Tutorial Beginner ⏱ 10 minutes

Intersection-Based Construction

Locate geometry precisely using line and circle intersections — the fundamental technique for all complex CNC profile construction.

What You Will Build

A demonstration profile built entirely from intersection points — showing how to locate any point in a drawing analytically without manual coordinate calculation.

Operators Used

Point/Intof,L,C Point/Intof,C,C Point/Intof,L,L Line/P,P Circle/Centre,R

Define Reference Lines

Create horizontal and vertical reference lines to establish the construction grid.

Line/Parlel, XAxis, D=0    → L0: X axis (Y=0)
Line/Parlel, XAxis, D=50   → L1: horizontal at Y=50
Line/Parlel, YAxis, D=0    → L2: Y axis (X=0)
Line/Parlel, YAxis, D=80   → L3: vertical at X=80

These four lines form the construction grid.

Find Line-Line Intersections

Create points at the intersections of the reference lines.

Point/Intof, L0, L2  → P0 = (0, 0)    [origin]
Point/Intof, L0, L3  → P1 = (80, 0)   [bottom-right]
Point/Intof, L1, L2  → P2 = (0, 50)   [top-left]
Point/Intof, L1, L3  → P3 = (80, 50)  [top-right]

No selector needed — two non-parallel lines intersect at exactly one point.

Create Reference Circles

Create circles at key points to define curved geometry.

Circle/Centre, P0, R=60, 0  → C0: large circle at origin
Circle/Centre, P3, R=30, 0  → C1: circle at top-right

These circles will be used to locate additional points.

Find Line-Circle Intersections

Use the reference lines and circles to locate points on curved geometry.

Point/XLarge, Intof, L1, C0  → P4 = (33.166, 50)
   [intersection of Y=50 line and large circle, right side]

Point/YSmall, Intof, L3, C1  → P5 = (80, 20)
   [intersection of X=80 line and small circle, lower side]

Point/XSmall, Intof, L0, C0  → P6 = (-60, 0)
   [intersection of X axis and large circle, left side]

Always use the selector on the axis where the two solutions differ. For horizontal line L1, solutions differ in X — use XLarge/XSmall.

Find Circle-Circle Intersections

Find where the two circles intersect to locate additional construction points.

Point/YLarge, Intof, C0, C1  → P7 = upper intersection
Point/YSmall, Intof, C0, C1  → P8 = lower intersection

For C0 (centre 0,0 r=60) and C1 (centre 80,50 r=30):
  dist(centres) = sqrt(80^2 + 50^2) = 94.34mm
  94.34 < 60 + 30 = 90... circles may not intersect
  [adjust circle sizes or positions as needed for your geometry]

Build the Final Profile from Intersection Points

Use the located intersection points to create the final profile geometry.

Line/P0, P4   → L4: diagonal line from origin to (33.166,50)
Line/P4, P3   → L5: horizontal from (33.166,50) to (80,50)
Line/P3, P5   → L6: vertical from (80,50) to (80,20)
Line/P5, P1   → L7: diagonal from (80,20) to (80,0)

All coordinates computed analytically — no manual trigonometry.

Every point in this profile was located by intersection — no coordinate was calculated manually. This is the power of deterministic analytic geometry.

Tips & Best Practices

Use construction lines (amber dashed) during Preview to create temporary reference geometry.
Always specify the selector (XLarge/XSmall/YLarge/YSmall) for line-circle and circle-circle intersections.
Delete construction points with Edit → Clear All Points when the profile is complete.
Verify intersection points using Measure → Entity Info before building the final profile.
The same technique works for any profile — replace the reference lines and circles with your actual geometry.

Operators Used in This Tutorial

Point/Intof,L,C

Intersection-Based Construction

Define Reference Lines

Find Line-Line Intersections

Create Reference Circles

Find Line-Circle Intersections

Find Circle-Circle Intersections

Build the Final Profile from Intersection Points

Tips & Best Practices

Operators Used in This Tutorial

Line Circle Intersection

Circle Circle Intersection

Line Line Intersection

Related Tutorials

Rectangle with Filleted Corners

Tangent Circle Chain

Perpendicular Bisector Construction