Mathematical methods for physical layout of printed circuit boards: an overview

Abstract

This article surveys mathematical models and methods used for the physical layout of printed circuit boards, in particular component placement and wire routing. The main concepts are briefly described together with relevant references.

Image Based Experiment Scene Builder for Virtual Educational Experiments

“…Use the existing connection lines efficiently, and reduce the count of lines segments in the whole virtual experiment scene. (5). The routing algorithm should be fast enough to meet the interactive requirements.…”

“…Many works have been done on routing algorithms with the development of Computer Aided Design [5]. The traditional maze routing algorithm [7], guarantee to find connection between two connector points if it exists and the routing line generated is guaranteed to be the shortest path.…”

Optimal Interconnection Trees in the Plane

“…2 1 The problem was suggested by Pierre de Fermat (1601Fermat ( -1665 in his celebrated essay on minima and maxima [152]: "Let he who does not approve of my method attempt the solution to the following problem: Given three points in the plane, find a fourth point such that the sum of its distances to the three given points is a minimum". The earliest known solution to Fermat's problem was a geometric construction due to the Italian physicist and mathematician Evangelista Torricelli (1608-1647) [371].…”

“…The following theorem summarises some basic properties of a minimum Steiner tree. The proofs of properties (1) and (4) are left as Exercises 1.5 and 1.6, respectively. …”

“…A Steiner point of degree 3 is called a T-point, and a Steiner point of degree 4 is called a cross. 1 The edge set E.T / consists of edges that connect pairs of nodes u and v by shortest rectilinear paths. The edge .u; v/ is a straight edge if uv is either a horizontal or a vertical line segment; otherwise, .u; v/ is a bent edge.…”