2.3 Tangency

• The meaning of a tangent, the point of contact, and the radius rule
• Constructing a tangent at a given point on a circle
• Constructing a tangent to a circle from a point outside it
• Tangents parallel to, and at right angles to, a given line
• External tangents to two circles, equal and unequal
• Internal tangents to two circles, equal and unequal
• Tangent length, and checking a tangent construction
• Tangency in belt and chain drives, gears, cams and wheels

• describe the methods of constructing tangents in drawing
• construct a tangent to a circle from a point
• construct external tangent to two circles
• construct interior tangent to two circles
• appreciate the importance of tangency in drawing

“Where is tangency applied in real life situation?”

The titles are open to everyone. Reading a lesson needs an account — sign in or create one.

1. What a tangent is
   40 min · 1 practice tasks · quiz
   Touching, cutting and missing; the point of contact; and the one rule that every tangent construction rests on.
2. A tangent at a point on the circle
   40 min · 1 practice tasks · quiz
   The simplest case: draw the radius, then erect a perpendicular to it at the given point.
3. A tangent from a point outside the circle
   40 min · 1 practice tasks · quiz
   The semicircle method: draw a circle on OT as diameter and it cuts the given circle at the two points of contact.
4. Why the semicircle method works
   40 min · 1 practice tasks · quiz
   The angle in a semicircle, why it puts the contact points where they are, and the proof that the two tangent lengths are equal.
5. Tangents parallel and square to a line
   40 min · 1 practice tasks · quiz
   How to draw the two tangents parallel to a given line, and the two at right angles to it, using only the radius rule.
6. External tangents to two equal circles
   40 min · 1 practice tasks · quiz
   The two outside tangents of equal circles are simply parallel to the line of centres, and each radius meets its tangent at 90 degrees.
7. External tangents to two unequal circles
   40 min · 1 practice tasks · quiz
   When the circles differ, a helper circle of radius R minus r turns the problem into a tangent from a point, which you already know.
8. Internal tangents to two equal circles
   40 min · 1 practice tasks · quiz
   The two inside tangents of equal circles always cross at the midpoint of the line of centres, and each is a tangent from that midpoint.
9. Internal tangents to two unequal circles
   40 min · 1 practice tasks · quiz
   The inside pair of unequal circles uses a helper of radius R plus r, and the smaller radius is drawn the opposite way to the larger one.
10. Tangent length
    40 min · 1 practice tasks · quiz
    Because the radius meets the tangent at 90 degrees, the length of a tangent from an outside point is just Pythagoras.
11. Checking a tangent construction
    40 min · 1 practice tasks · quiz
    Four checks that take half a minute and catch almost every tangent error before you ink the line.
12. Tangency in belt and chain drives
    40 min · 1 practice tasks · quiz
    An open belt sits on the two external tangents; a crossed belt sits on the two internal ones; a chain does the same on the sprocket pitch circles.
13. Tangency in gears, cams and wheels
    40 min · 1 practice tasks · quiz
    Wherever a round part touches something else - a meshing gear, a cam follower, a tyre on the road - the contact is a tangent.
14. Why tangency matters in drawing
    40 min · 1 practice tasks · quiz
    Pulling the sub strand together: six places a straight part must meet a round one on a tangent, and what a kink on the drawing becomes on the machine.