A horizontal curve is a piece of a circle dropped into the corner where two straights meet. Everything a crew needs to stake it is written as eight quantities, and the whole of curve design is knowing which one you are holding.

Three points name themselves. The PI is the point of intersection — the corner where the two straights would have met if the curve were not there. The PC is the point of curvature, where the road leaves the first straight; the PT is the point of tangency, where it rejoins the second.

Then the numbers. $R$ is the radius, centre of the circle out to the road's centreline, in metres or feet. $\Delta$ — read delta — is the deflection angle, the total turn between the two straights, in degrees; it is also the angle the whole arc subtends at the centre, and those two facts being the same thing is the hinge the rest of the geometry swings on. $T$ is the tangent length, PI back to PC along the straight. $L$ is the curve length, PC to PT along the arc. $C$ is the long chord, PC to PT in one straight line — always shorter than $L$, because a shortcut is a shortcut. $E$ is the external distance, PI in to the middle of the arc. $M$ is the middle ordinate, the offset from the midpoint of the chord out to the midpoint of the arc.

Note where $E$ and $M$ are measured from, because it is the one thing this lesson exists to fix: they sit on opposite sides of the same arc. $E$ comes in from the corner on the outside; $M$ goes out from the chord on the inside. They are never the same number.

Last, $D$ — the degree of curve, and the only element that is a convention rather than a measurement. On the arc definition, $D$ is the central angle that one 100 ft station of arc subtends, so $R = \dfrac{5729.578}{D}$ — R equals five seven two nine point five seven eight over D. The constant is simply $100 \times \frac{180}{\pi}$, feet per radian, done once. Because it is built on a 100 ft station it belongs to imperial practice and has no metric twin; this lesson and the next therefore work in feet, exactly as a North American plan sheet does. Everything afterward returns to metres.

The nugget worth carrying: $D$ describes the circle, and every other element describes this particular curve. Two curves can share a degree of curve and share nothing else at all.