circlesfrom

Circlesfrom is a term used in computational geometry and computer graphics to denote a function or operation that constructs one or more circle objects from a set of input constraints. The name reflects a programming convention of describing a task by combining descriptive words.

Common input forms include:

- circlesfrom(p1, p2, p3): returns the unique circle that passes through three non-collinear points.

- circlesfrom(p1, p2, r): returns the circles of radius r that pass through points p1 and p2 (generally two solutions).

- circlesfrom(l1, l2, l3): returns a circle tangent to three lines, when such a circle exists (the three-line tangent problem).

- circlesfrom(points, constraint): a generic form that computes a circle meeting a specified geometric constraint set.

Computation typically involves solving for the circle’s parameters (center coordinates and radius) by solving systems of equations. For three points, one solves for the circle through those points. For two points and a radius, one finds the intersection of two circles of radius r around each point. For tangency to lines, the center is found at equal distances from the relevant lines.

Applications include computer-aided design (CAD), vector graphics, collision detection, pattern generation, and geometry education. In software libraries, circlesfrom often appears as a factory function or constructor, sometimes exposed as specialized methods like Circle.fromTwoPointsAndRadius or Circle.fromThreePoints.

Limitations arise from input degeneracies: collinear points, incompatible constraints, or cases yielding no solution or multiple symmetric solutions. See also circle, circle through three points, and Apollonius problems in analytic and computational geometry contexts.