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A convex quadrilateral is a simple quadrilateral whose filled region is a convex set. In the nondegenerate case, each interior angle is strictly less than $\pi$, and its two diagonals of a polygon meet inside the region. A rectangle is a convex quadrilateral; a dart-shaped concave quadrilateral is not. If the diagonal lengths are $p,q$ and the angle between them is $\theta$, splitting about their intersection gives area $pq\sin \theta /2$. This is at most $pq/2$, with equality for perpendicular diagonals.

A cyclic polygon has all its vertices of a polygon on a single circle, its circumcircle. Every nondegenerate triangle is a cyclic polygon, but a general quadrilateral is not: a convex quadrilateral is cyclic exactly when opposite interior angles sum to $\pi$. Every regular polygon is cyclic. For a side subtending central angle $\theta$, the chord length is $2R\sin (\theta /2)$, where $R$ is the circumradius; this converts placement on a circle into side-length calculations.

Another convolution of probability densities gives, for $0\le s\le 1$,Only this part of the density is needed because the fourth length is at most one.

Four positive lengths $a,b,c,d$ can form a nondegenerate quadrilateral if and only if every length is less than the sum of the other three. Necessity follows by following the other three sides between the endpoints of the longest side and applying the triangle inequality. For sufficiency, select a diagonal length $r$ in the intersectionBoth intervals are nonempty, and their intersection is nonempty exactly when $|a-b|<c+d$ and $|c-d|<a+b$. These are precisely the four strict length inequalities. Construct triangles with sides $(a,b,r)$ and $(c,d,r)$ on opposite sides of their common diagonal. Their union has the required simple, nondegenerate quadrilateral as boundary.

The failure events, in which one length is at least the sum of the others, are disjoint except on sets of probability zero. For the specified length $W$, independence and the density above giveThere are four symmetric choices of the overly long side, so the probability isThis is the four-sided case of the polygon probability for independent uniform rods.

A polygon is a closed finite chain of line segments in a plane, with successive segments meeting at vertices of a polygon. In the usual simple case, nonadjacent sides of a polygon do not intersect and the chain bounds an interior region. A triangle has three sides, a quadrilateral four, and a hexagon six. A simple $n$-sided polygon has interior angle sum $(n-2)\pi$: splitting it into $n-2$ triangles explains the formula. A regular polygon has equal side lengths and equal interior angles. Self-intersecting regular star polygons require a separate convention for their interior and signed area.

A quadrilateral is a polygon with four sides and four vertices of a polygon. For a simple quadrilateral, its interior angles sum to $2\pi$ by dividing it along an interior diagonal of a polygon into two triangles. Rectangles, squares and parallelograms are special cases. The qualifier convex quadrilateral excludes a reflex interior angle; a self-crossing four-segment chain requires a different interior convention. Four side lengths alone generally leave freedom to vary the angles and area.

The upper half-plane is the open setIt contains precisely the complex numbers with positive imaginary part: $1+i$ is inside, $1$ lies on its boundary, and $1-i$ belongs to the lower half-plane. In the complex plane its boundary is the real line; in the Riemann sphere the boundary also includes infinity. It is a convex set, hence a simply connected domain, and the Cayley transform between the half-plane and disk makes it biholomorphic to the unit disc.

With the hyperbolic metric $ds=|dz|/\mathrm{Im}z$, the same domain becomes the Poincaré half-plane model. Its hyperbolic geodesics are vertical lines and upper semicircles whose centres lie on the real line. Under the Cayley transform, these become arcs of circles orthogonal to the unit circle, with diameters as the straight-line cases. ^[ref]

Matching colours identify each hyperbolic geodesic and its image. The vertical line through zero maps to a diameter; the vertical line through one maps to an arc ending at $-i$ and $1$. A semicircle from $-1$ to $1$ also maps to a diameter, whereas the semicircle from $0$ to $2$ maps to a curved arc. Their endpoints lie on the boundary, outside the domains: the endpoints are infinitely far from interior points in hyperbolic distance.

The upper half-plane is useful because its straight boundary and explicit upper half-plane automorphisms make conformal maps and hyperbolic geometry calculable. The picture explains why shortest hyperbolic geodesic routes can look curved in Euclidean coordinates; their lengths are measured with the position-dependent hyperbolic metric.

For historical context, in his letter to Klein of 22 September 1882, Poincaré constructs a quadrilateral bounded by four circles orthogonal to a fundamental circle, and studies transformations generating repeated copies. In modern language, the sides inside that circle are hyperbolic geodesics in a disk model. The letter directly exhibits the connection between transformation groups, geometry and complex functions. ^[ref]