  Liquid crystal elastomers and glasses can have significant shape change
determined by their director patterns. Cones deformed from circular director
patterns have non-trivial Gaussian curvature localised at tips, curved
interfaces, and intersections of interfaces. We employ a generalised metric
compatibility condition to characterize two families of interfaces between
circular director patterns -- hyperbolic and elliptical interfaces, and find
that the deformed interfaces are geometrically compatible. We focus on
hyperbolic interfaces to design complex topographies and non-isometric origami,
including n-fold intersections, symmetric and irregular tilings. The large
design space of three-fold and four-fold tiling is utilized to quantitatively
inverse design an array of pixels to display target images. Taken together, our
findings provide comprehensive design principles for the design of actuators,
displays, and soft robotics in liquid crystal elastomers and glasses.
