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CNC Offsets & Relief

2 min read•1 page

Cutter Compensation Invariant:

1. G41 / G42 Offset: Shifts cutter centerline to the left or right of the programmed contour. 2. Corner Arc Insertion: Generates circular lead-in arcs around sharp exterior corners. 3. Self-Intersection Clipping: Prunes offset loops where narrow features are smaller than cutter diameter.
python
1import numpy as np
2

3def compute_tool_offset_path(polygon, tool_radius=0.25):
4    """
5    Computes CNC center toolpath using 2D Minkowski sum with tool disk of radius R.
6    At convex corners, inserts circular arc fillets centered at the vertex.
7    """
8    offset_segments = []
9    n = len(polygon)
10    
11    for i in range(n):
12        p1 = polygon[i]
13        p2 = polygon[(i + 1) % n]
14        edge = p2 - p1
15        normal = np.array([-edge[1], edge[0]]) / np.linalg.norm(edge)
16        
17        # Parallel offset segment
18        p1_off = p1 + normal * tool_radius
19        p2_off = p2 + normal * tool_radius
20        offset_segments.append((p1_off, p2_off))
21        
22    return offset_segments

CNC Cutter Radius Compensation (Minkowski Offset): In subtractive CNC milling, a machine toolpath cannot follow the nominal workpiece contour directly because the rotary cutting tool possesses a finite radius R. The spindle center must be offset by the Minkowski sum of the part boundary and the circular cutter disk, inserting radial arc fillets around convex corners.

Tool Radius R
0.30
2 min read•1 page

Dogbone Clearance Geometry:

1. Rotary Tool Constraint: Cylindrical endmills leave an uncut corner fillet of radius R. 2. Bisector Overcut: Extends cutter center along the diagonal angle bisector by distance R. 3. Zero Interference: Allows square rectangular parts to seat fully flush against pocket walls.
python
1import math
2

3def compute_dogbone_relief(corner_vertex, in_dir, out_dir, tool_radius=0.25):
4    """
5    Computes CNC dogbone corner relief center.
6    A cylindrical bit cannot cut an internal 90-degree corner.
7    To clear the square mating tab, the tool must overcut diagonally into the corner.
8    Distance from corner: d = tool_radius.
9    """
10    # Bisector direction into material
11    bisector = -(in_dir + out_dir)
12    bisector = bisector / math.sqrt(bisector[0]**2 + bisector[1]**2)
13    
14    relief_center = corner_vertex + bisector * tool_radius
15    return relief_center

CNC Inside Corner Relief (Dogbone Fillets): Rotary milling cutters can never produce a sharp interior 90° corner; the bit radius leaves a residual circular corner fillet. When assembling friction-fit slot joinery (e.g. mortise and tenon in plywood or aluminum), mating square corners jam. A dogbone fillet deliberately overcuts into the corner along its angle bisector, clearing the square vertex.

Bit Radius R
0.30

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