Editing Transverse Mercator projection (section)

===Point scale===
In terms of the coordinates with respect to the rotated graticule the [[Scale (map)|point scale]] factor is given by ''k''&nbsp;=&nbsp;sec&nbsp;''φ′'': this may be expressed either in terms of the geographical coordinates or in terms of the projection coordinates:
:<math>
\begin{align}
  k(\lambda,\varphi)&=\frac{k_0}{\sqrt{1-\sin^2\lambda\cos^2\varphi}},\\[5px]
k(x,y)&=k_0\cosh\left(\frac{x}{k_0a}\right).
\end{align}
</math>
The second expression shows that the scale factor is simply a function of the distance from the central meridian of the projection. A typical value of the scale factor is ''k''{{sub|0}}&nbsp;=&nbsp;0.9996 so that ''k''&nbsp;=&nbsp;1 when ''x'' is approximately 180&nbsp;km. When ''x'' is approximately 255&nbsp;km and ''k''{{sub|0}}&nbsp;=&nbsp;1.0004: the scale factor is within 0.04% of unity over a strip of about 510&nbsp;km wide.

{{anchor|convergence}}