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Tutorial Intermediate ⏱ 12 minutes

Tangent Circle Chain

Create a chain of three circles where each circle is externally tangent to the next — a common construction in cam profiles and linkage geometry.

What You Will Build

Three circles of different radii (R=30, R=20, R=15) arranged in a chain where each circle is externally tangent to the next. Tangent points computed analytically.

Operators Used

Circle/Centre,R Point/Tanto,C,C Circle/Tanto,C,C,R
01

Create the First Circle

Place the first circle at the origin with radius 30mm.

P0 = (0, 0) [centre of first circle] Circle/Centre, P0, R=30, 0 → C0: centre=(0,0), radius=30
02

Place the Second Circle Tangent to the First

The second circle (R=20) must be externally tangent to C0. For external tangency, dist(centres) = R1 + R2 = 30 + 20 = 50mm. Place the second centre at (50, 0).

P1 = (50, 0) [centre of second circle] dist(P0, P1) = 50 = 30 + 20 ✓ [externally tangent] Circle/Centre, P1, R=20, 0 → C1: centre=(50,0), radius=20 Verify tangency: Point/Tanto, C0, C1, 0 → P_tangent = (30, 0) [tangent point between C0 and C1]
03

Find the Tangent Point Between C0 and C1

Create the tangent point between the first two circles to verify the construction.

Point/Tanto, C0, C1, 0 → P2 = (30, 0) [tangent point] Verification: dist(C0 centre, P2) = dist((0,0),(30,0)) = 30 = C0.radius ✓ dist(C1 centre, P2) = dist((50,0),(30,0)) = 20 = C1.radius ✓
04

Place the Third Circle Tangent to the Second

The third circle (R=15) must be externally tangent to C1. Place it above C1 for a chain that curves upward. dist(centres) = 20 + 15 = 35mm.

P3 = (50, 35) [centre of third circle] dist(P1, P3) = dist((50,0),(50,35)) = 35 = 20 + 15 ✓ Circle/Centre, P3, R=15, 0 → C2: centre=(50,35), radius=15 Verify tangency: Point/Tanto, C1, C2, 0 → P4 = (50, 20) [tangent point between C1 and C2]
05

Find All Tangent Points

Create tangent points at all connections in the chain.

Point/Tanto, C0, C1, 0 → P2 = (30, 0) [C0-C1 tangent] Point/Tanto, C1, C2, 0 → P4 = (50, 20) [C1-C2 tangent] The chain: C0 → [tangent at P2] → C1 → [tangent at P4] → C2
06

Create a Tangent Line Connecting the Chain

Optionally create a common external tangent line to C0 and C2 to complete the profile.

Line/YLarge, Tanto, C0, Tanto, C2 → L0: upper common external tangent to C0 and C2 Line/YSmall, Tanto, C0, Tanto, C2 → L1: lower common external tangent to C0 and C2 This creates a complete enclosed profile.
The common tangent lines connect the outer edges of the first and last circles, creating a closed cam-like profile.

Tips & Best Practices

  • For external tangency: dist(centres) = R1 + R2 exactly.
  • For internal tangency: dist(centres) = |R1 - R2| exactly.
  • Always verify tangency using Point/Tanto after placing each circle.
  • Use Point/Tanto to find the exact tangent point coordinates for CAM toolpath setup.
  • The tangent point is always on the line connecting the two circle centres.

Operators Used in This Tutorial

Circle/Centre,R

Circle by Centre and Radius

Creates a circle at a centre point.
Point/Tanto,C,C

Tangent Point Circle/Circle

Creates a point at the tangent location between two circles.
Line/Tanto,C,C

Common Tangent to Two Circles

Creates a line tangent to two circles.

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