GeometriaClipping Overview

This document contains information about concepts used by the GeometriaClipping extension library contained within this package.

GeometriaClipping is a 2-dimensional parametric clipping package that works on lines and circular arcs, reduced as ‘simplexes’, which compose larger objects called ‘parametric geometries’.

Parametric Simplex

• Represents the simplest drawing operation, a line or a circular arc;
• Contains a startPeriod/endPeriod, which defines when it should be stroked when drawing from say 0-1 in its parent geometry. Simplexes are chained in sequence, with the end period of a simplex connecting directly into the next simplex’s start period;
• Needs to be ‘clampable’ between a range (start, end], where the result is the same stroke operation, but cut to be within range cut to (max(startPeriod, start), min(endPeriod, end)]. This affects the end points of the stroke in relative terms, so e.g. a clamp of (startPeriod, (endPeriod - startPeriod) / 2] results in the same start point, but an ending point halfway around the stroke path;
• Needs to be ‘computable’ @ period ‘p’, effectively computing a global point on the line/arc at the given period relative to its startPeriod/endPeriod;
• Needs to have an intersection function defined that returns periods on any input pair of simplexes simplex1/simplex2, as pairs of periods (period1, period2) that must map to the same global point ‘p’ when computed with the rule above - this is required for the intersection process.

Parametric Geometry

• Composed of sequential simplexes joined end-to-end, looping back around to the start at ‘endPeriod’;
• Has a global startPeriod/endPeriod that matches the one of the simplexes;
• Periods are comparable and wrap around, so endPeriod + n is the same as startPeriod + n, assuming ‘n’ is < endPeriod - startPeriod - this only holds true for geometries and not simplexes as they don’t have any notion of ‘wrapping’ by themselves;
• Needs a ‘contains’ function that queries for a global point’s containment;
• Needs a ‘simplexes in range’ that performs a clamp of simplexes within a given range (start, end] using the ‘clampable’ property of simplexes, dropping simplexes that are completely out of the range;
• Has a ‘compute at’ function that takes a period ‘p’ and produces an appropriate global point using the simplexes and their periods.

Intersections

• With the intersection function defined for simplexes, intersecting parametric geometries results in a list of pairs for all possible intersections between each simplex in geometry1/geometry2.

Example: Union operation

• Takes as input two geometries, geometry1/geometry2, and returns up to two geometries;
• Start with collecting all intersections of geometry1/geometry2;
• Query if there are any intersections at all;
  • If not, check for containment using each period geometry’s ‘compute at’ and ‘contains’ function, and the result is the appropriate geometry not contained within the other;
  • If neither geometry is contained within the other, the result is a tuple of (geometry1, geometry2).
• Start by querying a random starting point on geometry1, and check if it is contained within geometry2;
  • If so, find the next intersection on geometry1 past the starting point;
  • If not, find instead the previous intersection on geometry1.
• Loop around the geometries, keeping track of which intersections have been seen, and repeat the following, starting with ‘current’ as geometry1 and ‘current point’ as the point computed above:
  • Compute the ‘next’ intersection point on the current geometry;
  • Store current’s ‘simplex in range’ (‘current point’, ‘next point’];
  • At the end, store ‘next’ as ‘current’, and flip the current geometry being stroked from geometry1/geometry2;
  • Repeat until ‘current’ has already been visited.
• The result is then a parametric geometry of all the simplexes collected.