How do I find the intersection of two circles in XyzPilot?

Use Point/Intof,C,C with a direction selector (XLarge, XSmall, YLarge, or YSmall). Two circles intersect at two points — the selector specifies which one to create.

What is the formula for circle circle intersection?

XyzPilot computes circle-circle intersections analytically using the radical axis method. The two circle equations are subtracted to find the radical line, then substituted back to find the exact intersection coordinates.

Point/Intof,C,C

Circle / Circle Intersection

Creates a point at the intersection of two circles. A direction selector is required to specify which of the two intersection points to create.

What It Does

Two circles can intersect at two points. The Point/Intof,C,C operator computes both intersection points analytically using the radical axis method and creates the one specified by the direction selector.

The result is an exact analytic point — computed directly from the two circle equations. No iteration, no approximation.

Construction Tier: The created point is disposable. Delete it freely once the product geometry built from it is complete.

Syntax

Point/XLarge|XSmall|YLarge|YSmall, Intof, C1, C2

Parameter	Description	Required
`Selector`	XLarge, XSmall, YLarge, or YSmall	✓ Yes
`C1`	First circle reference (e.g. C0, C1…)	✓ Yes
`C2`	Second circle reference (e.g. C1, C2…)	✓ Yes

Quick Input Panel

C:   [first circle reference]
C2:  [second circle reference]
Z:   0
Dir: ○ XL  ○ XS  ● YL  ○ YS

Direction Selector

Selector	Picks
`XLarge`	Intersection point with the largest X coordinate
`XSmall`	Intersection point with the smallest X coordinate
`YLarge`	Intersection point with the largest Y coordinate (upper)
`YSmall`	Intersection point with the smallest Y coordinate (lower)

Important: Use the selector on the axis where the two solutions differ most clearly. For circles with the same centre X coordinate, use YLarge/YSmall. For circles with the same centre Y coordinate, use XLarge/XSmall.

Worked Examples

Example 1 — Two Equal Circles

Setup:
  C0 = Centre (50, 50), Radius 40
  C1 = Centre (90, 50), Radius 40

Point/YSmall, Intof, C0, C1
→ P3 = (70.000, 15.359)   ← lower intersection

Point/YLarge, Intof, C0, C1
→ P4 = (70.000, 84.641)   ← upper intersection

Example 2 — Two Different Circles

Setup:
  C0 = Centre (0, 0), Radius 50
  C1 = Centre (60, 0), Radius 40

Point/YLarge, Intof, C0, C1
→ P5 = (28.583, 40.825)   ← upper intersection

Point/YSmall, Intof, C0, C1
→ P6 = (28.583, -40.825)  ← lower intersection

Example 3 — Vertically Offset Circles

Setup:
  C0 = Centre (0, 0), Radius 30
  C1 = Centre (0, 40), Radius 30

Point/XLarge, Intof, C0, C1
→ P7 = (22.913, 20.000)   ← right intersection

Point/XSmall, Intof, C0, C1
→ P8 = (-22.913, 20.000)  ← left intersection

Analytic Rule

Circle 1: (x - cx1)² + (y - cy1)² = r1²
Circle 2: (x - cx2)² + (y - cy2)² = r2²

Subtracting eliminates the quadratic terms,
giving the radical axis (a linear equation).

Substituting back into either circle equation
gives a quadratic with two exact roots.

The direction selector picks which root to use.

Result: Exact analytic coordinates.
        No iteration. No approximation.

FAQ

What if the circles do not intersect?

If the distance between centres is greater than the sum of radii (circles too far apart) or less than the absolute difference of radii (one circle inside the other), no intersection exists. XyzPilot returns an error. No point is created.

What if the circles are tangent?

If the circles are tangent (one intersection point), use Point/Tanto,C,C instead. It is the correct operator for tangent points between circles.

What if both circles are identical?

Identical circles have infinite intersections. XyzPilot returns an error — the operation is geometrically undefined.

Why does the canvas preview show the wrong point?

While the selector is being chosen, the transient preview may show a point in the wrong position. This is a preview only. Always confirm your selector before committing.