The Internal Construction Sequence

The Internal Construction Sequence

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Though its often natural and straightforward to describe a geometry problem in terms of constraints, internally, Geometry Expressions uses constructions.

  1. As you specify geometric objects and constraints, it creates a construction sequence for them.
  2. It executes those constructions to create a symbolic model of your geometry problem.
  3. It creates a drawing that includes the objects and conforms to the constraints. To do so, it assigns sample numeric values for the variables.

For example, in the drawing below, AB is constrained to be length c, BC is constrained to be length a, AB is constrained to be perpendicular to BC, BD is constrained to be perpendicular to AC, and D is constrained to lie on AC.

 

Given this input, Geometry Expressions internally creates a construction sequence such as:

  1. Place point A at an arbitrary location, creating variables to represent the coordinates of point A by default, (u0, v0).
  2. Put point B distance c from A in an arbitrary direction, creating another variable to represent the slope of line AB by default, θ0.
  3. Calculate a sample value for c by measuring the distance of the line AB, using the native coordinate system. (Though the drawings weve shown so far do not show the axes, coordinates, or grid, you can reveal or hide them easily by clicking on the coordinate tool on the toolbar. See the User Interface Reference or embedded Help system.)
  4. Create a line perpendicular to AB through B.
  5. Put point C on this line at distance a from B.

       Two points are possible at this distance either to the left or to the right of the line AB. Geometry Expressions takes its cue from where you drew the line segment and placed the point. If the drawing shows C to the right of AB, Geometry Expressions determines thats what you intended.

  1. Find a line perpendicular to AC through B.
  2. Put D at the intersection of this line and AC.

The existence of a construction sequence for Geometry Expressions is not equivalent to a problem being mathematically well defined. Unfortunately, Geometry Expressions toolbox of constructions cannot cope with a number of mathematically well defined problems. Nevertheless, with a little ingenuity, you can usually find an alternative way to describe the problem, one that the application can construct.

Application-added variables, resolving geometrical ambiguities, and what happens when Geometry Expressions cannot find a construction sequence are all discussed in more detail below.