cjerdonek / Hyperbolic Lambert quadrilateral c3f9c70a0b7cc058b03008c67a25f86b4b03c9de

A hyperbolic Lambert quadrilateral is a 4-sided polygon in the hyperbolic plane with at least three right angles.

Consider any hyperbolic Lambert quadrilateral. As in the picture on the right, label its sides \(a\), \(b\), \(c\), and \(d\) in the clockwise direction, with the not necessarily right angle \(\theta\) between \(c\) and \(d\).

It can be shown that any two of these five quantities determine the other three. Thus, any three of them satisfy a relation. There are six of these relations up to symmetry, as follows:

1. \(\sinh a \cdot \sinh b = \cos \theta\)
2. \(\cosh a = \cosh c \cdot \sin \theta\)
3. \(\tanh a \cdot \cosh b = \tanh c\)
4. \(\sinh a \cdot \cosh d = \sinh c\)
5. \(\tanh a \cdot \sinh d = \cot \theta\)
6. \(\tanh c \cdot \tanh d = \cos \theta\)

Consider equation (1). Since the hyperbolic sine \(\sinh x\) is positive for positive values of \(x\), the left side of the equation is always positive. This means that \(\theta\) must always be acute (less than \(\pi / 2\)), and in particular non-right. Thus, a hyperbolic Lambert quadrilateral always has exactly three right angles.

Also from equation (1), since \(\sinh x\) approaches \(0\) as \(x\) approaches \(0\), you can see that as \(a\) and \(b\) approach \(0\), the angle \(\theta\) approaches a right angle of \(\pi / 2\). Moreover, you can see from equation (2) that as \(\theta\) approaches \(\pi / 2\), the side lengths \(a\) and \(c\) approach each other (and similarly for \(b\) and \(d\)). Thus, as \(a\) and \(b\) approach \(0\), the hyperbolic Lambert quadrilateral approaches a Euclidean rectangle (which you can see illustrated on the right).

A hyperbolic Lambert quadrilateral is a 4-sided polygon in the hyperbolic plane with at least three right angles.

Consider any hyperbolic Lambert quadrilateral. As in the picture on the right, label its sides \(a\), \(b\), \(c\), and \(d\) in the clockwise direction, with the not necessarily right angle \(\theta\) between \(c\) and \(d\).

It can be shown that any two of these five quantities determine the other three. Thus, any three of them satisfy a relation. There are six of these relations up to symmetry, as follows:

1. \(\sinh a \cdot \sinh b = \cos \theta\)
2. \(\cosh a = \cosh c \cdot \sin \theta\)
3. \(\tanh a \cdot \cosh b = \tanh c\)
4. \(\sinh a \cdot \cosh d = \sinh c\)
5. \(\tanh a \cdot \sinh d = \cot \theta\)
6. \(\tanh c \cdot \tanh d = \cos \theta\)

Consider equation (1). Since the hyperbolic sine \(\sinh x\) is positive for positive values of \(x\), the left side of the equation is always positive. This means that \(\theta\) must always be acute (less than \(\pi / 2\)), and in particular non-right. Thus, a hyperbolic Lambert quadrilateral always has exactly three right angles.

Also from equation (1), since \(\sinh x\) approaches \(0\) as \(x\) approaches \(0\), you can see that as \(a\) and \(b\) approach \(0\), the angle \(\theta\) approaches a right angle of \(\pi / 2\). (This also holds if just one of the two approaches \(0\) while the other stays bounded.) Moreover, you can see from equation (2) that as \(\theta\) approaches \(\pi / 2\), the side lengths \(a\) and \(c\) approach each other (and similarly for \(b\) and \(d\)). Thus, as \(a\) and \(b\) approach \(0\), the hyperbolic Lambert quadrilateral approaches a Euclidean rectangle. This is illustrated on the right.