Geometric Constructions and Plane Geometry
Constructions you must do cold
Bisecting a line, an arc and an angle; dropping and erecting perpendiculars, including at the end of a line by the semicircle method; dividing a line into any number of equal parts using an oblique line; and drawing parallels at a given distance. Regular polygons are built either by the general method - draw the circle, step off the side, erect a semicircle on the side and divide it into n equal parts, ray 2 giving the second vertex for every polygon - or by the dedicated pentagon, hexagon and octagon constructions.
Tangency and curves
Tangents from an external point, external and internal tangent arcs of given radius, and fillets between lines and circles all depend on locating the centre as the intersection of two loci: a line parallel to a straight edge at distance r, or an arc concentric with the given circle of radius R ± r. Conic and special curves: ellipse by concentric circles, oblong and arcs-of-circle methods; parabola by the rectangle and tangent methods; hyperbola through a point; involute of a circle and cycloid obtained by dividing the rolling circle into 12 parts.
Worked example
To draw a regular hexagon of 40 mm side, inscribe it in a circle of diameter equal to the across-corners distance, 2 × 40 = 80 mm, and step 40 mm chords around the circumference. The across-flats distance is √3 × 40 = 69.3 mm, the diameter of the inscribed circle - the dimension the spanner actually grips on a nut.
Traps
Construction arcs drawn thick instead of thin and then not erased; tangency judged by eye rather than by intersection of loci; a hexagon dimensioned across corners when the spanner size (across flats) is meant; unequal divisions on a cycloid or involute.