Geometric Constructions and Tangency in Technical Drawing

Machine parts are not made of straight lines alone. Fillets, radii, tangent transitions, hexagon heads and elliptical openings appear on almost every drawing. All of them are built from a small set of compass-and-straightedge constructions — the same ones draughtsmen have used for centuries, and the same ones CAD software performs internally today.

You need them for two reasons. First, to draw correctly by hand. Second — and more importantly — to read a drawing: when you understand how a tangent arc is constructed, you understand why the drawing gives you a radius and a centre instead of a length.

Tangency

Two elements are tangent when they touch at exactly one point and pass into one another without a corner. That point is the tangent point, and it is where one element ends and the next begins.

Figure 3.1 — Line-to-arc, external arc-to-arc and internal arc-to-arc tangency

Line to arc. The centre of the arc lies on the perpendicular to the line at the tangent point. Draw the perpendicular first, then the centre falls on it at distance R.
Arc to arc, external. The distance between the two centres equals R1 + R2, and the tangent point lies on the line joining them.
Arc to arc, internal. The distance between the centres equals R1 − R2; the smaller arc sits inside the larger one.

Why this matters when reading a drawing. A drawing rarely tells you where a fillet starts and stops. It gives you a radius, and the geometry does the rest — the tangent points are wherever the construction puts them. If a drawing does give you a tangent point dimension as well as the radius, one of the two is redundant and the part may be over-dimensioned.

Bisecting an angle, dropping a perpendicular

These two constructions turn up constantly — finding a centre, laying out a symmetrical feature, setting out an axis. Neither needs a protractor.

Figure 3.2 — Bisecting an angle and dropping a perpendicular from an external point

The same idea underlies both: two equal arcs struck from two points always intersect on the perpendicular bisector of those points. Everything else follows from that one fact.

Regular polygons and the ellipse

Figure 3.3 — Constructing a regular hexagon, a regular pentagon and an ellipse with compass and straightedge

Hexagon

The hexagon is the easiest of all: in a regular hexagon the side length equals the radius of the circumscribed circle. Set the compass to R and step it six times around the circle. This is why hexagon bar stock and bolt heads are dimensioned across flats or across corners — both follow directly from R.

Pentagon

The pentagon has no such simple relation, so it is built in three steps: find the midpoint of a horizontal radius, open the compass from that midpoint to the top point of the circle, then step that distance five times around the circumference.

Ellipse

The two circle method gives a true ellipse. Draw two concentric circles whose diameters are the major and minor axes. Draw any ray from the centre; where it cuts the large circle drop a vertical, where it cuts the small circle draw a horizontal. Their intersection is a point on the ellipse. Repeat for twelve or so rays and join the points with a smooth curve.

Ellipses appear whenever a circular feature is seen at an angle — a hole on an inclined face, a cylinder in an isometric view. Being able to construct one properly keeps those views from looking wrong.

Worked Example: Drawing a Hexagon with Compasses

The polygon you will meet most in the shop is the hexagon (bolt heads, nuts). The construction takes three steps: draw the circle of the given diameter, step the compasses around the circle without changing the radius (exactly six points appear), join the points. This works because of one property: the side of an inscribed hexagon equals the circle’s radius:

Test yourself

Two arcs of R30 and R18 are externally tangent. How far apart are their centres?
48 mm — R1 + R2.
The same two arcs, internally tangent?
12 mm — R1 − R2.
Where does the centre of an arc tangent to a straight line lie?
On the perpendicular to that line at the tangent point, at a distance equal to the radius.
What is the side of a regular hexagon inscribed in a Ø60 circle?
30 mm — equal to the radius.
Can you bisect an angle without a protractor?
Yes — arc from the vertex, two equal arcs from the resulting points, join the vertex to their intersection.
Why does an ellipse appear on a drawing at all?
Because a circle seen from any direction other than square-on projects as an ellipse.

Next lesson. So far everything has been flat geometry on one plane. Lesson 4 introduces orthographic projection — how a three-dimensional part becomes two or three flat views, and the crucial difference between first angle and third angle.