Amogh N P
✦ In loving memory of Amogh N P — Architect · Designer · Visionary ✦
An Indian student's hands constructing geometry with a compass.
Lesson II25ARS123 · Architectural Graphics I

2D Geometry & Curves

Building shapes with compass and straightedge — and the curves architecture loves.

≈ 40 min

With nothing but a compass and a straightedge you can build almost any flat figure exactly — no measuring, just geometry. This lesson covers those constructions, the four conic sections, and the engineering curves that keep turning up in real buildings.

Learning objectives

By the end of this lesson, you will be able to — mapped to the course outcomes for Building Materials & Construction I:

1
CO2 · Apply

Perform the basic constructions — bisect, perpendicular, divide a line, regular polygon.

2
CO2 · Understand

Define the conic sections by eccentricity (e) and the focus-directrix rule.

3
CO2 · Apply

Construct an ellipse, a parabola and a hyperbola by instrument methods.

4
CO2 · Analyse

Recognise these curves in real buildings — and avoid the catenary/parabola confusion.

Compass and straightedge

Constructions, polygons & conics

Start with the basic moves, then the regular polygons, then the conic family defined by a single number — eccentricity. Select a topic.

The basic moves

All built with an un-marked straightedge and a compass: bisect a line (the perpendicular bisector), bisect an angle, drop a perpendicular, and divide a line into n equal parts using a slanted ray of equal steps (the intercept theorem).[1]

One cone, four curves Circlee = 0 (cut ⟂ axis) Ellipsee < 1 (cuts all sides) Parabolae = 1 (∥ to a side) Hyperbolae > 1 (∥ to the axis) e = distance to focus ÷ distance to directrix The same family appears when a cutting plane slices a solid cone — the angle of the cut sets the curve.
DiagramThe four conic sections cut from a cone, with their eccentricity values
Ellipse — concentric-circles method A radius cuts both circles; drop a vertical from the outer point and a horizontal from the inner — they meet on the ellipse.
DiagramConstructing an ellipse by the concentric-circles method
Engineering curves

Curves that roll and unwind

The cycloid, involute and spiral are all built as instrument constructions — and they show up as gear teeth, ramps and spiral stairs.[4]

Curves that roll and unwind Cycloid point on a rolling circle Involute taut string unwound Spiral constant spacing per turn Roulettes (cycloids), involutes (gear teeth) and spirals (ramps, the Guggenheim) all start as instrument constructions.
DiagramThree engineering curves: cycloid, involute and Archimedean spiral
Get it right

Two myths to drop

A freely hanging chain is a catenary, not a parabola — the famous Gateway Arch is a weighted catenary, and Gaudí designed with hanging-chain models. (And the Colosseum plan is most likely a polycentric oval, not a true ellipse.)[5, 6]

Catenary ≠ parabola (a useful myth to drop) catenary (hanging chain) parabola (uniform horizontal load) A uniform chain hangs as a catenary (y = a·cosh x/a). Inverted, it stands in pure compression — Gaudí's method. The Gateway Arch is a weighted catenary, NOT a parabola.
DiagramA hanging chain forms a catenary, overlaid with the slightly different parabola
A thin catenary masonry arch / shell against the sky.
PhotoA thin catenary masonry arch / shell against the sky.
A spiral staircase seen from above — a perfect helix.
PhotoA spiral staircase seen from above — a perfect helix.
Check your understanding

Self-assessment

1. A conic with eccentricity e = 1 is:

2. A freely hanging chain of uniform weight forms a:

3. The involute is the curve traced by:

In a nutshell

Recap

Master the basic constructions, then build any regular polygon by inscribing it in a circle.
Define conics by eccentricity (e); draw the ellipse by concentric circles or foci.
Know the engineering curves — cycloid, involute, spiral, helix — and their uses.
Drop two myths: hanging chain = catenary (not parabola); Colosseum = oval (not a true ellipse).
The evidence

References & further reading

  1. [1]Straightedge-and-compass constructions (bisect, perpendicular, divide, polygons). Math Open Reference. https://www.mathopenref.com/constructions.html
  2. [2]Conic section — definition, focus-directrix and eccentricity. Encyclopaedia. https://en.wikipedia.org/wiki/Conic_section
  3. [3]Engineering curves — ellipse / parabola / hyperbola construction methods. GRIET course notes. http://www.it.griet.ac.in/wp-content/uploads/2014/08/I-_UNIT-_CURVES.pdf
  4. [4]Cycloid, involute and Archimedean spiral — definitions. https://en.wikipedia.org/wiki/Involute
  5. [5]Catenary vs parabola; the Gateway Arch as a weighted catenary; Gaudí's hanging-chain models. https://en.wikipedia.org/wiki/Catenary_arch
  6. [6]The Colosseum plan — ellipse or polycentric oval? (scholarly debate). The-Colosseum.net. https://the-colosseum.net/wp/en/ellipse/

Further reading

  • Bhatt, N.D. (2014). Engineering Drawing — Plane and Solid Geometry (53rd ed.). Anand: Charotar — chapters on Geometrical Construction and Curves.
  • Osserman, R. (2010). Mathematics of the Gateway Arch. Notices of the AMS 57(2) — the catenary-vs-parabola authority.
  • Venugopal, K. & Prabhu Raja, V. Engineering Drawing and Graphics. New Delhi: New Age International.

Sources gathered and fact-checked June 2026. Published values vary by source, sample and method — treat as indicative and confirm against the cited standard before structural use.