Tutorial Intermediate ⏱ 12 minutes

Tangent Arc Between Two Circles

Create a smooth blending arc tangent to two existing circles — the fundamental technique for cam profiles, linkage geometry, and smooth transitions.

What You Will Build

A smooth blending arc of radius 40mm connecting two circles (R=20 and R=15) with external tangency — demonstrating the Arc/Tanto,C,C,R operator.

Operators Used

Circle/Centre,R Arc/Tanto,C,C,R Point/Tanto,C,C Trim/C,to,L

Create the Two Base Circles

Create the two circles that will be connected by the blending arc.

P0 = (0, 0)    [centre of first circle]
P1 = (80, 20)  [centre of second circle]

Circle/Centre, P0, R=20, 0  → C0: first circle
Circle/Centre, P1, R=15, 0  → C1: second circle

dist(P0, P1) = sqrt(80^2 + 20^2) = 82.462mm
82.462 > 20 + 15 = 35  [circles do not overlap]

Create the Blending Arc

Create an arc of radius 40mm that is externally tangent to both circles.

Arc/Tanto, C0, C1, R=40, YLarge, OUT
→ A0: blending arc, radius=40mm
   Externally tangent to C0 and C1
   Upper solution (YLarge)

Arc centre computed analytically:
  dist(A0 centre, C0 centre) = 40 + 20 = 60  [external]
  dist(A0 centre, C1 centre) = 40 + 15 = 55  [external]

A0 sweeps from tangent point on C0 to tangent point on C1.

The blending arc is tangent to both circles at its endpoints. The transition is perfectly smooth — no sharp corners.

Find the Tangent Points

Create points at the tangent locations for verification and CAM setup.

Point/Tanto, C0, A0_circle, 0  → P2: tangent point on C0
Point/Tanto, C1, A0_circle, 0  → P3: tangent point on C1

Verification:
  dist(C0 centre, P2) = 20 = C0.radius ✓
  dist(C1 centre, P3) = 15 = C1.radius ✓
  dist(A0 centre, P2) = 40 = A0.radius ✓
  dist(A0 centre, P3) = 40 = A0.radius ✓

Create the Lower Blending Arc

Create the lower blending arc for a complete cam-like profile.

Arc/Tanto, C0, C1, R=40, YSmall, OUT
→ A1: lower blending arc, radius=40mm
   Lower solution (YSmall)

The two blending arcs (A0 and A1) together with the
two circles form a complete closed cam profile.

Create Common Tangent Lines

Alternatively, connect the circles with common tangent lines instead of arcs.

Line/YLarge, Tanto, C0, Tanto, C1
→ L0: upper common external tangent

Line/YSmall, Tanto, C0, Tanto, C1
→ L1: lower common external tangent

With tangent lines instead of arcs:
  Profile = C0 arc + L0 + C1 arc + L1
  [closed profile with straight sides]

Verify the Blending Arc

Verify the arc is correctly tangent to both circles.

Measure → Entity Info on A0:
  type: arc
  radius: 40.000mm
  centre: (computed analytically)

Measure → Distance
Click A0 centre, Click C0 centre
→ Distance: 60.000mm = 40 + 20 ✓  [external tangency]

Click A0 centre, Click C1 centre
→ Distance: 55.000mm = 40 + 15 ✓  [external tangency]

Tips & Best Practices

The blending arc radius must be large enough to reach both circles — minimum R = (dist(centres) - R1 - R2) / 2.
Use YLarge for the upper arc and YSmall for the lower arc to create a complete profile.
External tangency (OUT): blending arc touches outside of both circles.
Internal tangency (IN): blending arc wraps around one or both circles.
This technique is used in cam profiles, linkage geometry, and smooth transitions.