, $M_4'$ are obtained by the intersection of the same quadruple of straight lines $m_1$, $m_2$, $m_3$, $m_4$, then |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713036.png" /></td> </tr></table>
| + | $$(M_1M_2M_3M_4)=(M_1'M_2'M_3'M_4').$$ |
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| − | The cross ratio is an invariant of projective transformations. A cross ratio equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027130/c02713037.png" /> is known as a harmonic ratio (cf. [[Harmonic quadruple|Harmonic quadruple]] of points). | + | The cross ratio is an invariant of projective transformations. A cross ratio equal to $-1$ is known as a harmonic ratio (cf. [[Harmonic quadruple|Harmonic quadruple]] of points). |
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double ratio, anharmonic ratio, of four points $M_1$, $M_2$, $M_3$, $M_4$ on a straight line
A number denoted by the symbol $(M_1M_2M_3M_4)$ and equal to
$$\frac{M_1M_3}{M_3M_2}:\frac{M_1M_4}{M_4M_2}.$$
Here, the ratio $M_1M_3/M_3M_2$ is considered to be positive if the directions of the segments $M_1M_3$ and $M_3M_2$ coincide, and is considered to be negative if these directions are opposite. The cross ratio depends on the numbering of the points, which may or may not be the same as the order of their appearance on the straight line. As well as the cross ratio of four points, one may consider the cross ratio of four straight lines passing through a point. This ratio, which is denoted by the symbol $(m_1m_2m_3m_4)$, is equal to
$$\frac{\sin(m_1m_3)}{\sin(m_3m_2}:\frac{\sin(m_1m_4)}{\sin(m_4m_2)},$$
and the angle $(m_im_j)$ between the straight lines $m_i$ and $m_j$ is considered together with its sign. If the points $M_1$, $M_2$, $M_3$, $M_4$ lie on the straight lines $m_1$, $m_2$, $m_3$, $m_4$, one has
$$(M_1M_2M_3M_4)=(m_1m_2m_3m_4).$$
If the points $M_1$, $M_2$, $M_3$, $M_4$ and $M_1'$, $M_2'$, $M_3'$, $M_4'$ are obtained by the intersection of the same quadruple of straight lines $m_1$, $m_2$, $m_3$, $m_4$, then
$$(M_1M_2M_3M_4)=(M_1'M_2'M_3'M_4').$$
The cross ratio is an invariant of projective transformations. A cross ratio equal to $-1$ is known as a harmonic ratio (cf. Harmonic quadruple of points).
References
| [a1] | H.S.M. Coxeter, "Projective geometry" , Univ. Toronto Press (1974) |