CNC Offsets & Relief

2 min read•1 page

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.

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

Tool Radius R

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2 min read•1 page

CNC Inside Corner Relief (Dogbone Fillets): Rotary milling cutters can never produce a sharp interior $90^\circ$ 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.

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

Bit Radius R

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