The Planar Field & Two Starting Points

Beginning from Nothing Save Minimal Distinction

Classical compass-and-straightedge geometry operates on a pure planar continuum using only two ideal instruments: an unmarked straightedge and a compass. Crucially, neither instrument can operate on a single isolated point—both require two points to project a locus.

Two Givens Establish Unit Measure

Every GEOMETOR construction begins on a blank slate. The only unconstructed givens are the first two starting points placed on the field—conventionally \(A = (0, 0)\) and \(B = (1, 0)\) (or symmetrically \((-\tfrac{1}{2}, 0)\) and \((\tfrac{1}{2}, 0)\) around the midpoint origin).

• Their separation defines the unit measure (\(1\)) and unit quadrance (\(Q = 1\)) of the entire model.

• No ruler ticks, protractor degrees, or arbitrary floating-point coordinates are imported from outside. Every subsequent magnitude—\(\frac{1}{2}\), \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\phi\)—must be earned constructively through the intersection of lines and circles.

Continuous Loci vs. Graphical Segments

In geometor.model and geometor.explorer:

• Lines [ A B ] and circles ( A B ) are infinite or closed continuous proportions projected across the entire plane. Extending lines beyond the local figure and drawing circles in full exposes remote intersections, harmonic conjugates, and symmetry axes that truncated textbook strokes conceal.

• Segments / A B /, polygons < A B C >, and wedges are bounded graphical and metric witnesses highlighted atop the underlying projections once their endpoint vertices have been constructively selected.

Cartesian Algebra Without Decimal Drift

While faithful to Euclid’s constructive postulates (G.14–G.15), GEOMETOR unites synthetic geometry with Descartes’ coordinate plane and SymPy symbolic algebra:

from geometor.model import Model

        model = Model("unit_field")
        A = model.set_point(0, 0, classes=["given"], ID="A")
        B = model.set_point(1, 0, classes=["given"], ID="B")

        # Project baseline and Vesica Piscis
        model.construct_line(A, B)
        model.construct_circle(A, B)
        model.construct_circle(B, A)

        # Every derived point carries exact SymPy coordinates and full ancestry
        for pt in model.points:
            print(f"{model[pt].ID}: ({pt.x}, {pt.y})")
        

Decimal approximations belong to applied measurement; inside the field, every point preserves its exact algebraic identity back to \(A\) and \(B\).