Technical Drawing · Lesson 4

Ellipse, parabola, hyperbola, cycloids, involute, spiral, helix and mechanisms

WAECNECO

What the syllabus asks

WAEC: straight line, circle, ellipse, parabola, hyperbola, helix, cycloid, trochoids, epicycloid, hypocycloid, involute, spiral, link mechanisms). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

NECO: straight line, circle, ellipse, parabola, hyperbola, helix, cycloid, trochoids, epicycloid, hypocycloid, involute, spiral, link mechanisms). (Not a JAMB UTME subject: Technical Drawing is absent from JAMB’s official IBASS e-syllabus list.)

What you will learn

Define a locus and solve problems combining two loci.
Construct the ellipse, parabola and hyperbola by several methods, including eccentricity.
Construct cycloids, trochoids, epicycloids and hypocycloids.
Construct involutes of circles and polygons.
Construct Archimedean spirals and helices and calculate pitch-related values.
Plot loci of points on link mechanisms.

Key points

Combine loci to find points satisfying two rules.
Ellipse e less than 1, parabola e = 1, hyperbola e greater than 1.
Ellipse foci: c² = a² − b²; focal sum = 2a.
Parabola: vertex midway between focus and directrix; rectangle method for arches.
Cycloid base πD and height D.
Epicycloid arc angle 360r/R.
Involute: tangent lengths equal arcs unwound.
Spiral: equal radius steps; helix: pitch per turn.
Slider crank stroke = 2 × crank radius.
Use 12 divisions, number points, join smoothly.

Start here: What a locus is

A locus (plural loci) is the path traced by a point that moves according to a fixed rule. Engineers use loci to design cams, gear teeth, linkages and springs; architects use them for arches and domes; physics students meet them as the path of a projectile or a planet. WAEC plane geometry item 11 lists the loci you must draw: straight line, circle, ellipse, parabola, hyperbola, helix, cycloid, trochoids, epicycloid, hypocycloid, involute, spiral and link mechanisms.

Four basic loci

Points at a fixed distance from a fixed point: a circle.
Points at a fixed distance from a straight line: a pair of parallel lines.
Points equidistant from two fixed points: the perpendicular bisector of the line joining them (td2).
Points equidistant from two intersecting lines: the bisectors of the angles between them.

Combining loci

Problems often ask for a point satisfying two rules at once; it lies where the two loci cross. Example: a borehole must be equidistant from two houses A and B, 60 m apart, and exactly 40 m from a road. Draw the perpendicular bisector of AB and a line parallel to the road at 40 m; the intersection is the site. If the loci cross twice, there are two possible positions; if they do not cross, there is no solution.

A worked two-loci problem

A security light at Ikeja must be placed within a rectangular compound 40 m × 30 m so that it is the same distance from the two long walls and at most 12 m from the gate at the middle of one short wall. The first locus is the centre line of the compound (parallel to the long walls, 15 m from each). The second is the region inside a circle of radius 12 m centred on the gate. The answer is the part of the centre line inside the circle: a segment 12 m long running in from the gate. Drawing to scale (say 1 : 200) and shading the region shows the examiner your reasoning clearly. Loci questions of this kind also appear in Further Mathematics, so practise stating the rule for each locus in words before drawing.

Plotting a locus

Most loci are curves that you plot point by point, then join with a smooth freehand curve or a French curve. Plot at least 12 points per turn or per half curve, and more where the curve bends sharply. Label points in order (1, 2, 3 ...) so that you join them correctly.

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Practice questions

1. A conic section with eccentricity less than 1 is

an ellipse
a parabola
a hyperbola
a circle only

Show answer

A: an ellipse
e < 1 ellipse; e = 1 parabola; e > 1 hyperbola.

2. The eccentricity of a parabola is

0
less than 1
greater than 1
1

Show answer

D: 1
Distance to focus equals distance to directrix.

3. For any point on an ellipse, the sum of the distances to the two foci equals

the distance between foci
the radius of curvature
the major axis
the minor axis

Show answer

C: the major axis
PF₁ + PF₂ = 2a.

4. An ellipse has major axis 120 mm and minor axis 80 mm. Each focus is from the centre about

20 mm
44.7 mm
40 mm
60 mm

Show answer

B: 44.7 mm
√(60² − 40²) ≈ 44.7 mm.

5. The locus of a point on the rim of a wheel rolling along a straight line is a

cycloid
involute
helix
spiral

Show answer

A: cycloid
Rolling on a straight line gives a cycloid.

Key words

Locus: Path of a point moving under a rule.
Conic section: Curve cut from a cone by a plane.
Focus: Fixed point in the definition of a conic.
Directrix: Fixed line in the definition of a conic.
Eccentricity: Ratio of focus distance to directrix distance.
Major axis: Longest diameter of an ellipse.
Minor axis: Shortest diameter of an ellipse.
Vertex: Point where a conic crosses its axis.
Asymptote: Line a curve approaches but never meets.
Cycloid: Path of a point on a circle rolling on a line.
Trochoid: Path of a point inside or outside a rolling circle.
Epicycloid: Path when a circle rolls outside another.

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