Geometrical Constructions & Engineering Curves Notes

Last reviewed 16 Sept 2026 · 8 min read

Basic geometrical constructions

Construction	Method (outline)
Bisect a line	Arcs of equal radius (> half the line) from both ends; join intersections — perpendicular bisector
Bisect an angle	Arc from vertex cutting both arms; equal arcs from these points; join vertex to intersection
Perpendicular from a point	Arcs cutting the line; bisect the chord
Parallel line	Set squares/mini-drafter, or equal arcs
Divide a line into n equal parts	Draw an inclined line from one end, step off n equal divisions, join last division to the other end, draw parallels
Regular polygon on a given side (general method)	Draw a semicircle on an extended side, divide it into n equal parts; the second division gives the next side; complete by arcs
Regular polygon in a circle	Divide the diameter into n equal parts and use arcs (approximate general method), or divide the circumference by 360°/n
Hexagon in a circle	Step off the radius six times around the circle
Tangent to a circle at a point	Perpendicular to the radius at that point
Tangent from an external point	Semicircle on the line joining the point and centre; intersection with circle gives the point of tangency
Arc tangent to two lines/circles	Centre lies at distance R from both (offset lines or concentric arcs)

Interior angle of a regular n-sided polygon $= \frac{( n - 2 ) \times 180°}{n}$ ; exterior angle $= 360°/ n$ .

Conic sections

Conics are curves obtained when a right circular cone is cut by a plane:

Section plane	Curve
Perpendicular to axis	Circle
Inclined to axis, cutting all generators	Ellipse
Parallel to one generator	Parabola
Parallel to the axis (or at a smaller angle with the axis than the generators make)	Hyperbola
Through the apex	Triangle (pair of straight lines)

Definition by eccentricity

A conic is the locus of a point moving so that the ratio of its distance from a fixed point (focus) to its distance from a fixed line (directrix) is constant — the eccentricity e.

e = \frac{distance from focus}{distance from directrix}

Curve	Eccentricity
Ellipse	$e < 1$
Parabola	$e = 1$
Hyperbola	$e > 1$
Circle	$e = 0$ (limiting case)

Ellipse

Locus of a point whose sum of distances from two foci is constant = major axis ( $2 a$ ).
Foci are located on the major axis at distance $c = a^{2} - b^{2}$ from the centre — equivalently, arcs of radius $a$ (half major axis) from the ends of the minor axis cut the major axis at the foci.
Eccentricity $e = c / a$ ; area $= π ab$ .

Methods of construction

Eccentricity (focus–directrix) method.
Concentric circles (auxiliary circles) method — circles on major and minor axes; radial lines; horizontal from minor-circle points and vertical from major-circle points intersect on the ellipse.
Rectangle (oblong) method — rectangle of sides equal to axes; divisions of sides and half-axes joined from the ends of the minor axis.
Arcs of circles (foci) method — using the sum-of-distances property.
Trammel method and parallelogram method; four-centre (approximate) method.

Tangent and normal at a point: the normal bisects the angle between the lines joining the point to the two foci; the tangent is perpendicular to the normal.

Applications: elliptical arches, manholes, gears, bridges, dams (elliptical cross-sections), planetary orbits.

Parabola

Locus of a point equidistant from the focus and directrix ( $e = 1$ ).
The vertex lies midway between focus and directrix.

Methods

Eccentricity method.
Rectangle method — given base (span) and axis height (rise).
Tangent (outline) method — divide two tangent lines into equal parts and join in reverse order.
Parallelogram method.

Applications: parabolic arches, cables of suspension bridges (under uniform load along the span), vertical curves in highways, reflectors, trajectory of projectiles, shape of bending moment diagram under UDL.

Hyperbola

Locus with $e > 1$ ; the difference of distances from the two foci is constant.
Rectangular hyperbola — asymptotes at right angles; $x y =$ constant — represents Boyle's law ( $p V =$ constant).

Applications: cooling towers (hyperboloid shells), gear design, Boyle's law graphs, some arches and dams.

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