Technical Glossary of Constructive Geometry

Analytical, Locus & Transformational Terminology (English · Dansk · Classical)

Analysis (Analytic Assumption) (Dansk: Analyse)

Petersen’s heuristic starting rule (p. 4): assume the construction problem already solved, draw a figure representing the solved configuration, and examine the relations linking given and sought elements until a constructible sub-figure or pair of loci emerges.

Centre of Similitude (Dansk: Lighedspunkt)

A fixed point P from which radii vectors to every point A of a curve k are multiplied by a constant signed ratio +-m/n to generate a homothetic curve K; two circles possess an external (+R1/R2) and an internal (-R1/R2) centre of similitude (pp. 22–24, 92–93).

Circle of Apollonius (Locus f) (Dansk: Apollonius’ Cirkel)

The locus of points P whose distances from two fixed points A and B have a constant ratio m : n (m != n): a circle whose diameter CC1 joins the internal and external harmonic division points of AB in the ratio m : n (pp. 11–12).

Determinate, Indeterminate, and Over-Determinate Problems (Dansk: Bestemt, ubestemt og overbestemt Opgave)

A problem is determinate when it admits a finite number of solutions, indeterminate when removing one condition leaves infinitely many solutions tracing a geometric locus, and more than determinate when independent conditions exceed the degrees of freedom (p. 4).

Discussion (Determinateness & Limits) (Dansk: Diskussion)

The fourth stage of solving a geometric construction problem (p. 4): determining the diorismos or algebraic/geometric limits on the given data for which the problem admits 0, 1, 2, …, or infinitely many real solutions.

Harmonical Rays (Locus g) (Dansk: Harmoniske Straaler)

Four concurrent straight lines through vertex O consisting of two given lines together with the pair of lines along which signed perpendicular distances to the two given lines stand in a constant ratio m : n, cutting every transversal in a harmonic quadruple (p. 12).

Inverse Figures (Circle Inversion) (Dansk: Inverse Figurer)

The conformal involutive transformation mapping each point A != P on ray PA to A1 such that PA * PA1 = I (the power of inversion), mapping circles through P to straight lines and preserving tangency and intersection angles (pp. 33–37).

Locus of Lines (Envelope) (Dansk: Geometrisk Sted for Linier)

A family of straight lines satisfying all conditions save one, enveloping a fixed curve (circle or point—including an infinitely distant point for a parallel pencil); intersecting two line-loci reduces to drawing common tangents (pp. 40–41).

Method of Similitude (Homothety) (Dansk: Lighedsmethoden)

Constructing a prototype figure of the required shape either by choosing an arbitrary scale unit (Case a, pp. 27–28) or by relaxing one boundary condition so a free vertex moves along a ray through a centre of similitude (Case b, pp. 28–33, e.g. Problem 155).

Method of Two Loci (Dansk: To geometriske Steder)

When a problem reduces to finding a single unknown point P subject to two independent conditions, suppress each condition in turn so P traces two geometric loci (straight lines or circles); the intersections of the two loci determine P (p. 5).

Parallel Translation (Dansk: Parallelforskydning)

Displacing disconnected segments, angles, or circles parallel to themselves by a known vector so that scattered givens gather into a single constructible triangle or parallelogram (pp. 46–55).

Pivot Point (Dansk: Drejningspunkt)

The unique finite fixed centre O (Miquel/spiral centre) around which one of two directly similar figures can be revolved through angle v and scaled by f to cover the other, found by intersecting the two-line intersection circle with corresponding vertices (pp. 65, 68).

Radical Axis & Radical Centre (Locus h, App. 1) (Dansk: Radikalaxe og Radikalcentrum)

The straight-line locus of points having equal power (equal tangent lengths) with respect to two given circles, derived from Locus h (PB^2 - PA^2 = const.); for three circles, the three pairwise radical axes concur at the radical centre (pp. 12–13).

Replacing (Dansk: Ombytning / Flytning)

Placing a sub-figure (such as a triangle inside a cyclic quadrilateral) in a flipped or rescaled congruent/similar position so that equal or proportional unknown edges coincide and eliminate each other (pp. 56–60).

Revolution Around an Axis (Axial Reflection) (Dansk: Drejning om en Axe)

Reflecting part of a figure across a line of symmetry—as in Hero’s light-ray problem (Problem 338) and polygonal billiards (Problem 339)—to straighten broken equal-angle paths or bring symmetric elements into incidence (pp. 60–64).

Straightedge-and-Compass Constructibility Criterion (Dansk: Konstruktion ved Lineal og Passer)

Petersen’s necessary and sufficient algebraic criterion (Appendix, p. 98): a problem is solvable by straightedge and compasses if and only if the sought quantities can be expressed from the givens rationally and by iterated square roots (degree 2^k).

Theory of Revolution (Spiral Similarity) (Dansk: Drejningstheorien)

Petersen’s unified planar similarity operator (O, v, f) rotating every ray OA from pivot point O through angle of revolution v while scaling its length by ratio of revolution f = OA/Oa (Chapter III, pp. 65–88).

Trapezium (Trapezoid) (Dansk: Trapez)

Following 19th-century British and Continental Danish usage (Trapez), Haagensen’s translation uses ‘trapezium’ for a quadrilateral with one pair of parallel sides (modern North American ‘trapezoid’; see Problem 259, p. 49 & Problem 302, p. 54).