Tangent constructions and links between curves and lines
Tangency is a precision topic: the examiner checks that the tangent point is genuinely located, not approximated. It covers tangents to circles and conics and the smooth joins (enlaces) that round a corner or blend two curves.
Tangents to a circle
- At a point of the circle: the perpendicular to the radius at that point.
- From an external point P: draw the circle with diameter PC (C the centre); its intersections with the given circle are the two tangent points.
- Parallel to a direction d: the radius perpendicular to d fixes the tangent point.
- Common tangents to two circles: external tangents are obtained from the centre of homothety found with the difference of radii, internal ones with the sum. Two separate circles have four common tangents; externally tangent, three; intersecting, two; internally tangent, one; nested, none.
Tangents to conics
Ellipse at P: the tangent bisects the exterior angle of the focal radii PF₁ and PF₂, equivalently the normal bisects the interior angle. The symmetric of F₁ about the tangent lies on the line F₂P, at distance 2a from F₂. Tangent to an ellipse from an external point Q: the circle centred on F₂ with radius 2a cuts the circle centred on Q with radius QF₁ at a point F₁′; the line F₂F₁′ meets the ellipse at the tangent point, and QF₁′ is the tangent. Parabola at P: the tangent bisects the angle between PF and the line through P parallel to the axis. Fast construction: drop the perpendicular from P to the axis to get P′ and place T on the axis so that the vertex is the midpoint of P′T; then TP is the tangent. Hyperbola at P: the tangent bisects the interior angle of the focal radii, and the segment it cuts between the asymptotes is bisected at P.
Links (enlaces)
- Two lines: the centre lies at the intersection of parallels drawn at distance r from each line; the tangent points are the feet of the perpendiculars from that centre.
- Line and circle: the centre lies on a parallel to the line at distance r and on a concentric circle of radius R + r (external link) or R − r (inner link).
- Two circles: distances R₁ + r and R₂ + r for an external blend, R₁ − r for an enclosing one; the tangent point lies on the line joining the centres.
Worked example
Blend a horizontal and a vertical line meeting at V with an arc of r = 15 mm. Parallels at 15 mm from each line meet at the centre O; the tangent points are the perpendicular feet from O onto each line, both 15 mm from V; the arc runs from one tangent point to the other, centred outside the corner.
Traps
Marking the tangent point where the centre-to-centre line crosses, instead of on the perpendicular of a line tangent; extending the arc beyond its tangent points; confusing tangent and normal in conics; forgetting that an internal common tangent crosses the line of centres.