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The Poincaré upper half plane model for
 is the upper half plane
 in which a point is similar to the Euclidean point and a line must be one of the following:
The Poincaré upper half plane model has the drawback that lines in the model do not necessarily resemble Euclidean lines; however, it has the advantage that it is angle preserving. That is, the Euclidean measure of an angle within the model is the angle measure in hyperbolic geometry. This model has the added bonus that analytic geometry is a useful tool for performing constructions. For example, consider the following:
Problem In the upper half plane model, determine and construct the common perpendicular to the lines  and
 .
Solution: The common perpendicular cannot be a vertical ray, so it must be a semicircle. Also, if a semicircle is to be perpendicular to  , it must have a center at  . Thus, the common perpendicular is of the form
 for some  .
Since
 must also be perpendicular to
 , their tangent lines at their point of intersection must be perpendicular. Let  denote this point of intersection. Thus, the line tangent to
 at  must pass through  . Let  denote the slope of this line. Then
 .
For
 ,
 . Thus,
 . Solving for  yields:
Now  can be found:
Finally,  can be found:
Hence, the common perpendicular is
 .
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